Difference between revisions of "Mathematical formula"

Latest revision as of 09:18, 13 July 2024

1 Addition
2 Subtraction
3 Doubling
4 Halving
5 Multiplication
6 Squaring
7 Cubing
8 Division
- 8.1 Smaller by greater
- 8.2 Greater by smaller
9 Fractions
10 Rule of Three
11 Roots
12 Multiplication of Algebraic Species
13 Linear Equation
14 Quadratic Equation
15 Cubic Equation
16 Biquadratic Equation
17 indeterminate equation
18 Euclid

Addition

\scriptstyle44+55

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\scriptstyle99+11

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\scriptstyle1+999

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\scriptstyle142+968

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\scriptstyle335+453

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\scriptstyle432+354

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\scriptstyle875+798

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\scriptstyle1098+9067

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\scriptstyle1215+2322

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\scriptstyle3372+3392

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\scriptstyle3372+9892

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\scriptstyle3465+5643

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\scriptstyle4373+2389

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\scriptstyle5243+8962

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\scriptstyle6503+7020

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\scriptstyle9208+3801

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\scriptstyle200+240+403

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\scriptstyle203+402+809

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\scriptstyle223+342+422

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\scriptstyle432+245+321

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\scriptstyle723+865+957

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\scriptstyle699+7156+867

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\scriptstyle209+3089+7639

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\scriptstyle205003+390005+625002

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\scriptstyle4204+212+12+30

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\scriptstyle1215+2322+9657+8563

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\scriptstyle1101+3931+9755+57052+28067

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\scriptstyle56''+\left(20'+40''+30'''\right)+\left(46'+27''+55'''\right)

Subtraction

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\scriptstyle39-20

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\scriptstyle46-24

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\scriptstyle107-59

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\scriptstyle234-122

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\scriptstyle245-123

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\scriptstyle246-135

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\scriptstyle304-253

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\scriptstyle349-207

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\scriptstyle468-382

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\scriptstyle654-321

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\scriptstyle956-867

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\scriptstyle1000-999

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\scriptstyle2000-1999

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\scriptstyle2040-1403

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\scriptstyle3333-2445

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\scriptstyle3735-425

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\scriptstyle4282-2432

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\scriptstyle4288-3268

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\scriptstyle4321-3456

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\scriptstyle5060-2304

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\scriptstyle5083-92

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\scriptstyle5114-4225

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\scriptstyle5432-2379

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\scriptstyle5787-3456

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\scriptstyle6475-2343

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\scriptstyle9385-5496

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\scriptstyle21333-221

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\scriptstyle100000-1

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\scriptstyle76540304-40438

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\scriptstyle90020235-63295223

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\scriptstyle\left(31080+46''+35'''+47^{iv}+53^{vi}\right)-\left(206+50'+37'''\right)

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Doubling

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\scriptstyle2\times5372

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\scriptstyle2\times795347

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\scriptstyle2\times974537

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\scriptstyle2\times95386349

Halving

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\scriptstyle54376\div2

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\scriptstyle262144\div2

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\scriptstyle698536\div2

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\scriptstyle1048876\div2

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\scriptstyle9835834\div2

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\scriptstyle9876374\div2

Multiplication

units by units

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\scriptstyle4\times3

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\scriptstyle4\times4

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\scriptstyle5\times6

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\scriptstyle6\times6

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\scriptstyle6\times8

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\scriptstyle7\times8

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\scriptstyle7\times9

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\scriptstyle8\times9

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\scriptstyle9\times8

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\scriptstyle9\times9

units by tens

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\scriptstyle5\times70

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\scriptstyle20\times5

units by hundreds

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\scriptstyle5\times300

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\scriptstyle9\times200

tens by tens

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\scriptstyle20\times30

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\scriptstyle30\times20

tens by hundreds

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\scriptstyle20\times200

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\scriptstyle30\times200

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\scriptstyle40\times600

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\scriptstyle80\times500

tens by thousands

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\scriptstyle50\times7000

hundreds by hundreds

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\scriptstyle200\times700

hundreds by thousands

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\scriptstyle600\times4000

units by tens and hundreds

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\scriptstyle5\times220

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\scriptstyle5\times320

tens by tens and hundreds

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\scriptstyle20\times230

tens and hundreds by tens and hundreds

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\scriptstyle240\times170

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\scriptstyle470\times580

tens by units and tens

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\scriptstyle10\times12

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\scriptstyle20\times35

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\scriptstyle33\times10

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\scriptstyle45\times20

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\scriptstyle15\times40

units by units and tens

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\scriptstyle5\times27

units and tens by units and tens

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\scriptstyle13\times28

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\scriptstyle13\times14

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\scriptstyle13\times16

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\scriptstyle18\times15

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\scriptstyle23\times17

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\scriptstyle23\times23

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\scriptstyle23\times24

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\scriptstyle24\times25

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\scriptstyle24\times26

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\scriptstyle25\times25

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\scriptstyle25\times28

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\scriptstyle25\times35

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\scriptstyle25\times43

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\scriptstyle27\times32

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\scriptstyle29\times31

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\scriptstyle34\times57

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\scriptstyle43\times57

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\scriptstyle54\times45

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\scriptstyle54\times66

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\scriptstyle66\times54

tens and hundreds by units and tens

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\scriptstyle250\times350

units by units, tens and hundreds

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\scriptstyle869\times6

units and tens by units, tens and hundreds

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\scriptstyle869\times46

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\scriptstyle234\times24

units, tens and hundreds by units, tens and hundreds

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\scriptstyle105\times224

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\scriptstyle122\times232

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\scriptstyle123\times456

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\scriptstyle125\times125

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\scriptstyle127\times355

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\scriptstyle212\times132

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\scriptstyle222\times333

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\scriptstyle230\times324

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\scriptstyle231\times342

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\scriptstyle236\times135

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\scriptstyle240\times368

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\scriptstyle242\times144

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\scriptstyle246\times140

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\scriptstyle321\times654

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\scriptstyle348\times235

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\scriptstyle352\times343

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\scriptstyle462\times323

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\scriptstyle464\times464

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\scriptstyle403\times230

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\scriptstyle755\times653

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\scriptstyle902\times246

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\scriptstyle15\times1080

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\scriptstyle209\times3030

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\scriptstyle253\times1335

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\scriptstyle6845\times327

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\scriptstyle1234\times4321

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\scriptstyle5432\times5323

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\scriptstyle54321\times54321

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\scriptstyle56023\times70235

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\scriptstyle7000030\times180640

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\scriptstyle9007500\times5400920

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\scriptstyle\left(355\times296\right)+\left(447\times178\right)+\left(396\times539\right)

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\scriptstyle\left(4-2\right)\times\left(8-3\right)

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\scriptstyle\left(7-2\right)\times\left(8-4\right)

Word Problems

[Expand]Word Problems

Permutations

[Expand]2!

[Expand]3!

[Expand]4!

[Expand]5!

[Expand]6!

[Expand]7!

[Expand]8!

Squaring

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\scriptstyle3^2

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\scriptstyle7^2

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\scriptstyle8^2

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\scriptstyle9^2

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\scriptstyle10^2

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\scriptstyle11^2

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\scriptstyle12^2

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\scriptstyle13^2

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\scriptstyle14^2

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\scriptstyle15^2

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\scriptstyle22^2

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\scriptstyle23^2

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\scriptstyle24^2

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\scriptstyle25^2

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\scriptstyle26^2

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\scriptstyle27^2

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\scriptstyle30^2

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\scriptstyle33^2

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\scriptstyle47^2

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\scriptstyle50^2

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\scriptstyle60^2

Cubing

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\scriptstyle3^3

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\scriptstyle4^3

Division

Smaller by greater

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\scriptstyle4\div12

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\scriptstyle5\div17

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\scriptstyle4\div50

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\scriptstyle6\div50

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\scriptstyle7\div40

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\scriptstyle11\div18

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\scriptstyle11\div19

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\scriptstyle23\div36

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\scriptstyle36\div48

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\scriptstyle37\div48

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\scriptstyle15\div100

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\scriptstyle70\div100

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\scriptstyle38\div101

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\scriptstyle73\div240

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\scriptstyle2447235\div50335084800

Greater by smaller

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\scriptstyle40\div7

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\scriptstyle100\div9

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\scriptstyle144\div8

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\scriptstyle321\div9

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\scriptstyle100\div12

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\scriptstyle100\div15

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\scriptstyle125\div11

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\scriptstyle218\div7

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\scriptstyle200\div50

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\scriptstyle245\div34

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\scriptstyle654\div70

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\scriptstyle891\div40

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\scriptstyle901\div32

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\scriptstyle140\div100

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\scriptstyle823\div278

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\scriptstyle1000\div72

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\scriptstyle4032\div30

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\scriptstyle4215\div14

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\scriptstyle4321\div23

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\scriptstyle5086\div19

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\scriptstyle9000\div70

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\scriptstyle9876\div12

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\scriptstyle5214\div108

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\scriptstyle8213\div353

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\scriptstyle9381\div296

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\scriptstyle20000\div90

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\scriptstyle11350\div110

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\scriptstyle12345\div234

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\scriptstyle20503\div304

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\scriptstyle70213\div136

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\scriptstyle83521\div903

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\scriptstyle54093\div2945

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\scriptstyle68921\div7053

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\scriptstyle104034\div114

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\scriptstyle204612\div289

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\scriptstyle583696\div764

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\scriptstyle583696\div1080

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\scriptstyle680402\div2009

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\scriptstyle777777777\div9999

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\scriptstyle987654321\div9437

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\scriptstyle4380408998\div46079

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\scriptstyle3123740520\div216

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Fractions

fraction of fraction

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\scriptstyle\frac{1}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}

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\scriptstyle\frac{1}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}\sdot\frac{1}{6}

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\scriptstyle\frac{1}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}\sdot\frac{1}{6}\sdot\frac{1}{7}

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\scriptstyle\frac{1}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}\sdot\frac{1}{6}\sdot\frac{1}{7}\sdot\frac{1}{8}

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\scriptstyle\frac{1}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}\sdot\frac{1}{6}\sdot\frac{1}{7}\sdot\frac{1}{8}\sdot\frac{1}{9}

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\scriptstyle\frac{1}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}\sdot\frac{1}{6}\sdot\frac{1}{7}\sdot\frac{1}{8}\sdot\frac{1}{9}\sdot\frac{1}{10}

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\scriptstyle\frac{1}{3}\sdot\frac{1}{4}

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\scriptstyle\frac{1}{3}\sdot\frac{1}{5}

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\scriptstyle\frac{1}{5}\sdot\frac{1}{9}

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\scriptstyle\frac{3}{8}\sdot\frac{2}{5}

division of fractions

fractions by fractions

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\scriptstyle\frac{1}{3}\div\frac{3}{4}

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\scriptstyle\frac{2}{3}\div\frac{3}{4}

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\scriptstyle\frac{2}{3}\div\frac{2}{7}

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\scriptstyle\frac{3}{4}\div\frac{2}{3}

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\scriptstyle\frac{3}{4}\div\frac{2}{5}

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\scriptstyle\frac{3}{5}\div\frac{2}{3}

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\scriptstyle\frac{6}{7}\div\frac{2}{5}

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\scriptstyle\frac{5}{8}\div\frac{1}{4}

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\scriptstyle\frac{3}{8}\div\frac{2}{5}

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\scriptstyle\frac{6}{8}\div\frac{2}{4}

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\scriptstyle\frac{3}{5}\div\frac{7}{9}

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\scriptstyle\frac{7}{9}\div\frac{2}{7}

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\scriptstyle\frac{2}{13}\div\frac{9}{10}

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\scriptstyle\frac{5}{11}\div\frac{9}{17}

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\scriptstyle\frac{2}{3}\div\left[\frac{9}{10}+\left(\frac{5}{8}\sdot\frac{1}{10}\right)\right]

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\scriptstyle\frac{7}{9}\div\left(\frac{3}{4}+\frac{7}{8}\right)

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\scriptstyle\left(\frac{1}{3}+\frac{1}{7}\right)\div\frac{10}{11}

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\scriptstyle\left[\frac{2}{8}+\left(\frac{1}{2}\sdot\frac{1}{8}\right)\right]\div\frac{9}{10}

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\scriptstyle\left[\frac{4}{9}+\left(\frac{1}{2}\sdot\frac{1}{9}\right)\right]\div\left[\frac{7}{8}+\left(\frac{1}{3}\sdot\frac{1}{8}\right)\right]

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\scriptstyle\left[\frac{3}{4}+\left(\frac{2}{3}\sdot\frac{1}{4}\right)\right]\div\left[\left[\frac{4}{9}+\left(\frac{5}{6}\sdot\frac{1}{9}\right)\right]\sdot\frac{2}{3}\right]

fractions by integers

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\scriptstyle\frac{1}{3}\div4

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\scriptstyle\frac{1}{4}\div2

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\scriptstyle\frac{3}{8}\div2

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\scriptstyle\frac{5}{6}\div8

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\scriptstyle\frac{5}{11}\div7

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\scriptstyle\frac{7}{9}\div17

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\scriptstyle\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\div5

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\scriptstyle\left[\frac{10}{11}+\left(\frac{3}{5}\sdot\frac{1}{11}\right)\right]\div63

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\scriptstyle\left(\frac{3}{5}+\frac{7}{8}\right)\div6

fractions by integers and fractions

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\scriptstyle\frac{2}{3}\div\left(4+\frac{1}{2}\right)

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\scriptstyle\frac{6}{7}\div\left(5+\frac{1}{2}\right)

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\scriptstyle\frac{10}{11}\div\left(6+\frac{2}{3}\right)

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\scriptstyle\frac{4}{5}\div\left[3+\frac{6}{7}+\left(\frac{1}{2}\sdot\frac{1}{7}\right)\right]

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\scriptstyle\frac{6}{13}\div\left(4+\frac{1}{3}+\frac{6}{11}\right)

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\scriptstyle\left[\frac{7}{8}+\left(\frac{1}{3}\sdot\frac{1}{8}\right)\right]\div\left(3+\frac{1}{2}\right)

fractions of fractions by fractions

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\scriptstyle\left(\frac{5}{9}\sdot\frac{1}{10}\right)\div\frac{3}{5}

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\scriptstyle\left(\frac{3}{4}\sdot\frac{1}{5}\right)\div\left[\frac{3}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]

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\scriptstyle\left(\frac{1}{2}\sdot\frac{1}{7}\right)\div\left(\frac{3}{4}+\frac{3}{5}\right)

fractions of fractions by integers

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\scriptstyle\left(\frac{2}{3}\sdot\frac{1}{5}\right)\div7

fractions of fractions by integers and fractions

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\scriptstyle\left(\frac{7}{8}\sdot\frac{1}{9}\right)\div\left(3+\frac{1}{2}\right)

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\scriptstyle\left(\frac{9}{10}\sdot\frac{1}{13}\right)\div\left[6+\frac{3}{8}+\left(\frac{1}{3}\sdot\frac{1}{8}\right)\right]

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\scriptstyle\left(\frac{3}{5}\sdot\frac{1}{7}\right)\div\left(1+\frac{5}{6}+\frac{2}{3}\right)

integers by fractions

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\scriptstyle4\div\frac{3}{4}

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\scriptstyle12\div\frac{4}{9}

integers by integers and fractions

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\scriptstyle1\div\left(3+\frac{1}{3}\right)

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\scriptstyle1\div\left(6+\frac{1}{4}\right)

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\scriptstyle2\div\left(5+\frac{1}{2}\right)

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\scriptstyle50\div\left(2+\frac{1}{2}\right)

integers and fractions by fractions

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\scriptstyle\left(4+\frac{1}{3}\right)\div\frac{5}{6}

integers and fractions by integers

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\scriptstyle\left(3+\frac{1}{3}\right)\div4

integers and fractions by integers and fractions

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\scriptstyle\left(4+\frac{1}{2}\right)\div\left(2+\frac{1}{4}\right)

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\scriptstyle\left(4+\frac{2}{3}\right)\div\left(2+\frac{2}{5}\right)

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\scriptstyle\left(6+\frac{1}{4}\right)\div\left(8+\frac{1}{3}\right)

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\scriptstyle\left(9+\frac{1}{2}\right)\div\left(3+\frac{4}{5}\right)

Combined Division

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\scriptstyle\left(\frac{2}{3}+\frac{3}{5}\right)\div\left(\frac{9}{10}+\frac{10}{11}\right)

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\scriptstyle\left(\frac{2}{3}+\frac{3}{4}\right)\div\left(\frac{2}{7}+\frac{1}{6}\right)

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\scriptstyle\left(\frac{3}{4}\div\frac{2}{3}\right)+\left(\frac{2}{7}\div\frac{1}{6}\right)

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\scriptstyle\left(\frac{2}{3}\times\frac{3}{4}\right)\div\left(\frac{2}{7}\times\frac{1}{6}\right)

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\scriptstyle\left(\frac{3}{4}\div\frac{2}{3}\right)\times\left(\frac{2}{7}\div\frac{1}{6}\right)

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\scriptstyle\left(\frac{3}{4}-\frac{2}{3}\right)\div\left(\frac{2}{7}-\frac{1}{6}\right)

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\scriptstyle\left(\frac{2}{7}\div\frac{1}{6}\right)-\left(\frac{3}{4}\div\frac{2}{3}\right)

multiplication of fractions

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\scriptstyle\frac{2}{3}\times\frac{5}{6}\times\frac{4}{7}

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\scriptstyle\frac{2}{5}\times\frac{1}{10}\times\frac{2}{3}

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\scriptstyle\left(\frac{5}{7}\sdot\frac{1}{3}\right)\times\frac{7}{8}

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\scriptstyle\frac{2}{3}\times\left(\frac{3}{4}\sdot\frac{1}{3}\right)

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\scriptstyle\frac{2}{3}\times\left(\frac{4}{5}\sdot\frac{1}{9}\right)

[Expand]

\scriptstyle\left(\frac{2}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}\right)\times\left(\frac{6}{7}\sdot\frac{1}{8}\right)

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\scriptstyle\left(\frac{3}{5}\sdot\frac{3}{7}\sdot\frac{2}{9}\sdot{\color{red}{\frac{1}{10}}}\right)\times\left(\frac{2}{3}\sdot\frac{6}{7}\sdot\frac{4}{5}\sdot\frac{7}{9}\right)

[Expand]

\scriptstyle\left(\frac{2}{3}\sdot\frac{3}{4}\sdot\frac{4}{5}\sdot\frac{5}{6}\sdot\frac{6}{7}\sdot\frac{7}{8}\sdot\frac{8}{9}\sdot\frac{9}{10}\sdot\frac{10}{11}\right)\times\left(\frac{2}{3}\sdot\frac{3}{4}\sdot\frac{4}{5}\sdot\frac{5}{6}\sdot\frac{6}{7}\sdot\frac{7}{8}\sdot\frac{8}{9}\sdot\frac{9}{10}\sdot\frac{10}{11}\right)

[Expand]

\scriptstyle\left[\frac{1}{5}+\left(\frac{1}{2}\sdot\frac{1}{5}\right)\right]\times12

[Expand]

\scriptstyle\left(\frac{1}{3}\sdot\frac{1}{7}\right)\times25

[Expand]

\scriptstyle\left(\frac{3}{4}+\frac{4}{5}\right)\times15

[Expand]

\scriptstyle2\times\left(3+\frac{1}{4}\right)

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\scriptstyle\left(4+\frac{1}{2}\right)\times\frac{2}{3}

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\scriptstyle\left(4+\frac{1}{3}\right)\times6

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\scriptstyle\left(3+\frac{4}{5}\right)\times\frac{3}{5}

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\scriptstyle\left(4+\frac{2}{5}\right)\times\frac{3}{4}

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\scriptstyle\frac{3}{5}\times\left(9+\frac{1}{3}\right)

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\scriptstyle\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\times\left(6+\frac{3}{5}\right)

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\scriptstyle\left(\frac{1}{2}\sdot\frac{1}{7}\right)\times\left(10+\frac{3}{5}\right)

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\scriptstyle\left(\frac{3}{4}+\frac{4}{5}\right)\times\left(5+\frac{5}{6}\right)

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\scriptstyle\frac{3}{4}\times\left[5+\frac{7}{8}+\left(\frac{1}{6}\sdot\frac{1}{8}\right)\right]

[Expand]

\scriptstyle\frac{5}{6}\times\left[8+\left(\frac{5}{7}\sdot\frac{1}{9}\right)\right]

[Expand]

\scriptstyle\frac{2}{3}\times\left[9+\left(\frac{3}{7}\sdot\frac{4}{5}\right)\right]

[Expand]

\scriptstyle\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\times\left[8+\frac{5}{7}+\left(\frac{1}{5}\sdot\frac{1}{7}\right)\right]

[Expand]

\scriptstyle\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\times\left[9+\left(\frac{1}{6}\sdot\frac{1}{7}\right)\right]

[Expand]

\scriptstyle\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\times\left[8+\left(\frac{9}{10}\sdot\frac{10}{11}\right)\right]

[Expand]

\scriptstyle\left(\frac{2}{3}\sdot\frac{1}{7}\right)\times\left[6+\frac{9}{10}+\left(\frac{4}{5}\sdot\frac{1}{10}\right)\right]

[Expand]

\scriptstyle\left(\frac{3}{4}\sdot\frac{1}{9}\right)\times\left[8+\left(\frac{5}{6}\sdot\frac{1}{7}\right)\right]

[Expand]

\scriptstyle\left(\frac{5}{6}\sdot\frac{1}{8}\right)\times\left(6+\frac{3}{4}+\frac{4}{5}\right)

[Expand]

\scriptstyle\left(\frac{3}{4}+\frac{4}{5}\right)\times\left[6+\frac{7}{8}+\left(\frac{1}{6}\sdot\frac{1}{8}\right)\right]

[Expand]

\scriptstyle\left(\frac{3}{4}+\frac{4}{5}\right)\times\left[6+\left(\frac{1}{7}\sdot\frac{1}{8}\right)\right]

[Expand]

\scriptstyle\left(\frac{1}{2}+\frac{2}{3}\right)\times\left(5+\frac{6}{7}+\frac{8}{9}\right)

[Expand]

\scriptstyle\left[2+\frac{3}{5}+\left(\frac{1}{2}\sdot\frac{1}{5}\right)\right]\times\left[4+\frac{7}{9}+\left(\frac{6}{8}\sdot\frac{1}{9}\right)\right]

[Expand]

\scriptstyle\left(6+\frac{2}{3}+\frac{3}{5}\right)\times\left(9+\frac{5}{6}+\frac{7}{8}\right)

[Expand]

\scriptstyle\left[2+\frac{3}{4}+\frac{4}{5}+\left(\frac{1}{2}\sdot\frac{1}{5}\right)\right]\times\left[6+\frac{6}{7}+\frac{9}{10}+\left(\frac{3}{8}\sdot\frac{1}{10}\right)\right]

[Expand]

\scriptstyle\left[2+\frac{1}{3}+\frac{1}{5}+\left(\frac{1}{7}\sdot\frac{1}{8}\right)\right]\times\left[3+\frac{1}{9}+\frac{1}{10}+\left(\frac{1}{6}\sdot\frac{1}{11}\right)\right]

[Expand]

\scriptstyle\left(2+\frac{1}{3}+\frac{1}{5}+\frac{1}{6}\right)\times\left(4+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}\right)

[Expand]

\scriptstyle\left[\frac{6}{7}+\left(\frac{1}{3}\sdot\frac{1}{7}\right)\right]\times\frac{8}{9}

[Expand]

\scriptstyle\left[\frac{1}{11}+\left(\frac{1}{2}\sdot\frac{1}{11}\right)\right]\times\frac{12}{13}

[Expand]

\scriptstyle\left(\frac{2}{3}+\frac{4}{7}\right)\times\frac{10}{11}

[Expand]

\scriptstyle\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\times\left[\frac{9}{10}+\left(\frac{7}{8}\sdot\frac{1}{10}\right)\right]

[Expand]

\scriptstyle\left(\frac{3}{4}+\frac{4}{5}\right)\times\left[\frac{10}{11}+\left(\frac{8}{9}\sdot\frac{1}{11}\right)\right]

[Expand]

\scriptstyle\left(\frac{1}{3}+\frac{3}{5}\right)\times\left(\frac{1}{7}+\frac{2}{9}\right)

[Expand]

\scriptstyle\left[\frac{2}{3}+\frac{5}{7}+\left(\frac{1}{6}\sdot\frac{1}{7}\right)\right]\times\left[\frac{4}{5}+\frac{9}{10}+\left(\frac{8}{9}\sdot\frac{1}{10}\right)\right]

[Expand]

\scriptstyle\left[\frac{3}{4}+\frac{4}{5}+\left(\frac{1}{2}\sdot\frac{1}{7}\right)\right]\times\left[\frac{7}{8}+\frac{8}{9}+\left(\frac{1}{3}\sdot\frac{1}{10}\right)\right]

[Expand]

\scriptstyle\left(\frac{2}{3}+\frac{3}{5}+\frac{4}{6}\right)\times\left(\frac{5}{7}+\frac{7}{8}+\frac{9}{11}\right)

[Expand]

\scriptstyle\left(\frac{3}{4}\sdot5\right)\times\left(\frac{5}{6}\sdot7\right)

[Expand]

\scriptstyle\left[\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\sdot8\right]\times\left[\left[\frac{8}{9}+\left(\frac{1}{5}\sdot\frac{1}{9}\right)\right]\sdot12\right]

[Expand]

\scriptstyle\left[\left(\frac{2}{3}+\frac{3}{5}\right)\sdot8\right]\times\left[\left(\frac{6}{7}+\frac{5}{9}\right)\sdot4\right]

[Expand]

\scriptstyle\left[\frac{3}{4}\sdot\left[5+\frac{6}{7}+\left(\frac{1}{6}\sdot\frac{1}{7}\right)\right]\right]\times\left[\frac{7}{8}\sdot\left[3+\frac{3}{10}+\left(\frac{1}{9}\sdot\frac{1}{10}\right)\right]\right]

[Expand]

\scriptstyle\left[\frac{1}{2}\sdot\left(2+\frac{1}{5}+\frac{1}{6}\right)\right]\times\left[\frac{1}{7}\sdot\left(3+\frac{1}{8}+\frac{1}{9}\right)\right]

[Expand]

\scriptstyle\left[\frac{6}{7}\sdot\left[5+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\right]\times\left[\frac{7}{8}\sdot\left[3+\left(\frac{3}{5}\sdot\frac{1}{9}\right)\right]\right]

[Expand]

\scriptstyle\left[\left(\frac{2}{3}\sdot\frac{1}{5}\right)\sdot\left(8+\frac{1}{6}\right)\right]\times\left[\left(\frac{5}{7}\sdot\frac{1}{8}\right)\sdot\left(6+\frac{1}{9}\right)\right]

[Expand]

\scriptstyle\left[\left(\frac{3}{4}\sdot\frac{1}{5}\right)\sdot\left[6+\frac{3}{11}+\left(\frac{5}{6}\sdot\frac{1}{11}\right)\right]\right]\times\left[\left(\frac{5}{7}\sdot\frac{1}{9}\right)\sdot\left[12+\frac{6}{13}+\left(\frac{3}{8}\sdot\frac{1}{13}\right)\right]\right]

[Expand]

\scriptstyle\left[\left(\frac{2}{3}\sdot\frac{1}{5}\right)\sdot\left(8+\frac{3}{6}+\frac{4}{7}\right)\right]\times\left[\left(\frac{4}{9}\sdot\frac{1}{10}\right)\sdot\left(18+\frac{6}{11}+\frac{5}{8}\right)\right]

[Expand]

\scriptstyle\left[\left(\frac{2}{3}+\frac{3}{5}\right)\sdot\left(9+\frac{5}{6}\right)\right]\times\left[\left(\frac{3}{7}+\frac{7}{8}\right)\sdot\left(12+\frac{7}{9}\right)\right]

[Expand]

\scriptstyle\left[\left(\frac{2}{3}+\frac{3}{5}\right)\sdot\left(7+\frac{3}{4}+\frac{3}{5}\right)\right]\times\left[\left(\frac{5}{6}+\frac{5}{8}\right)\sdot\left(4+\frac{5}{6}+\frac{9}{10}\right)\right]

[Expand]

\scriptstyle\left[\left[\frac{3}{5}+\left(\frac{1}{2}\sdot\frac{1}{5}\right)\right]\sdot\left[4+\frac{1}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\right]\times\left[\left[\frac{6}{7}+\left(\frac{2}{5}\sdot\frac{1}{7}\right)\right]\sdot\left[3+\frac{5}{11}+\left(\frac{5}{8}\sdot\frac{1}{11}\right)\right]\right]

[Expand]

\scriptstyle\left[\left(\frac{2}{3}\sdot\frac{1}{5}\right)\sdot\left[8+\left(\frac{1}{4}\sdot\frac{1}{6}\right)\right]\right]\times\left[\left(\frac{3}{7}\sdot\frac{1}{8}\right)\sdot\left[12+\left(\frac{1}{2}\sdot\frac{1}{13}\right)\right]\right]

[Expand]

\scriptstyle\left[\left(\frac{3}{5}\sdot5\right)+\left(\frac{5}{6}\sdot7\right)\right]\times\left[\left(\frac{2}{3}\sdot4\right)+\left(\frac{1}{2}\sdot5\right)\right]

[Expand]

\scriptstyle\left[\left(\frac{3}{5}\sdot3\right)+\left[\frac{3}{4}\sdot\left(2+\frac{2}{3}\right)\right]\right]\times\left[\left(\frac{4}{5}\sdot2\right)+\left[\frac{5}{7}\sdot\left(3+\frac{1}{2}\right)\right]\right]

[Expand]

\scriptstyle\left[\left(3+\frac{1}{2}\right)+\left(5+\frac{1}{3}\right)\right]\times\left[\left(4+\frac{3}{4}\right)+\left(6+\frac{4}{5}\right)\right]

[Expand]

\scriptstyle\left[2+\left[\left(\frac{2}{3}+\frac{3}{4}\right)\sdot3\right]+\left[\left(\frac{4}{5}+\frac{5}{6}\right)\sdot4\right]\right]\times\left[1+\left[\left(\frac{3}{5}+\frac{1}{2}\right)\sdot2\right]+\left[\left(\frac{3}{11}+\frac{{\color{red}{9}}}{10}\right)\sdot3\right]\right]

[Expand]

\scriptstyle\left[2+\left[\left(\frac{1}{2}\sdot\frac{1}{5}\right)\sdot3\right]+\left[\left(\frac{2}{3}\sdot\frac{1}{6}\right)\sdot5\right]\right]\times\left[3+\left[\left(\frac{1}{4}\sdot\frac{1}{7}\right)\sdot4\right]+\left[\left(\frac{5}{6}\sdot\frac{1}{8}\right)\sdot2\right]\right]

[Expand]

\scriptstyle\left[\left[\frac{3}{4}\sdot\left(5+\frac{1}{2}\right)\right]+\left[\frac{5}{6}\sdot\left(3+\frac{2}{5}\right)\right]\right]\times\left[\left[\frac{2}{3}\sdot\left(4+\frac{1}{7}\right)\right]+\left[\frac{3}{8}\sdot\left(2+\frac{3}{11}\right)\right]\right]

[Expand]

\scriptstyle\left[\left(\frac{3}{4}\sdot\frac{4}{5}\right)+\left(\frac{5}{6}\sdot\frac{6}{7}\right)+\left(\frac{7}{8}\sdot\frac{8}{9}\right)\right]\times\left[\left(\frac{2}{3}\sdot\frac{3}{4}\right)+\left(\frac{4}{5}\sdot\frac{5}{6}\right)+\left(\frac{6}{7}\sdot\frac{7}{8}\right)\right]

[Expand]

\scriptstyle\left[\left[\left(\frac{2}{5}\sdot\frac{5}{6}\right)+\frac{1}{2}\right]\sdot3\right]\times\left[\left[\left(\frac{2}{3}\sdot\frac{6}{7}\right)+\frac{3}{5}\right]\sdot10\right]

[Expand]

\scriptstyle\left[\left(\frac{2}{3}\sdot\frac{6}{9}\right)\sdot\left(6+\frac{3}{4}\right)\right]\times\left[\left(\frac{3}{4}\sdot\frac{8}{9}\right)\sdot\left(2+\frac{2}{5}\right)\right]

[Expand]

\scriptstyle\left[\left(\frac{3}{5}\sdot\frac{4}{7}\right)\sdot\left[3+\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\right]\times\left[\left(\frac{2}{3}\sdot\frac{5}{6}\right)\sdot\left[6+\frac{7}{8}+\left(\frac{2}{3}\sdot\frac{1}{8}\right)\right]\right]

[Expand]

\scriptstyle\left[3+\left(\frac{3}{5}\sdot\frac{5}{6}\right)\right]\times\left[4+\left(\frac{2}{7}\sdot\frac{3}{8}\right)\right]

[Expand]

\scriptstyle\left[2+\left(\frac{3}{7}\sdot\frac{3}{5}\right)+\frac{2}{3}\right]\times\left[3+\left(\frac{3}{4}\sdot\frac{5}{6}\right)+\frac{1}{2}\right]

[Expand]

\scriptstyle\left[\left(\frac{3}{4}\sdot\frac{3}{5}\right)\sdot\left[2+\left(\frac{3}{{\color{red}{7}}}\sdot\frac{5}{6}\right)\right]\right]\times\left[\left(\frac{1}{2}\sdot\frac{3}{4}\right)\sdot\left[3+\left(\frac{2}{5}\sdot\frac{4}{9}\right)\right]\right]

[Expand]

\scriptstyle\left(1+\frac{1}{2}\right)\times\left(1+\frac{1}{3}\right)\times\left(1+\frac{1}{4}\right)\times\left(1+\frac{1}{5}\right)\times\left(1+\frac{1}{6}\right)\times\left(1+\frac{1}{7}\right)\times\left(1+\frac{1}{8}\right)\times\left(1+\frac{1}{9}\right)\times\left(1+\frac{1}{10}\right)

[Expand]

\scriptstyle\left(3+\frac{1}{4}\right)\times\left[5+\left(\frac{1}{6}\sdot\frac{1}{7}\right)\right]

[Expand]

\scriptstyle\left[\frac{3}{4}\sdot\left(3+\frac{1}{3}\right)\right]\times\left(\frac{6}{7}\sdot5\right)

[Expand]

\scriptstyle\left[\left[\frac{4}{7}\sdot\left(\frac{5}{9}\sdot\frac{1}{8}\right)\right]\right]\times\left[\left(\frac{3}{5}\sdot\frac{1}{9}\right)\sdot\left(\frac{2}{3}\sdot5\right)\right]

multiplication of integer and fraction by integer and fraction

[Expand]

\scriptstyle\left(2+\frac{1}{2}\right)\times\left(3+\frac{1}{3}\right)

[Expand]

\scriptstyle\left(2+\frac{1}{2}\right)\times\left(4+\frac{3}{4}\right)

[Expand]

\scriptstyle\left(3+\frac{1}{3}\right)\times\left(4+\frac{1}{4}\right)

[Expand]

\scriptstyle\left(4+\frac{2}{5}\right)\times\left(5+\frac{3}{5}\right)

[Expand]

\scriptstyle\left(3+\frac{1}{7}\right)\times\left(5+\frac{1}{8}\right)

[Expand]

\scriptstyle\left(3+\frac{4}{5}\right)\times\left(2+\frac{3}{5}\right)

[Expand]

\scriptstyle\left(3+\frac{2}{5}\right)\times\left(2+\frac{4}{7}\right)

[Expand]

\scriptstyle\left(2+\frac{3}{4}\right)\times\left(3+\frac{2}{4}\right)

[Expand]

\scriptstyle\left(5+\frac{2}{3}\right)\times\left(2+\frac{3}{6}\right)

[Expand]

\scriptstyle\left(3+\frac{1}{5}\right)\times\left(7+\frac{3}{8}\right)

[Expand]

\scriptstyle\left(1+\frac{1}{4}\right)\times\left(1+\frac{1}{5}\right)

[Expand]

\scriptstyle\left(1+\frac{5}{7}\right)\times\left(1+\frac{4}{5}\right)

multiplication of fraction by fraction

[Expand]

\scriptstyle\frac{1}{3}\times\frac{1}{3}

[Expand]

\scriptstyle\frac{1}{4}\times\frac{1}{4}

[Expand]

\scriptstyle\frac{1}{5}\times\frac{1}{5}

[Expand]

\scriptstyle\frac{1}{4}\times\frac{1}{3}

[Expand]

\scriptstyle\frac{3}{4}\times\frac{3}{4}

[Expand]

\scriptstyle\frac{2}{4}\times\frac{3}{4}

[Expand]

\scriptstyle\frac{4}{5}\times\frac{4}{5}

[Expand]

\scriptstyle\frac{3}{4}\times\frac{4}{5}

[Expand]

\scriptstyle\frac{2}{3}\times\frac{2}{3}

[Expand]

\scriptstyle\frac{3}{5}\times\frac{4}{5}

[Expand]

\scriptstyle\frac{2}{3}\times\frac{2}{5}

[Expand]

\scriptstyle\frac{2}{3}\times\frac{4}{5}

[Expand]

\scriptstyle\frac{2}{3}\times\frac{3}{4}

[Expand]

\scriptstyle\frac{5}{7}\times\frac{7}{8}

[Expand]

\scriptstyle\frac{1}{9}\times\frac{1}{7}

[Expand]

\scriptstyle\frac{4}{7}\times\frac{5}{6}

[Expand]

\scriptstyle\frac{4}{7}\times\frac{5}{9}

[Expand]

\scriptstyle\frac{4}{7}\times\frac{7}{9}

[Expand]

\scriptstyle\frac{7}{9}\times\frac{4}{7}

[Expand]

\scriptstyle\frac{7}{8}\times\frac{9}{10}

[Expand]

\scriptstyle\frac{2}{11}\times\frac{2}{13}

[Expand]

\scriptstyle\frac{10}{11}\times\frac{12}{13}

[Expand]

\scriptstyle\frac{4}{5}\times\frac{3}{13}

[Expand]

\scriptstyle\frac{9}{15}\times\frac{11}{17}

[Expand]

\scriptstyle\frac{9}{13}\times\frac{17}{19}

[Expand]

\scriptstyle\frac{20}{3}\times\frac{19}{3}

multiplication of fraction of fraction by fraction of fraction

[Expand]

\scriptstyle\left(\frac{2}{5}\sdot\frac{1}{10}\right)\times\left(\frac{2}{8}\sdot\frac{1}{3}\right)

[Expand]

\scriptstyle\left(\frac{3}{4}\sdot\frac{3}{5}\right)\times\left(\frac{5}{6}\sdot\frac{3}{7}\right)

multiplication of fraction of integer and fraction by fraction of integer and fraction

[Expand]

\scriptstyle\left[\frac{7}{8}\sdot\left(5+\frac{1}{3}\right)\right]\times\left[\frac{3}{5}\sdot\left(6+\frac{1}{4}\right)\right]

[Expand]

\scriptstyle\left[\frac{2}{3}\sdot\left(5+\frac{5}{6}\right)\right]\times\left[\frac{5}{7}\sdot\left(8+\frac{4}{9}\right)\right]

multiplication of fraction by integer or integer by fraction

[Expand]

\scriptstyle\frac{1}{3}\times1

[Expand]

\scriptstyle\frac{1}{3}\times2

[Expand]

\scriptstyle2\times\frac{3}{4}

[Expand]

\scriptstyle\frac{2}{3}\times4

[Expand]

\scriptstyle4\times\frac{2}{3}

[Expand]

\scriptstyle4\times\frac{3}{5}

[Expand]

\scriptstyle5\times\frac{4}{6}

[Expand]

\scriptstyle\frac{3}{4}\times9

[Expand]

\scriptstyle\frac{5}{6}\times10

[Expand]

\scriptstyle\frac{5}{7}\times40

multiplication of fraction of fraction of integer by fraction of fraction of integer

[Expand]

\scriptstyle\left[\left(\frac{3}{4}\sdot\frac{1}{5}\right)\sdot8\right]\times\left[\left(\frac{5}{6}\sdot\frac{1}{7}\right)\sdot15\right]

[Expand]

\scriptstyle\left[\left(\frac{2}{3}\sdot\frac{3}{4}\right)\sdot6\right]\times\left[\left(\frac{3}{4}\sdot\frac{4}{7}\right)\sdot8\right]

[Expand]

\scriptstyle\left(\frac{2}{4}\times\frac{2}{6}\right)+\left(\frac{4}{5}\times\frac{5}{6}\right)

multiplication of sexagesimal fractions

[Expand]

\scriptstyle\left(2+24^\prime+43^{\prime\prime}\right)\times\left(3+3^\prime+8^{\prime\prime}\right)

[Expand]

\scriptstyle\left(4+6^\prime+8^{\prime\prime}+12^{\prime\prime\prime}\right)\times\left(5+15^\prime+10^{\prime\prime}+13^{\prime\prime\prime}\right)

addition of fractions

[Expand]

\scriptstyle\frac{1}{2}+\frac{1}{3}

[Expand]

\scriptstyle\frac{1}{3}+\frac{1}{2}

[Expand]

\scriptstyle\frac{1}{3}+\frac{1}{4}

[Expand]

\scriptstyle\frac{1}{3}+\frac{1}{5}

[Expand]

\scriptstyle\frac{1}{4}+\frac{1}{5}

[Expand]

\scriptstyle\frac{1}{3}+\frac{1}{7}

[Expand]

\scriptstyle\frac{1}{2}+\frac{1}{3}+\frac{1}{4}

[Expand]

\scriptstyle\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}

[Expand]

\scriptstyle\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}

[Expand]

\scriptstyle\frac{2}{3}+\frac{3}{4}

[Expand]

\scriptstyle\frac{2}{3}+\frac{4}{9}

[Expand]

\scriptstyle\frac{3}{4}+\frac{4}{5}

[Expand]

\scriptstyle\frac{3}{4}+\frac{4}{6}

[Expand]

\scriptstyle\frac{3}{4}+\frac{5}{6}

[Expand]

\scriptstyle\frac{3}{5}+\frac{4}{7}

[Expand]

\scriptstyle\frac{3}{7}+\frac{2}{3}

[Expand]

\scriptstyle\frac{3}{8}+\frac{7}{10}

[Expand]

\scriptstyle\frac{2}{3}+\frac{3}{4}+\frac{4}{5}

[Expand]

\scriptstyle\frac{3}{4}+\frac{4}{5}+\frac{5}{6}

[Expand]

\scriptstyle\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}

[Expand]

\scriptstyle\left(\frac{3}{4}+\frac{4}{5}\right)+\left(\frac{5}{6}+\frac{6}{7}\right)

[Expand]

\scriptstyle\frac{2}{3}+\frac{4}{5}+\frac{5}{6}+\left(\frac{3}{4}\sdot\frac{1}{6}\right)

[Expand]

\scriptstyle\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]+\left(\frac{7}{8}+\frac{8}{9}\right)

[Expand]

\scriptstyle\left[\frac{5}{11}+\left(\frac{3}{4}\sdot\frac{1}{11}\right)\right]+\left[\frac{10}{13}+\left(\frac{8}{9}\sdot\frac{1}{13}\right)\right]

[Expand]

\scriptstyle\left(3+\frac{1}{2}\right)+\frac{2}{5}

[Expand]

\scriptstyle\left(2+\frac{1}{2}\right)+\left(3+\frac{1}{4}\right)

[Expand]

\scriptstyle\left(2+\frac{1}{2}\right)+\left(3+\frac{2}{5}\right)

[Expand]

\scriptstyle\left[8+\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]+\left(6+\frac{5}{7}+\frac{10}{11}\right)

[Expand]

\scriptstyle\frac{5}{9}+\left(\frac{4}{5}\sdot\frac{1}{7}\right)

[Expand]

\scriptstyle\left(\frac{3}{4}\sdot5\right)+6

[Expand]

\scriptstyle\left(\frac{2}{3}\sdot7\right)+\left(\frac{7}{8}\sdot9\right)

[Expand]

\scriptstyle\left[\frac{3}{4}\sdot\left(5+\frac{1}{2}\right)\right]+\left[\frac{5}{6}\sdot\left(6+\frac{1}{3}\right)\right]

[Expand]

\scriptstyle\left[2+\frac{3}{8}+\left(\frac{2}{4}\sdot\frac{1}{8}\right)\right]+\frac{4}{5}+\left[\frac{6}{7}+\left(\frac{3}{8}\sdot\frac{1}{7}\right)\right]

[Expand]

\scriptstyle\left(\frac{2}{5}\sdot\frac{2}{9}\sdot2\right)+\left[\left[\frac{1}{8}+\left(\frac{2}{9}\sdot\frac{1}{7}\sdot\frac{1}{8}\right)\right]\sdot\left[\frac{1}{4}+\left(\frac{2}{6}\sdot\frac{1}{4}\right)\right]\right]

subtraction of fractions

[Expand]

\scriptstyle\frac{1}{3}-\frac{1}{4}

[Expand]

\scriptstyle\frac{1}{4}-\frac{1}{5}

[Expand]

\scriptstyle\frac{2}{4}-\frac{1}{5}

[Expand]

\scriptstyle\frac{3}{4}-\frac{1}{2}

[Expand]

\scriptstyle\frac{3}{4}-\frac{2}{3}

[Expand]

\scriptstyle\frac{1}{3}-\frac{1}{5}

[Expand]

\scriptstyle\frac{1}{3}-\frac{1}{7}

[Expand]

\scriptstyle\frac{2}{3}-\frac{3}{7}

[Expand]

\scriptstyle\frac{3}{4}-\frac{2}{8}

[Expand]

\scriptstyle\frac{2}{5}-\frac{3}{8}

[Expand]

\scriptstyle\frac{5}{8}-\frac{3}{8}

[Expand]

\scriptstyle\frac{5}{7}-\frac{4}{9}

[Expand]

\scriptstyle\frac{6}{8}-\frac{2}{4}

[Expand]

\scriptstyle\frac{12}{16}-\frac{4}{8}

[Expand]

\scriptstyle\frac{14}{16}-\frac{8}{12}

[Expand]

\scriptstyle\left(\frac{1}{3}+\frac{1}{4}\right)-\left(\frac{1}{5}+\frac{1}{6}\right)

[Expand]

\scriptstyle1-\frac{5}{8}

[Expand]

\scriptstyle\left(2+\frac{1}{2}\right)-\frac{3}{4}

[Expand]

\scriptstyle\left(3+\frac{3}{4}\right)-\left(2+\frac{1}{2}\right)

[Expand]

\scriptstyle\left[\left[\frac{8}{9}+\left(\frac{3}{7}\sdot\frac{1}{5}\sdot\frac{1}{9}\right)\right]\sdot\left(\frac{5}{6}\sdot3\right)\right]-\left[\left[\frac{3}{4}+\left(\frac{2}{5}\sdot\frac{1}{4}\right)\right]\sdot\frac{2}{9}\right]

[Expand]

\scriptstyle\left[3+\frac{5}{9}+\left(\frac{3}{7}\sdot\frac{1}{9}\right)+\left(\frac{2}{4}\sdot\frac{1}{7}\sdot\frac{1}{9}\right)\right]-\left[\frac{7}{9}+\left(\frac{5}{7}\sdot\frac{1}{9}\right)+\left(\frac{3}{4}\sdot\frac{1}{7}\sdot\frac{1}{9}\right)\right]

Rule of Three

[Expand]

\scriptstyle2:4=4:8

[Expand]

\scriptstyle4:6=6:9

[Expand]

\scriptstyle4:6=8:12

[Expand]

\scriptstyle2:3=5:X

[Expand]

\scriptstyle5:7=10:X

[Expand]

\scriptstyle3:7=5:X

[Expand]

\scriptstyle3:7=X:\left(11+\frac{2}{3}\right)

[Expand]

\scriptstyle5:\left(7+\frac{1}{2}\right)=2:X

[Expand]

\scriptstyle\left(5+\frac{2}{7}\right):4=20:X

[Expand]

\scriptstyle9:\frac{2}{3}=8:X

[Expand]

\scriptstyle\frac{2}{3}:8=9:X

[Expand]

\scriptstyle\frac{1}{3}:\frac{1}{4}=\frac{1}{5}:X

[Expand]

\scriptstyle\frac{2}{3}:\frac{3}{5}=\frac{1}{4}:X

[Expand]

\scriptstyle\frac{3}{7}:\frac{8}{9}=\frac{4}{5}:X

[Expand]

\scriptstyle\frac{2}{3}:\frac{4}{9}=\frac{4}{13}:X

[Expand]

\scriptstyle\left(\frac{2}{5}\sdot\frac{3}{4}\right):\left(\frac{3}{7}\sdot\frac{5}{6}\right)=\left(\frac{4}{5}\sdot\frac{5}{8}\right):X

[Expand]

\scriptstyle\frac{2}{3}:\left(4+\frac{1}{2}\right)=6:X

[Expand]

\scriptstyle6:\left(40+\frac{1}{2}\right)=\frac{2}{3}:X

[Expand]

\scriptstyle20:\left(15+\frac{5}{37}\right)=\left(5+\frac{2}{7}\right):X

[Expand]

\scriptstyle\frac{2}{3}:\left(7+\frac{4}{9}\right)=\frac{4}{13}:X

[Expand]

\scriptstyle\left(3+\frac{1}{3}\right):\left(4+\frac{1}{4}\right)=\left(5+\frac{1}{5}\right):X

[Expand]

\scriptstyle\left(5+\frac{2}{3}\right):\left(6+\frac{3}{4}\right)=\left(8+\frac{5}{12}\right):X

[Expand]

\scriptstyle\left[\frac{3}{4}\sdot\left(3-\frac{1}{4}\right)\right]:\left[\frac{4}{5}\sdot\left(5-\frac{1}{5}\right)\right]=\left[\frac{5}{6}\sdot\left(6-\frac{1}{6}\right)\right]:X

Roots

Extraction of Roots

[Expand]

\scriptstyle\sqrt{\frac{2}{8}}

[Expand]

\scriptstyle\sqrt{100}

[Expand]

\scriptstyle\sqrt{400}

[Expand]

\scriptstyle\sqrt{144}

[Expand]

\scriptstyle\sqrt{225}

[Expand]

\scriptstyle\sqrt{5625}

[Expand]

\scriptstyle\sqrt{7056}

[Expand]

\scriptstyle\sqrt{10375}

[Expand]

\scriptstyle\sqrt{164960}

[Expand]

\scriptstyle\sqrt{456789}

[Expand]

\scriptstyle\sqrt{583696}

[Expand]

\scriptstyle\sqrt{824464}

[Expand]

\scriptstyle\sqrt{973182}

[Expand]

\scriptstyle\sqrt{5499025}

[Expand]

\scriptstyle\sqrt{6169002849}

[Expand]

\scriptstyle\sqrt{344680129066}

Extraction of Cube Roots

[Expand]

\scriptstyle\sqrt[3]{\frac{8}{27}}

[Expand]

\scriptstyle\sqrt[3]{\frac{1}{5}}

[Expand]

\scriptstyle\sqrt[3]{\frac{1}{2}}

[Expand]

\scriptstyle\sqrt[3]{1+\frac{1}{2}}

[Expand]

\scriptstyle\sqrt[3]{2+\frac{1}{2}}

[Expand]

\scriptstyle\sqrt[3]{2+\frac{10}{27}}

[Expand]

\scriptstyle\sqrt[3]{3+\frac{3}{27}}

[Expand]

\scriptstyle\sqrt[3]{3+\frac{1}{4}}

[Expand]

\scriptstyle\sqrt[3]{10}

[Expand]

\scriptstyle\sqrt[3]{15}

[Expand]

\scriptstyle\sqrt[3]{10000}

[Expand]

\scriptstyle\sqrt[3]{12167}

[Expand]

\scriptstyle\sqrt[3]{571787}

[Expand]

\scriptstyle\sqrt[3]{1072000}

[Expand]

\scriptstyle\sqrt[3]{12812904}

Approximation

[Expand]

\scriptstyle\sqrt{2}

[Expand]

\scriptstyle\sqrt{20}

[Expand]

\scriptstyle\sqrt{200}

[Expand]

\scriptstyle\sqrt{2000}

[Expand]

\scriptstyle\sqrt{20000}

[Expand]

\scriptstyle\sqrt{4000}

[Expand]

\scriptstyle\sqrt{5}

[Expand]

\scriptstyle\sqrt{10}

[Expand]

\scriptstyle\sqrt{15}

[Expand]

\scriptstyle\sqrt{6}

[Expand]

\scriptstyle\sqrt{7}

[Expand]

\scriptstyle\sqrt{18}

[Expand]

\scriptstyle\sqrt{29}

[Expand]

\scriptstyle\sqrt{925}

[Expand]

\scriptstyle\sqrt{76543}

[Expand]

\scriptstyle\sqrt{6169004404}

[Expand]

\scriptstyle\sqrt{\frac{4}{6}\sdot\left(4+\frac{5}{9}\right)}

Addition of Roots

[Expand]

\scriptstyle\sqrt{9}+\sqrt{4}

[Expand]

\scriptstyle\sqrt{10}+\sqrt{2}

[Expand]

\scriptstyle\sqrt{3}+\sqrt{12}

[Expand]

\scriptstyle\sqrt{6}+\sqrt{7}

[Expand]

\scriptstyle\sqrt{12}+\sqrt{48}

[Expand]

\scriptstyle\sqrt{18}+\sqrt{8}

[Expand]

\scriptstyle\sqrt{8}+\sqrt{19}

[Expand]

\scriptstyle\sqrt{16}+\sqrt{36}

[Expand]

\scriptstyle2\sdot\sqrt{9}+3\sdot\sqrt{16}

[Expand]

\scriptstyle\frac{2}{3}\sdot\sqrt{9}+\frac{3}{4}\sdot\sqrt{16}

[Expand]

\scriptstyle\sqrt[3]{96}+\sqrt[3]{324}

[Expand]

\scriptstyle\left(4+\sqrt{12}\right)+\left(5+\sqrt{3}\right)

[Expand]

\scriptstyle\left(4+\sqrt{3}\right)+\left(\sqrt{12}-3\right)

[Expand]

\scriptstyle\left(4+\sqrt{3}\right)+\left(\sqrt{12}-2\right)

[Expand]

\scriptstyle\left(4-\sqrt{3}\right)+\left(\sqrt{12}-2\right)

[Expand]

\scriptstyle\sqrt{3}+\sqrt{6}+\sqrt{12}+\sqrt{24}

Subtraction of Roots

[Expand]

\scriptstyle\sqrt{9}-\sqrt{4}

[Expand]

\scriptstyle\sqrt{16}-\sqrt{9}

[Expand]

\scriptstyle\sqrt{12}-\sqrt{3}

[Expand]

\scriptstyle\sqrt{18}-\sqrt{8}

[Expand]

\scriptstyle\sqrt{7}-\sqrt{6}

[Expand]

\scriptstyle\sqrt{36}-\sqrt{16}

[Expand]

\scriptstyle3\sdot\sqrt{36}-2\sdot\sqrt{16}

[Expand]

\scriptstyle\frac{2}{3}\sdot\sqrt{36}-\frac{1}{2}\sdot\sqrt{16}

[Expand]

\scriptstyle19-\left(10-\sqrt{12}\right)

[Expand]

\scriptstyle10-\left(24-\sqrt{250}\right)

[Expand]

\scriptstyle16-\left(8+\sqrt{50}\right)

[Expand]

\scriptstyle\left(13-\sqrt{20}\right)-\left(6-\sqrt{5}\right)

Multiplication of Roots

[Expand]

\scriptstyle\sqrt{9}\times\sqrt{4}

[Expand]

\scriptstyle\sqrt{4}\times\sqrt{9}

[Expand]

\scriptstyle\sqrt{7}\times\sqrt{8}

[Expand]

\scriptstyle\sqrt{5}\times\sqrt{12}

[Expand]

\scriptstyle\sqrt{9}\times\sqrt{36}

[Expand]

\scriptstyle4\sdot\sqrt{9}\times3\sdot\sqrt{36}

[Expand]

\scriptstyle\frac{2}{3}\sdot\sqrt{9}\times\frac{1}{2}\sdot\sqrt{36}

[Expand]

\scriptstyle2\times\sqrt{16}

[Expand]

\scriptstyle4\times\sqrt{9}

[Expand]

\scriptstyle\frac{1}{2}\times\sqrt{9}

[Expand]

\scriptstyle\frac{1}{3}\times\sqrt{9}

[Expand]

\scriptstyle\frac{2}{3}\times\sqrt{9}

[Expand]

\scriptstyle\sqrt{6}\times3

[Expand]

\scriptstyle\sqrt{7}\times3

[Expand]

\scriptstyle\sqrt{7}\times8

[Expand]

\scriptstyle\sqrt{5}\times\left(\sqrt{7}+4\right)

[Expand]

\scriptstyle\sqrt{3}\times\left(6-\sqrt{8}\right)

[Expand]

\scriptstyle\left(3+\sqrt{5}\right)\times\left(3+\sqrt{5}\right)

[Expand]

\scriptstyle\left(5+\sqrt{6}\right)\times\left(5+\sqrt{6}\right)

[Expand]

\scriptstyle\left(3+\sqrt{5}\right)\times\left(4+\sqrt{7}\right)

[Expand]

\scriptstyle\left(3+\sqrt{4}\right)\times\left(4+\sqrt{9}\right)

[Expand]

\scriptstyle\left(3-\sqrt{5}\right)\times\left(4-\sqrt{7}\right)

[Expand]

\scriptstyle\left(3-\sqrt{5}\right)\times\left(3-\sqrt{5}\right)

[Expand]

\scriptstyle\left(5+\sqrt{3}\right)\times\left(5-\sqrt{3}\right)

[Expand]

\scriptstyle\left(3+\sqrt{4}\right)\times\left(5-\sqrt{9}\right)

[Expand]

\scriptstyle\sqrt{8}\times\left(\sqrt{8}-2\right)

[Expand]

\scriptstyle\left(\sqrt{8}-2\right)\times\left(\sqrt{10}-3\right)

[Expand]

\scriptstyle\left(\sqrt{12}-2\right)\times\left(\sqrt{12}-2\right)

[Expand]

\scriptstyle\left(\sqrt{32}-3\right)\times\left(\sqrt{32}-3\right)

[Expand]

\scriptstyle\left(\sqrt{15}-3\right)\times\left(\sqrt{12}+2\right)

[Expand]

\scriptstyle\left(\sqrt{8}+2\right)\times\left(\sqrt{8}-2\right)

[Expand]

\scriptstyle\left(2\times\sqrt{10}\right)\times\left(\frac{1}{2}\times\sqrt{5}\right)

[Expand]

\scriptstyle\sqrt{5}\times\left(\sqrt{7}+\sqrt{10}\right)

[Expand]

\scriptstyle\sqrt{5}\times\left(\sqrt{12}-\sqrt{8}\right)

[Expand]

\scriptstyle\left(\sqrt{5}+\sqrt{7}\right)\times\left(\sqrt{10}+\sqrt{15}\right)

[Expand]

\scriptstyle\left(\sqrt{5}+\sqrt{7}\right)\times\left(\sqrt{5}+\sqrt{7}\right)

[Expand]

\scriptstyle\left(\sqrt{5}+\sqrt{7}\right)\times\left(\sqrt{10}-\sqrt{6}\right)

[Expand]

\scriptstyle\left(\sqrt{10}+\sqrt{7}\right)\times\left(\sqrt{10}-\sqrt{7}\right)

[Expand]

\scriptstyle\left(\sqrt{8}+\sqrt{4}\right)\times\left(\sqrt{8}-\sqrt{4}\right)

[Expand]

\scriptstyle\left(\sqrt{48}+\sqrt{10}\right)\times\left(\sqrt{48}-\sqrt{10}\right)

[Expand]

\scriptstyle\left(\sqrt{12}-\sqrt{7}\right)\times\left(\sqrt{15}-\sqrt{10}\right)

[Expand]

\scriptstyle\left(\sqrt{12}-\sqrt{7}\right)\times\left(\sqrt{12}-\sqrt{7}\right)

[Expand]

\scriptstyle3\times\sqrt{4}

[Expand]

\scriptstyle3\times\sqrt[3]{8}

[Expand]

\scriptstyle3\times\sqrt[3]{5}

[Expand]

\scriptstyle\sqrt{4}\times\sqrt[3]{8}

[Expand]

\scriptstyle\sqrt{9}\times\sqrt[3]{8}

[Expand]

\scriptstyle\sqrt[3]{5}\times\sqrt[3]{6}

[Expand]

\scriptstyle2\times\sqrt[4]{5}

[Expand]

\scriptstyle\sqrt[3]{8}\times\sqrt[4]{16}

[Expand]

\scriptstyle\sqrt[3]{3}\times\sqrt[4]{4}

[Expand]

\scriptstyle\sqrt[4]{4}\times\sqrt[4]{7}

Division of Roots

[Expand]

\scriptstyle\sqrt{9}\div\sqrt{4}

[Expand]

\scriptstyle\sqrt{4}\div\sqrt{9}

[Expand]

\scriptstyle\sqrt{10}\div\sqrt{2}

[Expand]

\scriptstyle\sqrt{30}\div\sqrt{6}

[Expand]

\scriptstyle\sqrt{36}\div\sqrt{9}

[Expand]

\scriptstyle5\sdot\sqrt{9}\div2\sdot\sqrt{36}

[Expand]

\scriptstyle\frac{2}{3}\sdot\sqrt{36}\div\frac{2}{3}\sdot\sqrt{9}

[Expand]

\scriptstyle4\div\sqrt{9}

[Expand]

\scriptstyle12\div\sqrt{4}

[Expand]

\scriptstyle20\div\sqrt{10}

[Expand]

\scriptstyle8\div\left(3+\sqrt{4}\right)

[Expand]

\scriptstyle19\div\left(2+\sqrt{16}\right)

[Expand]

\scriptstyle\sqrt{64}\div\left(\sqrt{8}-\sqrt{4}\right)

[Expand]

\scriptstyle\sqrt{64}\div\left(\sqrt{8}+\sqrt{4}\right)

[Expand]

\scriptstyle\left(5+\sqrt{16}\right)\div3

[Expand]

\scriptstyle20\div\left(4-\sqrt{9}\right)

[Expand]

\scriptstyle\left(19+\sqrt{25}\right)\div\left(5+\sqrt{9}\right)

[Expand]

\scriptstyle\left(2\times\sqrt{20}\right)\div\left(3\times\sqrt{6}\right)

[Expand]

\scriptstyle36\div\left(\sqrt{4}+\sqrt{9}+\sqrt{16}\right)

[Expand]

\scriptstyle70\div\left(\sqrt{4}+\sqrt{9}+\sqrt{16}+\sqrt{25}\right)

[Expand]

\scriptstyle\sqrt{6}\div\sqrt[3]{10}

[Expand]

\scriptstyle\sqrt[3]{18}\div\sqrt[4]{10}

Multiplication of Algebraic Species

[Expand]

\scriptstyle3x\times6

[Expand]

\scriptstyle2x\times2x

[Expand]

\scriptstyle\left(10+x\right)\times x

[Expand]

\scriptstyle\left(10-x\right)\times x

[Expand]

\scriptstyle\left(10+x\right)\times\left(10+x\right)

[Expand]

\scriptstyle\left(10-x\right)\times\left(10-x\right)

[Expand]

\scriptstyle\left(10+x\right)\times\left(10-x\right)

[Expand]

\scriptstyle\left(10+x\right)\times\left(x-10\right)

[Expand]

\scriptstyle\left(10+\frac{2}{3}x\right)\times\left(3-6x\right)

Linear Equation

[Expand]

\scriptstyle bx=\sqrt[3]{c}

[Expand]

\scriptstyle c=\sqrt[3]{bx}

Quadratic Equation

ax²=bx

[Expand]squares equal roots

\scriptstyle ax^2=bx

[Expand]

\scriptstyle x^2=5x

[Expand]

\scriptstyle\frac{1}{2}x^2=10x

[Expand]

\scriptstyle5x^2=20x

ax²=c

[Expand]squares equal numbers

\scriptstyle ax^2=c

[Expand]

\scriptstyle x^2=16

[Expand]

\scriptstyle5x^2=45

[Expand]

\scriptstyle\frac{1}{3}x^2=27

bx=c

[Expand]roots equal numbers

\scriptstyle bx=c

ax²+bx=c

[Expand]

\scriptstyle ax^2+bx=c

[Expand]

\scriptstyle x^2+10x=39

[Expand]

\scriptstyle2x^2+10x=48

[Expand]

\scriptstyle3x^2+15x=72

[Expand]

\scriptstyle\frac{1}{2}x^2+5x=28

[Expand]

\scriptstyle\frac{1}{3}x^2+3x=30

ax²+c=bx

[Expand]

\scriptstyle ax^2+c=bx

[Expand]

\scriptstyle x^2+21=10x

[Expand]

\scriptstyle x^2+25=10x

[Expand]

\scriptstyle3x^2+21=10x

[Expand]

\scriptstyle\frac{1}{3}x^2+21=10x

bx+c=ax²

[Expand]

\scriptstyle bx+c=ax^2

[Expand]

\scriptstyle3x+4=x^2

Compound Quadratic Equations

[Expand]

\scriptstyle ax^2=\sqrt[3]{c}

[Expand]

\scriptstyle c=\sqrt[3]{ax^2}

[Expand]

\scriptstyle\left[x^2-\left(\frac{1}{3}x^2+2\right)\right]^2=x^2+24

[Expand]

\scriptstyle3\sqrt{x^2}+4\sqrt{x^2-3\sqrt{x^2}}=20

[Expand]

\scriptstyle\left(x^2-\frac{1}{3}x^2\right)\sdot3\sqrt{x^2}=x^2

[Expand]

\scriptstyle\left(x^2-\frac{1}{3}x^2\right)\sdot3\sqrt{x^2-\frac{1}{3}x^2}=x^2

[Expand]

\scriptstyle3\sqrt{x^2}+2\sqrt{x^2-3\sqrt{x^2}}=x^2

[Expand]

\scriptstyle3\sqrt{x^2}+4\sqrt{x^2-3\sqrt{x^2}}=x^2+4

[Expand]

\scriptstyle x^2\sdot\left(x^2+\sqrt{10}\right)=9x^2

[Expand]

\scriptstyle\left(\sqrt{8x^2}\sdot\sqrt{3x^2}\right)+20=\left(x^2\right)^2

[Expand]

\scriptstyle\left(\sqrt{6x^2}\sdot\sqrt{5x^2}\right)+10x^2+20=\left(x^2\right)^2

[Expand]

\scriptstyle\left(x^2+10\right)\sdot\sqrt{5}=\left(x^2\right)^2

[Expand]

\scriptstyle\left(\sqrt{x^2\sdot2x^2}+2\right)\sdot x^2=30

[Expand]

\scriptstyle\left[x^2-\left(2\sqrt{x^2}+10\right)\right]^2=8x^2

[Expand]

\scriptstyle2\sqrt{x^2}+\sqrt{\frac{1}{2}x^2}+\sqrt{\frac{1}{3}x^2}=x^2

[Expand]

\scriptstyle2\sqrt{x^2}+\sqrt{\frac{1}{2}x^2}+\sqrt{\frac{1}{3}x^2}=20

[Expand]

\scriptstyle x^2+4\sqrt{x^2}+\sqrt{\frac{1}{2}x^2}+\sqrt{\frac{1}{3}x^2}=10

[Expand]

\scriptstyle\left(x^2+\sqrt{x^2}+\sqrt{\frac{1}{2}x^2}\right)^2=5x^2

[Expand]

\scriptstyle\left(x^2+\sqrt{x^2}+\sqrt{\frac{1}{2}x^2}\right)^2=20

[Expand]

\scriptstyle\left(x^2+\sqrt{\frac{1}{2}x^2}\right)^2=4x^2

[Expand]

\scriptstyle\left(x^2+7\right)\sdot\sqrt{3x^2}=10x^2

[Expand]

\scriptstyle\left(x^2+\sqrt{3x^2}\right)\sdot\sqrt{{\color{red}{2}}x^2}=4x^2

[Expand]

\scriptstyle\left(\sqrt{\frac{1}{2}x^2}+3\right)\sdot\left(\sqrt{\frac{1}{3}x^2}+2\right)=20

[Expand]

\scriptstyle\frac{\sqrt{10}x^2}{2+\sqrt{3}}=x^2-10

[Expand]

\scriptstyle\sqrt{x^2}+\sqrt{\sqrt{x^2}}+\sqrt{2\sqrt{x^2}}+\sqrt{5x^2}=10

quadratic equation in two variables

[Expand]

\scriptstyle\begin{cases}\scriptstyle b^2=3a^2\\\scriptstyle\left(a^2+\sqrt{a^2}\right)\sdot\left(b^2+\sqrt{b^2}\right)=10b^2\end{cases}

quadratic equation in three variables

[Expand]

\scriptstyle\begin{cases}\scriptstyle a^2+b^2=c^2\\\scriptstyle a\sdot c=b^2\\\scriptstyle a\sdot b=10\end{cases}

Cubic Equation

[Expand]

\scriptstyle ax^3=\sqrt[3]{c}

Biquadratic Equation

[Expand]

\scriptstyle4\sdot\left(x^2+8\right)=x^4

[Expand]

\scriptstyle c=ax^4+\sqrt{bx^4}

[Expand]

\scriptstyle ax^4+bx^2=c

[Expand]

\scriptstyle bx^2=ax^4+c

[Expand]

\scriptstyle ax^4=bx^2+c

indeterminate equation

[Expand]

\scriptstyle x^2+5=n^2

[Expand]

\scriptstyle x^2-10=n^2

[Expand]

\scriptstyle x^2+3x=n^2

[Expand]

\scriptstyle x^2-6x=n^2

[Expand]

\scriptstyle x-x^2=n^2

[Expand]

\scriptstyle x^2+10x+20=n^2

[Expand]

\scriptstyle x^2-8x-30=n^2

[Expand]

\scriptstyle 8x+109-x^2=n^2

[Expand]

\scriptstyle 2x+49-x^2=n^2

[Expand]

\scriptstyle 10x-8-x^2=n^2

[Expand]

\scriptstyle 260-6x-x^2=n^2

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2+2x=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2+3x=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2-2x=n^2\\\scriptstyle x^2-3x=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2-x=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2+2x=n^2\\\scriptstyle x^2-3x=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle 3-x^2=n^2\\\scriptstyle 2+x^2=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle 10-x^2=n^2\\\scriptstyle 20+x^2=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle 20+x^2=n^2\\\scriptstyle 30+x^2=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle 10+x^2=n^2\\\scriptstyle 10-x^2=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x+x^2=n^2\\\scriptstyle x-x^2=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle 8x+x^2=n^2\\\scriptstyle 2x-x^2=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2+2x=n^2\\\scriptstyle x^2+2x+3\sqrt{x^2+2x}=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2+3x=n^2\\\scriptstyle x^2+3x+6\sqrt{x^2+3x}=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2+2x=n^2\\\scriptstyle x^2+2x+\sqrt{x^2+2x}=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2+4x=n^2\\\scriptstyle x^2+4x+2\sqrt{x^2+4x}=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2-4x=n^2\\\scriptstyle x^2-4x+2\sqrt{x^2-4x}=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2-5=n^2\\\scriptstyle x^2-5+\sqrt{x^2-5}=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2-2x=n^2\\\scriptstyle x^2-2x+\sqrt{x^2-2x}=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2+1=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2+x+1=n^2\\\scriptstyle x^2+2x+2=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2+4x=n^2\\\scriptstyle x^2-\left(2x+1\right)=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2+3x+1=n^2\\\scriptstyle x^2-\left(3x-2\right)=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2-\left(x-1\right)=n^2\\\scriptstyle x^2-\left(1-x\right)=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2+\left(2-x\right)=n^2\\\scriptstyle x^2-\left(3-x\right)=m^2\end{cases}

[Expand]

\scriptstyle x^2+y^2=n^2

[Expand]

\scriptstyle\begin{cases}\scriptstyle x^2-y=n^2\\\scriptstyle x^2-1\frac{1}{2}y=m^2\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle 49+x=n^2\\\scriptstyle 49+2x=m^2\end{cases}

[Expand]

\scriptstyle x^2+1=10x-8

Euclid

Elements - Introduction

[Expand]common notions

Elements II

Elements II-1

[Expand]proposition

[Expand]

\scriptstyle a\sdot\left(\sum_{i=1}^n b_i\right)=\sum_{i=1}^n \left(a\sdot b_i\right)

Category	Comment	Link	Annotated text
Elements/Elements II-1	a·∑bᵢ=∑(a·bᵢ)	ספר_המספר_/_אליהו_מזרחי#me96	For every number divided into many parts randomly, the number that is generated from the product of a number by the whole given divided number is equal to the number generated from the sum of the products of that number by each part of the divided number. : $\scriptstyle a\sdot\left(\sum_{i=1}^n b_i\right)=\sum_{i=1}^n \left(a\sdot b_i\right)$ השנית שכל מספר נחלק לחלקים רבים איך מה שקרה הנה המספר ההווה מהכאת מספר מה עם המספר המונח הנחלק בכללו הוא שוה למספר ההווה מהכאת המספר ההוא עם כל אחד מחלקי המספר הנחלק כאשר יקובצו
Elements/Elements II-1	a·∑bᵢ=∑(a·bᵢ)	לקוטים_מספר_פראלוקא#t9Uk	If you have two numbers and you divide one of the two numbers, then you multiply each part by the second number, the total sum is the same as the product of the first number by the second. : $\scriptstyle\sum_{i=1}^n \left(a\sdot b_i\right)=a\sdot\left(\sum_{i=1}^n b_i\right)$ ‫156 אם יש לך ב' מספרים ותחלק א' מהב' מספרים ותכפול כל חלק על המספר השני הנה הסך העולה הוא כמו מה שיעלה מכפל המספר הראשון על השני
Elements/Elements II-1	a·∑bᵢ=∑(a·bᵢ)	ספר_המלכים#Sjh3	For every two numbers, such that one of them is divided into as many parts as there are, [the product of] the number that is not divided by the divided number is equal to the sum of its products by each part of the divided number. : $\scriptstyle{\color{OliveGreen}{\left(a_1+a_2+\ldots+a_n\right)\sdot b=\left(a_1\sdot b\right)+\left(a_2\sdot b\right)+\ldots+\left(a_n\sdot b\right)}}$ כל שני מספרים יחלק אחד מהם בחלקים כמו שיהיו הנה המספר שלא חולק במספר שחולק כמו הכאתו בכל חלקי המספר הנחלק כאשר יקובצו
Elements/Elements II-1	a·∑bᵢ=∑(a·bᵢ)	ספר_מעשה_חושב#4J1r	When there are two given numbers and one of them is divided into parts, as many as they may be, the product of the first number by the second is equal to the [sum of] the products of each of the parts of the first number by the second. : $\scriptstyle a\sdot\left(\sum_{i=1}^n b_i\right)=\sum_{i=1}^n \left(a\sdot b_i\right)$ ב כאשר היו שני מספרים מונחים וחולק המספר האחד לחלקים כמה שיהיו הנה שטח המספר האחד בשני שוה לשטחי כל אחד מחלקי המספר האחד בשני מקובצים

Examples

[Expand]

\scriptstyle\left(2\sdot4\right)+\left(3\sdot4\right)+\left(5\sdot4\right)=10\sdot4

[Expand]

\scriptstyle\left(2\sdot10\right)+\left(3\sdot10\right)+\left(5\sdot10\right)+\left(2\sdot10\right)=12\sdot10

Elements II-2

[Expand]

\scriptstyle\sum_{k=1}^n \left[\left(\sum_{i=1}^n a_i\right)\sdot a_k\right]=\left(\sum_{i=1}^n a_i\right)^2

Examples

[Expand]

\scriptstyle\left(7\sdot10\right)+\left(3\sdot10\right)=10^2

[Expand]

\scriptstyle\left(3\sdot12\right)+\left(4\sdot12\right)+\left(5\sdot12\right)=12^2

Elements II-3

[Expand]

\scriptstyle\left(a+b\right)\sdot b=\left(a\sdot b\right)+b^2

Category	Comment	Link	Annotated text
Elements/Elements II-3	(a+b)·b=(a·b)+b²	ספר_המספר_/_אליהו_מזרחי#oIDS	For any number that you divide into two parts randomly, the product of the whole number by any of its two parts is equal to the product of the one part by the other plus the square of the part by which you have multiplied the whole number. : $\scriptstyle{\color{OliveGreen}{\left(a+b\right)\sdot b=\left(a\sdot b\right)+b^2}}$ עוד כל מספר שחלקת אותו לשני חלקים איך שקרה הנה הכאת המספר כלו עם כל אחד משני חלקיו איזה שיהיה שוה להכאת החלק האחד עם השני ולמרובע החלק אשר הכית עם כל המספר
Elements/Elements II-3	(a+b)·b=(a·b)+b²	ספר_מעשה_חושב#BvNU	When a number is divided into two parts, the product of the whole number by one of its parts is equal to the product of the one part by the other plus the square of the mentioned part. : $\scriptstyle \left(a+b\right)\sdot b=\left(a\sdot b\right)+b^2$ ד כאשר חולק מספר מה בשני חלקים הנה שטח כל המספר באחד מחלקיו שוה לשטח החלק האחד באחר ולמרובע החלק אשר זכרנו
Elements/Elements II-3	(a+b)·b=(a·b)+b²	ספר_החשבון_והמדות#Eu5r	For any number divided into two parts as you wish, the product of the whole number by any of its two parts is equal to the product of the one part by the other plus the square of the part by which you multiplied the whole number. : $\scriptstyle{\color{OliveGreen}{\left(a+b\right)\sdot b= \left(a\sdot b\right)+b^2}}$ עוד כל מספר שחלקת אותו בשני חלקים איך שקרה הנה כפל המספר כולו על אחד משני חלקיו איזה שיהיה שוה לכפול החלק האחד על השני ולמרובע החלק משניהם אשר כפלת על כל המספר
Elements/Elements II-3	(a+b)·b=(a·b)+b²	קצת_מענייני_חכמת_המספר#PVyt	For any number divided into two parts as you wish, the product of the whole number by any of its two parts is equal to the product of the one part by the other plus the square of the part by which you multiplied the whole number. : $\scriptstyle{\color{OliveGreen}{\left(a+b\right)\sdot b=\left(a\sdot b\right)+b^2}}$ ב כל מספר שחלקת אותו לב' חלקים איך שקרה הנה הכאת המספר כלו עם כל אחד מב' חלקיו איזה שיהיה שוה להכאת החלק האחד עם האחר ולמרובע החלק אשר בו הכית כל המספר
Elements/Elements II-3	(a+b)·b=(a·b)+b²	לקוטים_מספר_פראלוקא#fYtF	If you divide any number into two parts, then multiply any of the two parts by the divided number and keep the product, so will be the result if you multiply the same part by itself, then add to it the product of the one part by the other. : $\scriptstyle\left(a+b\right)\sdot b=b^2+\left(a\sdot b\right)$ ‫158 אם תחלק איזה מספר על ב' חלקים ותכפול החלק שיהיה מב' החלקים על המספר המחולק ושמור העולה כך יעלה אם תכפול אותו חלק על עצמו ותוסיף בו העולה מהכפל מהחלק האחד על השני

Examples

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\scriptstyle10\sdot3=\left(3\sdot7\right)+3^2

[Expand]

\scriptstyle12\sdot8=\left(4\sdot8\right)+8^2

Elements II-4

[Expand]

\scriptstyle\left(a+b\right)^2=a^2+b^2+2\sdot\left(a\sdot b\right)

Category	Comment	Link	Annotated text
Elements/Elements II-4	(a+b)²=a²+b²+2(a·b)	ספר_המספר_/_אליהו_מזרחי#oC2j	For any number that you divide into two parts randomly, the square of the whole number is equal to [the sum of] the squares of the two parts and twice the product of the one part by the other. : $\scriptstyle{\color{OliveGreen}{\left(a+b\right)^2=a^2+b^2+\left[2\sdot\left(a\sdot b\right)\right]}}$ עוד כל מספר שחלקת אותו לשני חלקים איך שקרה הנה מרובע כל המספר שוה לשני המרובעי' ההווים משני החלקים ולהכאת החלק האחד עם חברו פעמים
Elements/Elements II-4	(a+b)²=a²+b²+2(a·b)	לקוטים_מספר_פראלוקא#lb40	If you divide any number into two and multiply each part by itself, you multiply also one [part] by the other and multiply [the product] by 2, then you sum all, the result is equal to the divided number multiplied by itself. : $\scriptstyle a^2+b^2+\left[2\sdot\left(a\sdot b\right)\right]=\left(a+b\right)^2$ ‫159 אם תחלק איזה מספר שיהיה על ב' ותכפול כל חלק על עצמו וג"כ כפול הא' על הב' וכפלם על ב' ותקבץ הכל הנה העולה יהיה שוה אל המספר המחולק כפול על עצמו
Elements/Elements II-4	(a+b)²=a²+b²+2(a·b)	ספר_מעשה_חושב#lUMx	When a number is added to a number, the square of the two numbers that are summed together is equal to [the sum of] the squares of these numbers and twice the product of the one by the other. : $\scriptstyle\left(a+b\right)^2=a^2+b^2+\left[2\sdot\left(a\sdot b\right)\right]$ ו כאשר נוסף על מספר מונח מספר מה הנה מרובע שני המספרים מחוברים שוה למרובעי המספרים ההם ולכפל שטח זה בזה
Elements/Elements II-4	(a+b)²=a²+b²+2(a·b)	ספר_החשבון_והמדות#q5ib	For any number divided into two parts as you wish, the square of the whole number is equal to [the sum of] the squares of the two parts and twice the product of the one part by the other. : $\scriptstyle{\color{OliveGreen}{\left(a+b\right)^2=a^2+b^2+\left[2\sdot\left(a\sdot b\right)\right]}}$ עוד כל מספר שחלקת אותו לשני חלקים איך שקרה הנה מרובע כל המספר שוה לשני המרובעים ההוים משני החלקים ולכפל החלק האחד על חברו פעמים
Elements/Elements II-4	(a+b)²=a²+b²+2(a·b)	קצת_מענייני_חכמת_המספר#v7Dx	For any number divided into two parts as you wish, the square of the whole number is equal to [the sum of] the squares of the two parts and twice the product of the one part by the other. : $\scriptstyle{\color{OliveGreen}{\left(a+b\right)^2=a^2+b^2+\left[2\sdot\left(a\sdot b\right)\right]}}$ ג כל מספר שחלקת אותו לשני חלקים איך שקרה הנה מרובע כל המספר שוה לב' המרובעים ההווים מב' החלקים ולהכאת החלק האחד עם חבירו ב' פעמים
Elements/Elements II-4	(a+b)²=a²+b²+2(a·b)	לקוטים_מספר_פראלוקא#lAxq	If you have a number and you divide it into two unequal parts, I say that if you sum the [parts] that are multiplied by themselves, then you multiply the smaller [part] by the greater, multiply the product by 2 and sum it with the reserved, the result is equal to the [original] number multiplied by itself. : $\scriptstyle\left(a+b\right)^2=a^2+b^2+\left[2\sdot\left(a\sdot b\right)\right]$ ‫169 אם יש לך מספר ותחלקהו על ב' חלקים בלתי שוים אומר כי אם תקבץ המספרי' המוכפלים על עצמם ואח"כ תכפול המספר הקטון על הגדול והעולה כפול על ב' ותקבץ זה עם השמור יעלה הכל כמו המספר כפול על עצמו
Elements/Elements II-4	(a+b)²=a²+b²+2(a·b)	ספר_המספר_/_אליהו_מזרחי#TApz	For any number divided into two parts randomly, the square of the whole number is equal to [the sum of] the squares of the two parts and twice the product of the one part by the other. : $\scriptstyle\left(a+b\right)^2=a^2+b^2+\left[2\sdot\left(a\sdot b\right)\right]$ האחת שכל מספר נחלק לשנים חלקים איך מה שקרה הנה המרובע ההווה מן המספר כלו הוא שוה לשני המרובעים ההווים משני חלקיו עם כפל המספר ההווה מהכאת החלק האחד עם האחר
Elements/Elements II-4	(a+b)²=a²+b²+2(a·b)	ספר_המלכים#wUeW	For any number divided into [two] parts, whichever they may be, the product of the whole number by itself is equal to the sum of the products of each of the two parts by itself and double the product of one of the two parts by the other. : $\scriptstyle{\color{OliveGreen}{\left(a+b\right)^2=a^2+b^2+\left[2\sdot\left(a\sdot b\right)\right]}}$ \|style="width:45%; text-align:right;"\|כל מספר יחלק בחלקים כמו שיהיו הנה הכאת המספר כלו בעצמו כמו הכאת כל אחד משני החלקים בעצמו וכפל הכאת אחד משני החלקים באחר כאשר יקובצו

Examples

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\scriptstyle10^2=3^2+7^2+2\sdot\left(3\sdot7\right)

[Expand]

\scriptstyle12^2=4^2+8^2+2\sdot\left(4\sdot8\right)

Elements II-5

[Expand]

\scriptstyle\left[\frac{1}{2}\sdot\left(a+b\right)\right]^2=\left(b\sdot b\right)+\left[b-\left[\frac{1}{2}\sdot\left(a\sdot b\right)\right]\right]^2=\left(b\sdot b\right)+\left[\left[\frac{1}{2}\sdot\left(a\sdot b\right)\right]-a\right]^2

Category	Comment	Link	Annotated text
Elements/Elements II-5	(½(a+b))²=(a·b)+(b-½(a+b))²=(a·b)+(½(a+b)-a)²	ספר_המספר_/_אליהו_מזרחי#yKYx	For any number, when you divide it into two equal parts and into two unequal parts, [the sum of] the product of one of the unequal parts by the other and the square of the difference between the two parts, i.e. between the equal part [= the half of the whole number] and the unequal [part] is equal to the square of half the [whole] number. : $\scriptstyle{\color{OliveGreen}{\left(a\sdot b\right)+\left[\left[\frac{1}{2}\sdot\left(a+b\right)\right]-a\right]^2=\left(a\sdot b\right)+\left[b-\left[\frac{1}{2}\sdot\left(a+b\right)\right]\right]^2=\left[\frac{1}{2}\sdot\left(a+b\right)\right]^2}}$ עוד כל מספר כאשר תחלקהו לשני חלקים שוים ולשני חלקים בלתי שוים הנה הכאת החלק האחד עם חברו מהחלקים הבלתי שוים ומרובע מה שבין שני החלקים ר"ל בין החלק השוה ובלתי שוה שוה למרובע חצי המספר
Elements/Elements II-5	(½(a+b))²=(a·b)+(b-½(a+b))²=(a·b)+(½(a+b)-a)²	ספר_החשבון_והמדות#43lj	For any number divided into two equal parts and into two unequal parts, [the sum of] the product of one of the unequal parts by the other and the square of the difference between the two parts, i.e. between the equal part [= the half of the whole number] and the unequal [part] is equal to the square of half the [whole] number. : $\scriptstyle{\color{OliveGreen}{\left(a\sdot b\right)+\left[\left[\frac{1}{2}\sdot\left(a+b\right)\right]-a\right]^2=\left(a\sdot b\right)+\left[b-\left[\frac{1}{2}\sdot\left(a+b\right)\right]\right]^2=\left[\frac{1}{2}\sdot\left(a+b\right)\right]^2}}$ עוד כל מספר כאשר תחלקהו לשני חלקים שוים ולשני חלקים בלתי שוים הנה כפל החלק האחד אל חברו מהחלקים הבלתי שוים ומרובע מה שבין שני ‫32vהחלקים ר"ל בין החלק השוה ובלתי שוה שוה למרובע חצי המספר
Elements/Elements II-5	(½(a+b))²=(a·b)+(b-½(a+b))²=(a·b)+(½(a+b)-a)²	קצת_מענייני_חכמת_המספר#9QNa	For any number divided into two equal parts and into two unequal parts, [the sum of] the product of one of the unequal parts by the other and the square of the difference between the two parts, i.e. between the equal part [= the half of the whole number] and the unequal [part] is equal to the square of half the [whole] number. : $\scriptstyle{\color{OliveGreen}{\left(a\sdot b\right)+\left[\left[\frac{1}{2}\sdot\left(a+b\right)\right]-a\right]^2=\left[\frac{1}{2}\sdot\left(a+b\right)\right]^2}}$ ד כל מספר כאשר תחלקהו לב' חלקים שוים ולב' חלקים בלתי שוים הנה הכאת החלק הא' עם חבירו מהחלקים הבלתי שוים ומרוב' מה שבין ב' חלקים ר"ל בין החלק השוה ובלתי שוה שוה למרובע חצי המספר
Elements/Elements II-5	(½(a+b))²=(a·b)+(b-½(a+b))²=(a·b)+(½(a+b)-a)²	לקוטים_מספר_פראלוקא#24rA	If you divide any number into two equal parts and into two unequal parts, the product of the equal parts one by the other is as the product of the unequal parts one by the other, when you add to it the [square] of the difference between the [equal] part and the [unequal part]. ‫160 אם תחלק מספר אחד לב' חלקים שוים ולב' חלקים בלתי שוים הנה כל כך יעלה כפל החלקים השוים זה על זה כמו כפל החלקים הבלתי שוים זה על זה ותוסיף בם כפל היתרון שיש מהחלק האחד על האחר
Elements/Elements II-5	(½(a+b))²=(a·b)+(b-½(a+b))²=(a·b)+(½(a+b)-a)²	ספר_מעשה_חושב#AHZr	The product of half the given number by itself is equal to [the sum of] the product of a part of that number by the other part and the square of the difference between one of the [unequal] parts and half of the [whole] given number. ח השטח ההוה מחצי המספר המונח בעצמו שוה לשטח ההוה מחלק מה מהמספר ההוא בחלק השני ולמרובע יתרון אחד מן החלקים על חצי המספר המונח
Elements/Elements II-5	(½(a+b))²=(a·b)+(b-½(a+b))²=(a·b)+(½(a+b)-a)²	ספר_המלכים#4PpZ	For any even number divided into halves and into [two] unequal parts, the product of half the [whole] number by itself is equal to [the sum of] the product of the greater part by the smaller [part] and the product of the excess of the half of the [whole] number over the smaller part by itself. : $\scriptstyle{\color{OliveGreen}{\left[\frac{1}{2}\sdot\left(a+b\right)\right]^2=\left(a\sdot b\right)+\left[\left[\frac{1}{2}\sdot\left(a+b\right)\right]-a\right]^2}}$ כל מספר זוג יחלק לחצאים ולחלקים מתחלפים הנה אשר יהיה מהכאת [חצי]‫M om. המספר בעצמו כמו ההווה מהכאת החלק הגדול בקטן עם הכאת מותר חצי המספר על החלק ה^הקטן [בכמהו]‫M וכמוהו

Examples

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\scriptstyle5^2=\left(3\sdot7\right)+\left(7-5\right)^2

[Expand]

\scriptstyle6^2=\left(4\sdot8\right)+\left(8-6\right)^2

Elements II-6

[Expand]

\scriptstyle\left[\left(a+b\right)\sdot b\right]+\left(\frac{1}{2}\sdot a\right)^2=\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2

Category	Comment	Link	Annotated text
Elements/Elements II-6	(a+b)·b+(½a)²=(½a+b)²	ספר_החשבון_והמדות#A9xw	If you divide any number into half and add to it another number, [the sum of] the product of the whole number plus the additional [number] by the additional [number] and the square of half the number is equal to the square of half the number and the additional [number] together. : $\scriptstyle{\color{OliveGreen}{\left[\left(a+b\right)\sdot b\right]+\left(\frac{1}{2}\sdot a\right)^2=\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2}}$ עוד כל מספר כאשר חלקת אותו לחצאים והוספת עליו מספר אחר הנה כפל המספר כלו מקובץ עם התוספת בתוספת והמרובע ההוה מחצי המספר שוה למרובע חצי המספר והתוספת ביחד
Elements/Elements II-6	(a+b)·b+(½a)²=(½a+b)²	ספר_המספר_/_אליהו_מזרחי#tcs6	For any number, when you divide it into half and add to it another number, [the sum of] the product of the whole number plus the additional [number] by the additional [number] and the square of half the number is equal to the square of half the number and the additional [number] together. : $\scriptstyle{\color{OliveGreen}{\left[\left(a+b\right)\sdot b\right]+\left(\frac{1}{2}\sdot a\right)^2=\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2}}$ עוד כל מספר כאשר חלקת אותו לחצי והוספת עליו מספר אחר הנה הכאת המספר כלו מקובץ עם התוספת בתוספת והמרובע ההווה מחצי המספר שוה למרובע חצי המספר והתוספת ביחד
Elements/Elements II-6	(a+b)·b+(½a)²=(½a+b)²	קצת_מענייני_חכמת_המספר#nmaV	If we divide any number into half and add to it another number, [the sum of] the product of the whole number plus the additional [number] by the additional [number] and the square of half the number is equal to the square of half the number and the additional [number] together. : $\scriptstyle{\color{OliveGreen}{\left[\left(a+b\right)\sdot b\right]+\left(\frac{1}{2}\sdot a\right)^2=\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2}}$ ה כל מספר כאשר חלקנו אותו לחצי והוספת עליו מספר אחר הנה הכאת המספר כלו מחובר עם התוספ' בתוספת ומרובע חצי המספר שוה למרובע חצי המספר והתוספת ביחד
Elements/Elements II-6	(a+b)·b+(½a)²=(½a+b)²	ספר_המלכים#c7yY	For any number divided into two halves and another number is added to it, the product of half the number and the additional [number] together by itself is equal to [the sum of] the product of the [whole] number plus the additional [number] by the additional [number] and the product of half the original number by itself. : $\scriptstyle{\color{OliveGreen}{\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2=\left[\left(a+b\right)\sdot b\right]+\left(\frac{1}{2}\sdot a\right)^2}}$ כל מספר זוג יחלק לשני חצאים ויתוסף בו מספר אחר הנה הכאת חצי המספר עם התוספת בכמהו כהכאת המספר עם התוספת בתוספת והכאת חצי המספר הראשון בעצמו
Elements/Elements II-6	(a+b)·b+(½a)²=(½a+b)²	ספר_מעשה_חושב#vsiV	When a number is divided into two halves and a number is added to it, [the sum of] the product of the additional [number] by the whole number plus the additional [number] and the square of half the number is equal to the square of half the number and the additional [number] summed together. : $\scriptstyle\left[\left(a+b\right)\sdot b\right]+\left(\frac{1}{2}\sdot a\right)^2=\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2$ ה כאשר חולק מספר מה לחציין והוסף עליו מספר מה הנה שטח התוספת במספר כלו עם התוספת עם מרובע חצי המספר שוה למרובע חצי המספר והתוספת מקובצים
Elements/Elements II-6	(a+b)·b+(½a)²=(½a+b)²	לקוטים_מספר_פראלוקא#77le	If you divide any number into two equal parts, then add an additional [number] to the divided number, multiply [the sum] by the additional [number] and add [the product] to the square of one of the parts, I say that it is equal to the square of half [the number] with the additional [number]. : $\scriptstyle\left[\left(a+b\right)\sdot b\right]+\left(\frac{1}{2}\sdot a\right)^2=\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2$ ‫161 אם תחלק איזה מספר לב' חלקים שוים ותוסיף בם א' מהחלקי' כפול על עצמו ותוסיף בם התוספת על כל המספר המחולק ותכפול אותו המספר אשר הוספת ותחברם עם כפל אחד מהחלקים אומר כי הוא שוה אל כפל המחצית עם התוספת

Examples

[Expand]

\scriptstyle\left[\left(10+2\right)\sdot2\right]+\left(\frac{1}{2}\sdot10\right)^2=\left[\left(\frac{1}{2}\sdot10\right)+2\right]^2

[Expand]

\scriptstyle\left[\left(12+4\right)\sdot4\right]+\left(\frac{1}{2}\sdot12\right)^2=\left[\left(\frac{1}{2}\sdot12\right)+4\right]^2

Elements II-7

[Expand]

\scriptstyle\left(a+b\right)^2+a^2=\left[2\sdot\left(a+b\right)\sdot a\right]+b^2

Category	Comment	Link	Annotated text
Elements/Elements II-7	(a+b)²+a²=(2·(a+b)·a)+b²	לקוטים_מספר_פראלוקא#lBZs	If you have a number and you divide it into two unequal parts, you multiply the greater part by itself and multiply also the divided number by itself, then sum both products, the result is equal to the product of the greater number by the divided number multiplied by two, then you add to it the square of smaller part. : $\scriptstyle\left(a+b\right)^2+a^2=2\sdot\left[\left(a+b\right)\sdot a\right]+b^2$ ‫162 אם יש לך מספר ותחלקהו לב' חלקים בלתי שוים ותכפול החלק הגדול על עצמו גם תכפול המספר המחולק על עצמו ותחבר הב' הכפלות הנה העולה יהיה שוה אל כפל המספר הגדול על המספר המחולק ותכפלהו אח"כ על ב' גם תוסיף על זה כפל החלק הקטן ותחבר הכל יהיה שוה
Elements/Elements II-7	(a+b)²+a²=(2·(a+b)·a)+b²	ספר_המלכים#isjp	For any number divided into two parts, [the sum of] the product of the [whole] number by itself and [the product of] one of the two parts by itself is equal to twice the product of the [whole] number by the part that is multiplied by itself plus the [product of the] other part by itself. : $\scriptstyle{\color{OliveGreen}{\left(a+b\right)^2+a^2=\left[2\sdot\left(a+b\right)\sdot a\right]+b^2}}$ כל מספר יחלק לשני חלקים ~~[..]~~ הנה הכאת המספר בכמוהו ואחד ‫59vמשני החלקים בכמוהו כמו ההווה מהכאת המספר בחלק המוכה בכמוהו שני פעמים והחלק השני בכמוהו
Elements/Elements II-7	(a+b)²+a²=(2·(a+b)·a)+b²	קצת_מענייני_חכמת_המספר#DEFA	For any number divided into two parts, the sum of the square of the whole number and the square of one of the parts is equal to twice the product of this part by the whole number plus the product of the other part by itself : $\scriptstyle{\color{OliveGreen}{\left(a+b\right)^2+a^2=2\sdot\left[\left(a+b\right)\sdot a\right]+b^2}}$ ו כל מספר שתחלקהו בב' חלקים איך שקרה המרובע ההווה מהמספר כלו והמרובע ההווה מא' מב' אלו החלקים כאשר התקבצו שוה להכאת החלק הנזכ' עם המספר כלו ולהכאת החלק הב' הנשאר בעצמו
Elements/Elements II-7	(a+b)²+a²=(2·(a+b)·a)+b²	ספר_החשבון_והמדות#hdYZ	For any number divided into two parts, the sum of the square of the whole number and the square of one of the parts is equal to twice the product of this part by the whole number plus the product of the other part by itself. : $\scriptstyle{\color{OliveGreen}{\left(a+b\right)^2+a^2=2\sdot\left[\left(a+b\right)\sdot a\right]+b^2}}$ עוד כל מספר כשתחלקהו בשני חלקים איך שיקרה הנה המרובע ההוה מן המספר כלו והמרובע ההוה מאחד משני חלקים כאשר {{#annot:term\|178,1216\|tLsc}}התקבצו{{#annotend:tLsc}} שוים לכפל המספר כלו עם החלק הנזכר פעמים והמרובע ההוה מן החלק השני
Elements/Elements II-7	(a+b)²+a²=(2·(a+b)·a)+b²	ספר_המספר_/_אליהו_מזרחי#WZG6	For any number, when you divide it into two parts randomly, [the sum of] the square of the whole number and the square of one of the two parts is equal to twice the product of the whole number by the mentioned part plus the square of the second part. : $\scriptstyle{\color{OliveGreen}{\left(a+b\right)^2+a^2=2\sdot\left[\left(a+b\right)\sdot a\right]+b^2}}$ עוד כל מספר כאשר תחלקהו בשני חלקים איך שיקרה המרובע ההווה מן המספר כלו והמרובה ההווה מאחד משנים החלקים כאשר התקבצו שוים להכאת המספר כלו עם החלק הנזכר פעמים והמרובע ההווה מן החלק השני

Examples

[Expand]

\scriptstyle10^2+7^2=\left(2\sdot10\sdot7\right)+3^2

[Expand]

\scriptstyle12^2+8^2=\left(2\sdot12\sdot8\right)+4^2

Elements II-8

[Expand]

\scriptstyle4\sdot\left[\left(a+b\right)\sdot a\right]+b^2=\left[\left(a+b\right)+a\right]^2

Category	Comment	Link	Annotated text
Elements/Elements II-8	(4·(a+b)·a)+b²=((a+b)+a)²	ספר_המספר_/_אליהו_מזרחי#LHo4	For any number, when you divide it into two parts randomly and you multiply the whole number by one of the two parts four times, then sum [the product] with the square of the other part [it] is equal to the [square] of the whole number plus the mentioned part when you sum them together. : $\scriptstyle{\color{OliveGreen}{4\sdot\left[\left(a+b\right)\sdot a\right]+b^2=\left[\left(a+b\right)+a\right]^2}}$ עוד כל מספר כאשר חלקת אותו בשני חלקים איך שיקרה והכית המספר כלו עם אחד משני חלקיו ארבעה פעמים וחברת אותו עם מרובע החלק הנשאר שוה למרובע ההווה מן המספר כלו והחלק הנזכר כאשר תחברם ביחד
Elements/Elements II-8	(4·(a+b)·a)+b²=((a+b)+a)²	ספר_החשבון_והמדות#EAD6	For any number divided into two parts as you wish, if you multiply the whole number by one of the parts four times, the sum of the product with the square of the other part is equal to the [square] of the whole number plus the one part. : $\scriptstyle{\color{OliveGreen}{4\sdot\left[\left(a+b\right)\sdot a\right]+b^2=\left[\left(a+b\right)+a\right]^2}}$ עוד כל מספר כאשר חלקת אותו בשני חלקים איך שיקרה וכפלת המספר כלו עם אחד משני חלקיו ארבעה פעמים ועם מרובע החלק הנשאר שוה למרובע ההוה מן המספר כלו והחלק הנזכר כאשר תחברם ביחד ותקח מרובעם
Elements/Elements II-8	(4·(a+b)·a)+b²=((a+b)+a)²	לקוטים_מספר_פראלוקא#TRuX	If you have a number and you divide it into unequal two parts, if you add one of the parts to the whole number, i.e. the divided number, [its square] is equal to the same part that you added multiplied by the whole number, then multiplied by four, when you add to it the square of the second part. : $\scriptstyle\left[\left(a+b\right)+a\right]^2=4\sdot\left[\left(a+b\right)\sdot a\right]+b^2$ ‫163 אם יש לך מספר ותחלקהו לב' חלקים בלתי שוים אם תוסיף א' מהחלקים על כל המספר ר"ל על המספר המחולק יהיה שוה אל אותו החלק אשר הוספת כפול על כל המספר אח"כ העולה תכפול על ד' ותוסיף על זה מרובע החלק השני
Elements/Elements II-8	(4·(a+b)·a)+b²=((a+b)+a)²	ספר_המלכים#Ocvs	For any number divided into two parts and one of the two parts is added to it, the product of the [whole] number plus the additional [part] by itself is equal to [the sum of] the product of the [whole] number by the additional [part] four times and the product of the other part by itself. : $\scriptstyle{\color{OliveGreen}{\left[\left(a+b\right)+a\right]^2=\left[4\sdot\left(a+b\right)\sdot a\right]+b^2}}$ כל מספר יחלק בשני חלקים ונוסף עליו כמו אחד משני החלקים הנה הכאת המספר עם התוספת בכמהו כהכאת המספר בתוספת ד' פעמים והכאת החלק האחר בכמהו
Elements/Elements II-8	(4·(a+b)·a)+b²=((a+b)+a)²	קצת_מענייני_חכמת_המספר#R68t	For any number divided into two parts as you wish, if you multiply the whole number by one of the parts four times, the sum of the product with the square of the other part is equal to the [square] of the whole number plus the one part : $\scriptstyle{\color{OliveGreen}{4\sdot\left[\left(a+b\right)\sdot a\right]+b^2=\left[\left(a+b\right)+a\right]^2}}$ ז כל מספר שחלקת אותו לב' חלקים איך שקרה אם הכית המספר כלו עם חלק אח' מהם ד' פעמים וקבצת הכל עם מרובע החלק הב' הנשאר היה שוה להכאת מספרו החלק הנזכר כאשר תחברם יחד

Examples

[Expand]

\scriptstyle4\sdot\left(10\sdot7\right)+3^2=\left(10+7\right)^2

[Expand]

\scriptstyle4\sdot\left(12\sdot4\right)+8^2=\left(12+4\right)^2

Elements II-9

[Expand]

\scriptstyle a^2+b^2=2\sdot\left[\left[\frac{1}{2}\sdot\left(a+b\right)\right]^2+\left[a-\left[\frac{1}{2}\sdot\left(a+b\right)\right]\right]^2\right]

Category	Comment	Link	Annotated text
Elements/Elements II-9	a²+b²=2·((½·(a+b))²+(a-(½·(a+b)))²)	ספר_המספר_/_אליהו_מזרחי#ClRv	For any number that you divide into two equal parts and into two unequal parts, [the sum of] the two squares of the unequal parts is equal to twice [the sum of] the square of half the [whole] number and [the square] of the excess of the large part over the half [of the whole number]. : $\scriptstyle{\color{OliveGreen}{a^2+b^2=2\sdot\left[\left[\frac{1}{2}\sdot\left(a+b\right)\right]^2+\left[a-\left[\frac{1}{2}\sdot\left(a+b\right)\right]\right]^2\right]}}$ עוד כל מספר שחלקת אותו לשני חלקים שוים ושני חלקים בלתי שוים הנה שני המרובעים אשר יהיו מהחלקים הבלתי שוים הם כפל שני המרובעים אשר יהיו מחצי המספר ומהתוספת אשר לחלק הגדול על הה' שהוא המחצית
Elements/Elements II-9	a²+b²=2·((½·(a+b))²+(a-(½·(a+b)))²)	ספר_המלכים#g6A2	For any even number divided into two halves and into two unequal parts, [the sum of the products of] each of the two unequal parts by themselves is equal to [the sum of] twice the product of half the [whole] number by itself and twice the product of the excess of half the [whole] number over the smaller part by itself. : $\scriptstyle{\color{OliveGreen}{a^2+b^2=\left[2\sdot\left[\frac{1}{2}\sdot\left(a+b\right)\right]^2\right]+\left[2\sdot\left[\left[\frac{1}{2}\sdot\left(a+b\right)\right]-a\right]^2\right]}}$ כל מספר זוג יחלק בשני חצאים ובשני חלקים מתחלפים הנה כל אחד משני החלקים המתחלפים בכמהו כהכאת חצי המספר בכמהו שני פעמים והכאת מותר חצי המספר על החלק הקטן בכמהו שני פעמים
Elements/Elements II-9	a²+b²=2·((½·(a+b))²+(a-(½·(a+b)))²)	לקוטים_מספר_פראלוקא#E95L	If you divide any line into two equal parts and into two unequal parts, then you multiply each of the unequal parts [by itself] and sum them, they are twice [the sum of] the product of the equal parts plus the [square of] excess of the greater [part] over [half the number]. : $\scriptstyle a^2+b^2=2\sdot\left[\left[\frac{1}{2}\sdot\left(a+b\right)\right]^2+\left[a-\left[\frac{1}{2}\sdot\left(a+b\right)\right]\right]^2\right]$ ‫164 אם תחלק איזה קו לב' חלקים שוים ולב' חלקים בלתי שוים וכפול כל אחד מהחלקים בלתי שוים ותחברם הם כפל מהחלקים השוים נוסף בם יתרון החלק הגדול על הקטן
Elements/Elements II-9	a²+b²=2·((½·(a+b))²+(a-(½·(a+b)))²)	ספר_החשבון_והמדות#CM8j	For any number divided into two equal parts and into two unequal parts, [the sum of] the squares of the unequal parts is equal to twice [the sum of] the square of half the [whole] number and the square of the difference between the large part and the half [of the whole number]. : $\scriptstyle{\color{OliveGreen}{a^2+b^2=2\sdot\left[\left[\frac{1}{2}\sdot\left(a+b\right)\right]^2+\left[a-\left[\frac{1}{2}\sdot\left(a+b\right)\right]\right]^2\right]}}$ עוד כל מספר שחלקת אותו לשני חלקים שוים ושני חלקים בלתי שוים הנה שני המרובעים אשר יהיו מהחלקים הבלתי שוים הם כפל שני המרובעים אשר יהיו מחצי המספר ומהתוספת אשר לחלק הגדול על המחצית

Examples

[Expand]

\scriptstyle 7^2+3^2=2\sdot\left[\left(\frac{1}{2}\sdot10\right)^2+\left[7-\left(\frac{1}{2}\sdot10\right)\right]^2\right]

[Expand]

\scriptstyle 8^2+4^2=2\sdot\left[\left(\frac{1}{2}\sdot12\right)^2+\left[8-\left(\frac{1}{2}\sdot12\right)\right]^2\right]

Elements II-10

[Expand]

\scriptstyle\left(a+b\right)^2+b^2=2\sdot\left[\left(\frac{1}{2}\sdot a\right)^2+\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2\right]

Category	Comment	Link	Annotated text
Elements/Elements II-10	(a+b)²+b²=2·((½a)²+((½a)+b)²)	לקוטים_מספר_פראלוקא#ijBn	If you divide any number into two equal parts, then add another number to the divided number, square it and add to it the square of the [number] you added, it is twice [the sum of] the square of the additional [number] plus half [the number] with the square of half [the number]. : $\scriptstyle\left(a+b\right)^2+b^2=2\sdot\left[\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2+\left(\frac{1}{2}\sdot a\right)^2\right]$ ‫165 אם תחלק איזה מספר על ב' חלקים שוים ואחר תוסיף מספר אחר על המספר המחולק ותרבענו ותוסיף על זה מרובע שהוספת יהיה ב' פעמים כמו כפל ממה שיעלה התוספת הנוסף על החצי עם מרובע החצי נוספים
Elements/Elements II-10	(a+b)²+b²=2·((½a)²+((½a)+b)²)	ספר_החשבון_והמדות#w9pW	If you divide any number into half and add to it another number, [the sum of] the square of the whole number plus the additional [number] and the square of the additional [number] is equal to twice [the sum of] the square of half the number and the square of half the number plus the additional [number] together. : $\scriptstyle{\color{OliveGreen}{\left(a+b\right)^2+b^2=2\sdot\left[\left(\frac{1}{2}\sdot a\right)^2+\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2\right]}}$ עוד כל מספר שחלקת אותו לשני חציים והוספת עליו מספר אחר הנה מרובע המספר עם התוספת יחד ומרובע התוספת בעצמו הם כפל שני המרובעים שהם מרובע חצי המספר ומרובע חצי המספר עם התוספת יחד כאשר {{#annot:term\|178,2083\|Ycx9}}יחוברו{{#annotend:Ycx9}}
Elements/Elements II-10	(a+b)²+b²=2·((½a)²+((½a)+b)²)	ספר_המלכים#CVU6	For any even number divided into half and another number is added to it, [the sum of] twice the product of half the number by itself and twice the product of half the number plus the additional [number] by itself is equal to [the sum of] the product of the [whole] number plus the additional [number] by itself and [the product] of the additional [number] by itself. : $\scriptstyle{\color{OliveGreen}{\left[2\sdot\left(\frac{1}{2}\sdot a\right)^2\right]+\left[2\sdot\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2\right]=\left(a+b\right)^2+b^2}}$ כל מספר זוג יחלק לחציים ונוסף בו מספר אחר הנה ההווה מהכאת חצי המספר בכמהו שני פעמים והכאת חצי המספר עם התוספת בכמהו שני פעמים הכהכאת המספר עם התוספת בכמהו והתוספת בכמהו
Elements/Elements II-10	(a+b)²+b²=2·((½a)²+((½a)+b)²)	ספר_המספר_/_אליהו_מזרחי#FJSe	Any number that you divide into half and add to it another number, [the sum of] the square of the [whole] number plus the additional [number] and the square of the additional [number] is equal to twice [the sum of] the square of half the number and the square of half the number plus the additional [number] when they are summed together. : $\scriptstyle{\color{OliveGreen}{\left(a+b\right)^2+b^2=2\sdot\left[\left(\frac{1}{2}\sdot a\right)^2+\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2\right]}}$ עוד כל מספר שחלקת אותו לשני חצאים והוספת עליו מספר אחר הנה מרובע המספר עם התוספת יחד ומרובע התוספת בעצמו הם כפל שני המרובעים שהם מרובע חצי המספר ומרובע חצי המספר עם התוספת יחד כאשר יחוברו

Examples

[Expand]

\scriptstyle\left(10+2\right)^2+2^2=2\sdot\left[\left(\frac{1}{2}\sdot10\right)^2+\left[\left(\frac{1}{2}\sdot10\right)+2\right]^2\right]

[Expand]

\scriptstyle\left(12+8\right)^2+8^2=2\sdot\left[\left(\frac{1}{2}\sdot12\right)^2+\left[\left(\frac{1}{2}\sdot12\right)+8\right]^2\right]

Elements V

[Expand]Elements V-9

Elements V-15

[Expand]

\scriptstyle\left(n\sdot a\right):\left(n\sdot b\right)=a:b

Elements VII

[Expand]Elements VII-11

Elements VIII

[Expand]Elements VIII-11

Elements IX

[Expand]Elements IX-21

[Expand]Elements IX-22

[Expand]Elements IX-23

[Expand]Elements IX-28

[Expand]Elements IX-29

Category	Comment	Link	Annotated text
basic operations/subtraction	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#U9zP	Another example with zeros: the amount of the debt is two hundred thousand weights of wool and he paid 123 thousand and 345. : $\scriptstyle200000-123345$ משל אחר כולו ספרות הנה מספר החיוב הוא מאתים אלפים משקלי צמר ופרע קכ"ג אלפים שמ"ה
basic operations/subtraction	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#jQ7J	I say that Reuven owed Shimon 573 dinars and he paid him 352. We want to subtract what he paid from the debt. : $\scriptstyle573-352$ ואומר כי ראובן חייב לשמעון תקע"ג די' ופרע לו שנ"ב והנה רצינו לגרע מה שפרע מן החיוב
basic operations/subtraction	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#RwHV	Another example: Reuven owed Shimon 5047 liter and he paid him 958 liter. : $\scriptstyle5047-958$ משל אחר ראובן חיב לשמעון ה' אלפים ומ"ז לטרי ופרע לו תתקנ"ח לטרין
basic operations/subtraction	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#pBrg	[Illegible]. : $\scriptstyle900-854$ ‫30r / י"ד[...] החיוב [...]עון והשיור [...]עון תת"ק כורי חיטה [...] רצינו לדעת הנשאר

Category	Comment	Link	Annotated text
basic operations/multiplication	4×4	ספר_ציפרא#ynhp	Example: 4 times 4. :: $\scriptstyle4\times4$ דמיון ד'פ'ד‫'
basic operations/multiplication	4×4	ספר_ציפרא#yXdJ	I will teach you an example: if you want to multiply 4 times 4. :: $\scriptstyle4\times4$ דומיון אם תרצה לחשוב ד'פ'ד‫’

Category	Comment	Link	Annotated text
basic operations/multiplication	600×4000	ספר_יסודי_התבונה_ומגדל_האמונה#nEsS	As if you wish to multiply six hundred by four thousand. :: $\scriptstyle600\times4000$ וכאלו היית רוצה לחשוב שש מאות בד' אלפים
basic operations/multiplication	600×4000	Anonymous#ZjcQ	As if you want to [multiply] six hundred by four thousand. :: $\scriptstyle600\times4000$ כמו שתרצה שש מאות על ארבעת אלפים

Category	Comment	Link	Annotated text
basic operations/multiplication	29×31	ספר_המספר_/_אברהם_אבן_עזרא#jFgw	Example: we wish to multiply 29 by 31. :: $\scriptstyle29\times31$ דמיון רצינו לכפול כ"ט על ל"א
basic operations/multiplication	29×31	ספר_המספר_/_אליהו_מזרחי#2iNx	Example: we wish to multiply 29 by 31. : $\scriptstyle29\times31$ המשל בזה רצינו להכות כ"ט עם ל"א

Category	Comment	Link	Annotated text
basic operations/multiplication	250×350	ספר_המספר_/_אליהו_מזרחי#8ywW	Also if we wish to multiply 250 by 350. : $\scriptstyle250\times350$ וכן אם רצינו להכות ר"נ עם ש"נ
basic operations/multiplication	250×350	ספר_המספר_/_אברהם_אבן_עזרא#jWUi	Another example: one number is 250 and the other number 350. :: $\scriptstyle250\times350$ דמיון אחר {{#annot:term\|35\|PgHE}}המספר{{#annotend:PgHE}} האחד ר"נ והמספר האחר ש"נ

@@ Line 191: / Line 191: @@
 	[[category: #subtraction[0] ]]
 	[[comment: 468-382]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle654-321</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #subtraction[0] ]]
+	[[comment: 654-321]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle956-867</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #subtraction[0] ]]
+	[[comment: 956-867]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle1000-999</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #subtraction[0] ]]
+	[[comment: 1000-999]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle2000-1999</math><div class="mw-collapsible-content">
@@ Line 286: / Line 301: @@
 	[[category: #subtraction[0] ]]
 	[[comment: (31080+46ⁱⁱ+35ⁱⁱⁱ+47ⁱᵛ+53ᵛⁱ)-(206+50ⁱ+37ⁱⁱⁱ)]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">WP<div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #subtraction[0] ]]
+	[[comment: WP]]
 }}</div></div><br>
@@ Line 345: / Line 365: @@
 == Multiplication ==
 === units by units ===
@@ Line 755: / Line 776: @@
 	[[category: #multiplication[0] ]]
 	[[comment: 246×140]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle321\times654</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #multiplication[0] ]]
+	[[comment: 321×654]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle348\times235</math><div class="mw-collapsible-content">
@@ Line 770: / Line 796: @@
 	[[category: #multiplication[0] ]]
 	[[comment: 462×323]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle464\times464</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #multiplication[0] ]]
+	[[comment: 464×464]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle403\times230</math><div class="mw-collapsible-content">
@@ Line 800: / Line 831: @@
 	[[category: #multiplication[0] ]]
 	[[comment: 253×1335]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle6845\times327</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #multiplication[0] ]]
+	[[comment: 6845×327]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle1234\times4321</math><div class="mw-collapsible-content">
@@ Line 847: / Line 883: @@
 }}</div></div><br>
-== Squaring ==
+=== Word Problems ===
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle3^2</math><div class="mw-collapsible-content">
+<div class="mw-collapsible mw-collapsed">Word Problems<div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #squared number[0] ]]
+	[[category: #multiplication[0] ]]
-	[[comment: 3²]]
+	[[comment: WP]]
 }}</div></div><br>
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle7^2</math><div class="mw-collapsible-content">
+=== Permutations ===
+<div class="mw-collapsible mw-collapsed">2!<div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #squared number[0] ]]
+	[[category: #permutation[0] ]]
-	[[comment: 7²]]
+	[[comment: 2]]
 }}</div></div><br>
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle8^2</math><div class="mw-collapsible-content">
+<div class="mw-collapsible mw-collapsed">3!<div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #squared number[0] ]]
+	[[category: #permutation[0] ]]
-	[[comment: 8²]]
+	[[comment: 3]]
 }}</div></div><br>
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle9^2</math><div class="mw-collapsible-content">
+<div class="mw-collapsible mw-collapsed">4!<div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #squared number[0] ]]
+	[[category: #permutation[0] ]]
-	[[comment: 9²]]
+	[[comment: 4]]
 }}</div></div><br>
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle10^2</math><div class="mw-collapsible-content">
+<div class="mw-collapsible mw-collapsed">5!<div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #squared number[0] ]]
+	[[category: #permutation[0] ]]
-	[[comment: 10²]]
+	[[comment: 5]]
 }}</div></div><br>
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle11^2</math><div class="mw-collapsible-content">
+<div class="mw-collapsible mw-collapsed">6!<div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #permutation[0] ]]
+	[[comment: 6]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">7!<div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #permutation[0] ]]
+	[[comment: 7]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">8!<div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #permutation[0] ]]
+	[[comment: 8]]
+}}</div></div><br>
+== Squaring ==
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle3^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #squared number[0] ]]
+	[[comment: 3²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle7^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #squared number[0] ]]
+	[[comment: 7²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle8^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #squared number[0] ]]
+	[[comment: 8²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle9^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #squared number[0] ]]
+	[[comment: 9²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle10^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #squared number[0] ]]
+	[[comment: 10²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle11^2</math><div class="mw-collapsible-content">
 {{#annotask:
 	[[category: #squared number[0] ]]
@@ Line 953: / Line 1,035: @@
 	[[category: #squared number[0] ]]
 	[[comment: 60²]]
+}}</div></div><br>
+== Cubing==
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle3^3</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #a³[0] ]]
+	[[comment: 3³]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle4^3</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #a³[0] ]]
+	[[comment: 4³]]
 }}</div></div><br>
@@ Line 960: / Line 1,055: @@
 === Smaller by greater ===
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle5\div17</math><div class="mw-collapsible-content">
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle4\div12</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #division[0] ]]
+	[[comment: 4÷12]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle5\div17</math><div class="mw-collapsible-content">
 {{#annotask:
 	[[category: #division[0] ]]
 	[[comment: 5÷17]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle4\div50</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #division[0] ]]
+	[[comment: 4÷50]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle6\div50</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #division[0] ]]
+	[[comment: 6÷50]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle7\div40</math><div class="mw-collapsible-content">
@@ Line 1,064: / Line 1,174: @@
 	[[category: #division[0] ]]
 	[[comment: 125÷11]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle218\div7</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #division[0] ]]
+	[[comment: 218÷7]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle200\div50</math><div class="mw-collapsible-content">
@@ Line 1,252: / Line 1,367: @@
 	[[category: #division[0] ]]
 	[[comment: 3123740520÷216]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">WP<div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #division[0] ]]
+	[[comment: WP]]
 }}</div></div><br>
@@ Line 1,359: / Line 1,480: @@
 	[[category: #division of fractions]]
 	[[comment: ⅜÷⅖]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{6}{8}\div\frac{2}{4}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #division of fractions]]
+	[[comment: ⁶/₈÷²/₄]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{3}{5}\div\frac{7}{9}</math><div class="mw-collapsible-content">
@@ Line 1,664: / Line 1,790: @@
 	[[category: #multiplication of fractions[0] ]]
 	[[comment: (⁵/₇·⅓)×⅞]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{2}{3}\times\left(\frac{3}{4}\sdot\frac{1}{3}\right)</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #multiplication of fractions[0] ]]
+	[[comment: ⅔×(¾·⅓)]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{2}{3}\times\left(\frac{4}{5}\sdot\frac{1}{9}\right)</math><div class="mw-collapsible-content">
@@ Line 2,107: / Line 2,238: @@
 	[[category: #multiplication of fractions[0] ]]
 	[[comment: ⅕×⅕]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{1}{4}\times\frac{1}{3}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #multiplication of fractions[0] ]]
+	[[comment: ¼×⅓]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{3}{4}\times\frac{3}{4}</math><div class="mw-collapsible-content">
@@ Line 2,261: / Line 2,397: @@
 	[[category: #multiplication of fractions[0] ]]
 	[[comment: 2×¾]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{2}{3}\times4</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #multiplication of fractions[0] ]]
+	[[comment: ⅔×4]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle4\times\frac{2}{3}</math><div class="mw-collapsible-content">
@@ Line 2,319: / Line 2,460: @@
 	[[category: #multiplication of sexagesimal fractions[0] ]]
 	[[comment: (2+24'+43'')×(3+3'+8'')]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(4+6^\prime+8^{\prime\prime}+12^{\prime\prime\prime}\right)\times\left(5+15^\prime+10^{\prime\prime}+13^{\prime\prime\prime}\right)</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #multiplication of sexagesimal fractions[0] ]]
+	[[comment: (4+6'+8''+12''')×(5+15'+10''+13''')]]
 }}</div></div><br>
@@ Line 2,373: / Line 2,519: @@
 	[[category: #addition of fractions]]
 	[[comment: ⅔+¾]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{2}{3}+\frac{4}{9}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #addition of fractions]]
+	[[comment: ⅔+⁴/₉]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{3}{4}+\frac{4}{5}</math><div class="mw-collapsible-content">
@@ Line 2,379: / Line 2,530: @@
 	[[comment: ¾+⅘]]
 }}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{3}{4}+\frac{4}{6}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #addition of fractions]]
+	[[comment: ¾+⁴/₆]]
+}}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{3}{4}+\frac{5}{6}</math><div class="mw-collapsible-content">
 {{#annotask:
@@ Line 2,549: / Line 2,704: @@
 	[[category: #subtraction of fractions]]
 	[[comment: ⅖-⅜]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{5}{8}-\frac{3}{8}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #subtraction of fractions]]
+	[[comment: ⅝-⅜]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{5}{7}-\frac{4}{9}</math><div class="mw-collapsible-content">
@@ Line 2,554: / Line 2,714: @@
 	[[category: #subtraction of fractions]]
 	[[comment: ⁵/₇-⁴/₉]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{6}{8}-\frac{2}{4}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #subtraction of fractions]]
+	[[comment: ⁶/₈-²/₄]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{12}{16}-\frac{4}{8}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #subtraction of fractions]]
+	[[comment: ¹²/₁₆-⁴/₈]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{14}{16}-\frac{8}{12}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #subtraction of fractions]]
+	[[comment: ¹⁴/₁₆-⁸/₁₂]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(\frac{1}{3}+\frac{1}{4}\right)-\left(\frac{1}{5}+\frac{1}{6}\right)</math><div class="mw-collapsible-content">
@@ Line 2,559: / Line 2,734: @@
 	[[category: #subtraction of fractions]]
 	[[comment: (⅓+¼)-(⅕+⅙)]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle1-\frac{5}{8}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #subtraction of fractions]]
+	[[comment: 1-⅝]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(2+\frac{1}{2}\right)-\frac{3}{4}</math><div class="mw-collapsible-content">
@@ Line 2,587: / Line 2,767: @@
 	[[category: #rule of three]]
 	[[comment: 2÷4=4÷8]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle4:6=6:9</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #rule of three]]
+	[[comment: 4÷6=6÷9]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle4:6=8:12</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #rule of three]]
+	[[comment: 4÷6=8÷12]]
 }}</div></div><br>
@@ Line 2,641: / Line 2,833: @@
 	[[category: #rule of three]]
 	[[comment: ⅓÷¼=⅕÷X]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{2}{3}:\frac{3}{5}=\frac{1}{4}:X</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #rule of three]]
+	[[comment: ⅔÷⅗=¼÷X]]
 }}</div></div><br>
@@ Line 2,653: / Line 2,850: @@
 	[[category: #rule of three]]
 	[[comment: ⅔÷⁴/₉=⁴/₁₃÷X]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(\frac{2}{5}\sdot\frac{3}{4}\right):\left(\frac{3}{7}\sdot\frac{5}{6}\right)=\left(\frac{4}{5}\sdot\frac{5}{8}\right):X</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #rule of three]]
+	[[comment: (⅖·¾)÷(³/₇·⅚)=(⅘·⅝)÷X]]
 }}</div></div><br>
@@ Line 2,717: / Line 2,920: @@
 }}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sqrt{144}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #root[0] ]]
+	[[comment: √144]]
+}}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sqrt{225}</math><div class="mw-collapsible-content">
 {{#annotask:
@@ Line 2,756: / Line 2,964: @@
 	[[category: #root[0] ]]
 	[[comment: √824464]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sqrt{973182}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #root[0] ]]
+	[[comment: √973182]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sqrt{5499025}</math><div class="mw-collapsible-content">
@@ Line 3,733: / Line 3,946: @@
 	[[category: #S+R equal N]]
 	[[comment: ½x²+5x=28]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{1}{3}x^2+3x=30</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #S+R equal N]]
+	[[comment: ⅓x²+3x=30]]
 }}</div></div>
 <br>
@@ Line 3,756: / Line 3,975: @@
 }}</div></div>
 <br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle3x^2+21=10x</math><div class="mw-collapsible-content">
-=== bx+c=ax² ===
+{{#annotask:
+	[[category: #S+N equal R]]
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle bx+c=ax^2</math><div class="mw-collapsible-content">
+	[[comment: 3x²+21=10x]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{1}{3}x^2+21=10x</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #S+N equal R]]
+	[[comment: ⅓x²+21=10x]]
+}}</div></div>
+<br>
+=== bx+c=ax² ===
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle bx+c=ax^2</math><div class="mw-collapsible-content">
 {{#annotask:
 	[[category: #R+N equal S]]
@@ Line 4,005: / Line 4,236: @@
 	[[comment: x²+3x=n²]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle x^2-6x=n^2</math><div class="mw-collapsible-content">
 {{#annotask:
@@ Line 4,011: / Line 4,241: @@
 	[[comment: x²-6x=n²]]
 }}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle x-x^2=n^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x-x²=n²]]
+}}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle x^2+10x+20=n^2</math><div class="mw-collapsible-content">
 {{#annotask:
@@ Line 4,017: / Line 4,251: @@
 	[[comment: x²+10x+20=n²]]
 }}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle x^2-8x-30=n^2</math><div class="mw-collapsible-content">
 {{#annotask:
@@ Line 4,023: / Line 4,256: @@
 	[[comment: x²-8x-30=n²]]
 }}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle 8x+109-x^2=n^2</math><div class="mw-collapsible-content">
-== Euclid ==
-=== Elements - Introduction ===
-<div class="mw-collapsible mw-collapsed">common notions<div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #Elements-Introduction]]
+	[[category: #indeterminate equation]]
-	[[comment: a=b,a-c=b-c]]
+	[[comment: 8x+109-x²=n²]]
 }}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle 2x+49-x^2=n^2</math><div class="mw-collapsible-content">
-=== Elements II ===
-==== Elements II-1 ====
-<div class="mw-collapsible mw-collapsed">proposition<div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #Elements II-1]]
+	[[category: #indeterminate equation]]
-	[[comment: definition]]
+	[[comment: 2x+49-x²=n²]]
 }}</div></div><br>
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\sdot\left(\sum_{i=1}^n b_i\right)=\sum_{i=1}^n \left(a\sdot b_i\right)</math><div class="mw-collapsible-content">
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle 10x-8-x^2=n^2</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #Elements II-1]]
+	[[category: #indeterminate equation]]
-	[[comment: a·∑bᵢ=∑(a·bᵢ)]]
+	[[comment: 10x-8-x²=n²]]
 }}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle 260-6x-x^2=n^2</math><div class="mw-collapsible-content">
-Examples
+{{#annotask:
+	[[category: #indeterminate equation]]
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(2\sdot4\right)+\left(3\sdot4\right)+\left(5\sdot4\right)=10\sdot4</math><div class="mw-collapsible-content">
+	[[comment: 260-6x-x²=n²]]
-{{#annotask:
+}}</div></div><br>
-	[[category: #Elements II-1]]
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2+2x=m^2\end{cases}</math><div class="mw-collapsible-content">
-	[[comment: 4,2+3+5]]
+{{#annotask:
-}}</div></div><br>
+	[[category: #indeterminate equation]]
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(2\sdot10\right)+\left(3\sdot10\right)+\left(5\sdot10\right)+\left(2\sdot10\right)=12\sdot10</math><div class="mw-collapsible-content">
+	[[comment: x²+x=n²,x²+2x=m²]]
-{{#annotask:
+}}</div></div><br>
-	[[category: #Elements II-1]]
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2+3x=m^2\end{cases}</math><div class="mw-collapsible-content">
-	[[comment: 10,2+3+5+2]]
+{{#annotask:
-}}</div></div><br>
+	[[category: #indeterminate equation]]
+	[[comment: x²+x=n²,x²+3x=m²]]
-==== Elements II-2 ====
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle x^2-2x=n^2\\\scriptstyle x^2-3x=m^2\end{cases}</math><div class="mw-collapsible-content">
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{k=1}^n \left[\left(\sum_{i=1}^n a_i\right)\sdot a_k\right]=\left(\sum_{i=1}^n a_i\right)^2</math><div class="mw-collapsible-content">
+{{#annotask:
-{{#annotask:
+	[[category: #indeterminate equation]]
-	[[category: #Elements II-2]]
+	[[comment: x²-2x=n²,x²-3x=m²]]
-	[[comment: ∑ₖ((∑ᵢaᵢ)·aₖ)=(∑ᵢaᵢ)²]]
+}}</div></div><br>
-}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2-x=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
-Examples
+	[[category: #indeterminate equation]]
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(7\sdot10\right)+\left(3\sdot10\right)=10^2</math><div class="mw-collapsible-content">
+	[[comment: x²+x=n²,x²-x=m²]]
-{{#annotask:
+}}</div></div><br>
-	[[category: #Elements II-2]]
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle x^2+2x=n^2\\\scriptstyle x^2-3x=m^2\end{cases}</math><div class="mw-collapsible-content">
-	[[comment: 10,7+3]]
+{{#annotask:
-}}</div></div><br>
+	[[category: #indeterminate equation]]
+	[[comment: x²+2x=n²,x²-3x=m²]]
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(3\sdot12\right)+\left(4\sdot12\right)+\left(5\sdot12\right)=12^2</math><div class="mw-collapsible-content">
+}}</div></div><br>
-{{#annotask:
+<div class="mw-collapsible mw-collapsed">
-	[[category: #Elements II-2]]
+<math>\scriptstyle\begin{cases}\scriptstyle 3-x^2=n^2\\\scriptstyle 2+x^2=m^2\end{cases}</math><div class="mw-collapsible-content">
-	[[comment: 12,3+4+5]]
+{{#annotask:
-}}</div></div><br>
+	[[category: #indeterminate equation]]
+	[[comment: 3-x²=n²,2+x²=m²]]
-==== Elements II-3 ====
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(a+b\right)\sdot b=\left(a\sdot b\right)+b^2</math><div class="mw-collapsible-content">
+<math>\scriptstyle\begin{cases}\scriptstyle 10-x^2=n^2\\\scriptstyle 20+x^2=m^2\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #Elements II-3]]
+	[[category: #indeterminate equation]]
-	[[comment: (a+b)·b=(a·b)+b²]]
+	[[comment: 10-x²=n²,20-x²=m²]]
 }}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
-Examples
+<math>\scriptstyle\begin{cases}\scriptstyle 20+x^2=n^2\\\scriptstyle 30+x^2=m^2\end{cases}</math><div class="mw-collapsible-content">
-<div class="mw-collapsible mw-collapsed"><math>\scriptstyle10\sdot3=\left(3\sdot7\right)+3^2</math><div class="mw-collapsible-content">
+{{#annotask:
-{{#annotask:
+	[[category: #indeterminate equation]]
-	[[category: #Elements II-3]]
+	[[comment: 20+x²=n²,30+x²=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle 10+x^2=n^2\\\scriptstyle 10-x^2=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: 10+x²=n²,10-x²=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle x+x^2=n^2\\\scriptstyle x-x^2=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x+x²=n²,x-x²=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle 8x+x^2=n^2\\\scriptstyle 2x-x^2=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: 8x+x²=n²,2x-x²=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle x^2+2x=n^2\\\scriptstyle x^2+2x+3\sqrt{x^2+2x}=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²+2x=n²,x²+2x+3√(x²+2x)=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle x^2+3x=n^2\\\scriptstyle x^2+3x+6\sqrt{x^2+3x}=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²+3x=n²,x²+3x+6√(x²+3x)=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle x^2+2x=n^2\\\scriptstyle x^2+2x+\sqrt{x^2+2x}=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²+2x=n²,x²+2x+√(x²+2x)=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle x^2+4x=n^2\\\scriptstyle x^2+4x+2\sqrt{x^2+4x}=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²+4x=n²,x²+4x+2√(x²+4x)=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle x^2-4x=n^2\\\scriptstyle x^2-4x+2\sqrt{x^2-4x}=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²-4x=n²,x²-4x-2√(x²-4x)=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle x^2-5=n^2\\\scriptstyle x^2-5+\sqrt{x^2-5}=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²-5=n²,x²-5+√(x²-5)=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle x^2-2x=n^2\\\scriptstyle x^2-2x+\sqrt{x^2-2x}=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²-2x=n²,x²-2x+√(x²-2x)=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2+1=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²+x=n²,x²+1=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle x^2+x+1=n^2\\\scriptstyle x^2+2x+2=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²+x+1=n²,x²+2x+2=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle x^2+4x=n^2\\\scriptstyle x^2-\left(2x+1\right)=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²+4x=n²,x²-(2x+1)=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle x^2+3x+1=n^2\\\scriptstyle x^2-\left(3x-2\right)=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²+3x+1=n²,x²-(3x-2)=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle x^2-\left(x-1\right)=n^2\\\scriptstyle x^2-\left(1-x\right)=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²-(x-1)=n²,x²-(1-x)=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed">
+<math>\scriptstyle\begin{cases}\scriptstyle x^2+\left(2-x\right)=n^2\\\scriptstyle x^2-\left(3-x\right)=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²+(2-x)=n²,x²-(3-x)=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle x^2+y^2=n^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²+y²=n²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle x^2-y=n^2\\\scriptstyle x^2-1\frac{1}{2}y=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²-y=n²,x²-1½y=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle 49+x=n^2\\\scriptstyle 49+2x=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: 49+x=n²,49+2x=m²]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle x^2+1=10x-8</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #indeterminate equation]]
+	[[comment: x²+1=10x-8]]
+}}</div></div><br>
+== Euclid ==
+=== Elements - Introduction ===
+<div class="mw-collapsible mw-collapsed">common notions<div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #Elements-Introduction]]
+	[[comment: a=b,a-c=b-c]]
+}}</div></div><br>
+=== Elements II ===
+==== Elements II-1 ====
+<div class="mw-collapsible mw-collapsed">proposition<div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #Elements II-1]]
+	[[comment: definition]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\sdot\left(\sum_{i=1}^n b_i\right)=\sum_{i=1}^n \left(a\sdot b_i\right)</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #Elements II-1]]
+	[[comment: a·∑bᵢ=∑(a·bᵢ)]]
+}}</div></div><br>
+Examples
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(2\sdot4\right)+\left(3\sdot4\right)+\left(5\sdot4\right)=10\sdot4</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #Elements II-1]]
+	[[comment: 4,2+3+5]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(2\sdot10\right)+\left(3\sdot10\right)+\left(5\sdot10\right)+\left(2\sdot10\right)=12\sdot10</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #Elements II-1]]
+	[[comment: 10,2+3+5+2]]
+}}</div></div><br>
+==== Elements II-2 ====
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{k=1}^n \left[\left(\sum_{i=1}^n a_i\right)\sdot a_k\right]=\left(\sum_{i=1}^n a_i\right)^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #Elements II-2]]
+	[[comment: ∑ₖ((∑ᵢaᵢ)·aₖ)=(∑ᵢaᵢ)²]]
+}}</div></div><br>
+Examples
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(7\sdot10\right)+\left(3\sdot10\right)=10^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #Elements II-2]]
+	[[comment: 10,7+3]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(3\sdot12\right)+\left(4\sdot12\right)+\left(5\sdot12\right)=12^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #Elements II-2]]
+	[[comment: 12,3+4+5]]
+}}</div></div><br>
+==== Elements II-3 ====
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(a+b\right)\sdot b=\left(a\sdot b\right)+b^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #Elements II-3]]
+	[[comment: (a+b)·b=(a·b)+b²]]
+}}</div></div><br>
+Examples
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle10\sdot3=\left(3\sdot7\right)+3^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #Elements II-3]]
 	[[comment: 10,3+7]]
 }}</div></div><br>
@@ Line 4,266: / Line 4,676: @@
 	[[category: #Elements V-15]]
 	[[comment: (n·a)÷(n·b)=a÷b]]
+}}</div></div><br>
+=== Elements VII ===
+<div class="mw-collapsible mw-collapsed">Elements VII-11<div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #Elements VII-11]]
+	[[comment: a÷b=c÷d→(c-a)÷(d-b)]]
 }}</div></div><br>

Category	Comment	Link	Annotated text
basic operations/multiplication	348×235	ספר_החשבון_והמדות#a8Vc	Example: we wish to multiply three hundred forty-eight by two hundred thirty-five. : $\scriptstyle348\times235$ דמיון רצינו לכפול שלש מאות וארבעים ושמנה על מאתים ושלשים וחמשה
basic operations/multiplication	348×235	ספר_החשבון_והמדות#VJUW	Example: we wish to multiply three hundred and forty-eight by two hundred and thirty-five. : $\scriptstyle348\times235$ דמיון רצינו לכפול שלש מאות וארבעים ושמנה על מאתים ושלשים וחמשה

Category	Comment	Link	Annotated text
basic operations/multiplication	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#poq3	It is the same as saying 245 liṭra of pepper at 75; the total price is the mentioned amount. והוא הדין כאומר רמ"ה לטרין פלפל לסך ע"ה שיעלה הסך הנז' מערוך
basic operations/multiplication	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#Qt03	Or, as saying: 26 weights of a thing for 26 dinar, or peraḥim, or liter; how much will it cost? או כמו האומר כ"ו משקלי דבר מה לסך כ"ו די' או פרחים או לטרין כמה יעלה
basic operations/multiplication	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#8edj	Another example: If one asks for 245 kor of wheat at 19 dinar. : $\scriptstyle245\times19$ משל אחר אם שאל רמ"ה כורי חיטה לסך י"ט די‫'
basic operations/multiplication	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#u3Eg	As saying: 26 cubits of cloth at 26 dinar for one cubit; how much will it cost? והוא כאומר כ"ו אמות בגד לסך כ"ו די' האמה כמה יעלה
basic operations/multiplication	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#tv8T	It is also as saying two hundred kikkar of copper at two hundred dinar for one kikkar. והוא גם כן כאומר מאתים ככרי נחושת לסך מאתים די' כל כיכר

Category	Comment	Link	Annotated text
combinatorics/permutation	2	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#GuHR	The example of 2 [terms] that generate 2 [permutations]: והמשל ב' בונות ב‫'
combinatorics/permutation	2	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#HudC	I say that 2 generate two permutations [lit. houses], which is [the 2] in the second row beneath the 2. ואומר כי ב' בונות שתי {{#annot:term\|2241,1951\|Dgyh}}בתים{{#annotend:Dgyh}} והוא תחת ב' בטור השנית

Category	Comment	Link	Annotated text
combinatorics/permutation	3	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#9VTh	[...] 3 terms [...] [...] מג' אותיות [...]
combinatorics/permutation	3	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#04e4	To know the permutations of 3: ועתה לדעת {{#annot:term\|2255,2246\|pHd4}}בניין{{#annotend:pHd4}} ג‫'

Category	Comment	Link	Annotated text
combinatorics/permutation	4	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#1Yfq	To know the permutations of 4: ולדעת בניין ד‫'
combinatorics/permutation	4	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#uGjG	The example of 4 terms: והמשל בד' אבנים

Category	Comment	Link	Annotated text
exponentiation/squared number	7²	ספר_המספר_/_אברהם_אבן_עזרא#FUPY	Example: we wish to know the square of 7. :: $\scriptstyle7^2$ דמיון בקשנו לדעת מרובע ז‫'
exponentiation/squared number	7²	קצת_מענייני_חכמת_המספר#kNhC	Such as: you wish to know the square of 7. כגון רצית לידע מרובע ז‫'

Category	Comment	Link	Annotated text
exponentiation/squared number	9²	ספר_החשבון_והמדות#b4f5	Example: we wish to know the square of 9. : $\scriptstyle9^2$ דמיון רצינו לדעת מרובע ט‫'
exponentiation/squared number	9²	קצת_מענייני_חכמת_המספר#2n2J	Example: if you wish to know how much is the square of 9. המשל אם תרצה לידע כמה הוא מרובע הט‫'

Category	Comment	Link	Annotated text
exponentiation/squared number	11²	ספר_החשבון_והמדות#YlLr	Example: we wish to know the square of 11. : $\scriptstyle11^2$ דמיון רצינו לדעת מרובע י"א
exponentiation/squared number	11²	ספר_החשבון_והמדות#CGBe	Example: we wish to multiply 11 by 11. : $\scriptstyle11^2$ דמיון רצינו לכפול י"א על י"א

Category	Comment	Link	Annotated text
exponentiation/squared number	12²	קצור_המספר#aLsp	For example: we wish to multiply twelve by twelve: they are 144: ומשל לזה רצינו לכפול שנים עשר על שנים עשר והוא קמ"ד
exponentiation/squared number	12²	ספר_החשבון_והמדות#MxZM	We wish to multiply 12 by 12. : $\scriptstyle12^2$ רצינו לכפול י"ב על י"ב

Category	Comment	Link	Annotated text
exponentiation/squared number	23²	ספר_המספר_/_אליהו_מזרחי#4gKU	Example of a number [that is less than the number that has a third]: 23. :: $\scriptstyle23^2$ ומשל המספר היתר הוא כ"ג
exponentiation/squared number	23²	ספר_המספר_/_אברהם_אבן_עזרא#eFIa	Example: we wish to know how much is the square of 23. :: $\scriptstyle23^2$ דמיון בקשנו לדעת כמה מרובע כ"ג
exponentiation/squared number	23²	ספר_המספר_/_אליהו_מזרחי#EzUv	Example [when the required number] is smaller [than the number that has a fifth] by 2: 23. :: $\scriptstyle23^2$ ומשל הפחות מספר ב' הכ"ג

Category	Comment	Link	Annotated text
exponentiation/squared number	24²	ספר_המספר_/_אליהו_מזרחי#Divz	Example: we wish to multiply 24 by itself. :: $\scriptstyle24^2$ המשל בזה רצינו להכות כ"ד עם עצמו
exponentiation/squared number	24²	ספר_המספר_/_אליהו_מזרחי#kgbD	Example [when the required number] is smaller [than the number that has a fifth] by one: 24. :: $\scriptstyle24^2$ ומשל הפחות אחד הוא מספר כ"ד
exponentiation/squared number	24²	ספר_המספר_/_אברהם_אבן_עזרא#Ny3j	Another example: we wish to know how much is the square of 24. :: $\scriptstyle24^2$ דמיון אחר בקשנו לדעת כמה מרובע כ"ד

Category	Comment	Link	Annotated text
exponentiation/squared number	25²	ספר_המספר_/_אליהו_מזרחי#yJQV	Example: 25. :: $\scriptstyle25^2$ המשל בזה מספר כ"ה
exponentiation/squared number	25²	ספר_המספר_/_אליהו_מזרחי#kQVs	Example of a number [that exceeds the number that has a third]: 25. :: $\scriptstyle25^2$ המשל במספר הפחות הוא כ"ה

Category	Comment	Link	Annotated text
exponentiation/squared number	33²	ספר_מעשה_חושב#hibu	Example: you wish to know the square of thirty-three. : $\scriptstyle33^2$ דמיון זה אם רצית לדעת מרובע שלשים ושלשה
exponentiation/squared number	33²	ספר_מעשה_חושב#dhRJ	Example: you wish to know the square of 33. : $\scriptstyle33^2$ דמיון רצית לדעת מרובע ל"ג

Category	Comment	Link	Annotated text
basic operations/division	4215÷14	ספר_החשבון_והמדות#dIVO	Example: we wish to divide 4 thousand, two hundred and 15 by 14. : $\scriptstyle4215\div14$ דמיון רצינו לחלק ד' אלפים ומאתים וט"ו על י"ד
basic operations/division	4215÷14	ספר_החשבון_והמדות#kF9U	Example: we wish to divide four thousand, two hundred and fifteen by fourteen. : $\scriptstyle4215\div14$ דמיון רצינו לחלק ארבעת אלפים ומאתים ‫33vוחמשה עשר על ארבעה עשר

Category	Comment	Link	Annotated text
basic operations/division	104034÷114	ספר_החשבון_והמדות#XR8w	Another example: we wish to divide one hundred thousand, four thousand, a zero and thirty-four by 114. : $\scriptstyle104034\div114$ דמיון אחר רצינו לחלק מאת אלף וארבעת אלפים וספרא ושלשים וארבעה על קי"ד
basic operations/division	104034÷114	ספר_החשבון_והמדות#WKuz	Another example: we wish to divide 104034 by 114. : $\scriptstyle104034\div114$ דמיון אחר רצינו לחלק מאת אלף וסיפרא וארבעת אלפים וסיפרא ושלשים וארבעה על קי"ד

Category	Comment	Link	Annotated text
basic operations/division	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#g5Nb	The third example: we wish to divide 497 liters of money by 34. : $\scriptstyle497\div34$ משל שלשי רצינו לחלק תצ"ז לטרי ממון על ל"ד
basic operations/division	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#hspE	The fifth example: we wish to divide 5 thousand 852 peraḥim by 56. : $\scriptstyle5852\div56$ משל חמשי רצינו לחלק ה' אלפים תתנ"ב פרחים לנ"ו
basic operations/division	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#9mgM	The seventh example: we wish to divide two hundred and 85 thousands and 976 kikkar of wool by 372. : $\scriptstyle285976\div372$ משל שביעי רצינו לחלק מאתים ופ"ה אלפים ותתקע"ו כיכרי צמר על שע"ב
basic operations/division	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#S8zi	The second example: we wish to divide 246 pešiṭim by 12. : $\scriptstyle246\div12$ משל שני רצינו לחלק רמ"ו פשיטין לי"ב
basic operations/division	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#rzDk	The first example: we wish to divide [...] : $\scriptstyle879\div7$ משל ראשון [...] רצינו לחלק [...] הצורה
basic operations/division	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#3h3o	The sixth example: we wish to divide 156 liters of pepper by 14. : $\scriptstyle156\div14$ משל ששי רצינו לחלק קנ"ו לטרי פלפל לי"ד
basic operations/division	WP	Arithmetic_Textbook_by_R._Aharon_ben_Isaac#3wFN	The fourth example: we wish to divide 2 thousand 987 liters by 39. : $\scriptstyle2987\div39$ המשל רביעי רצינו לחלק ב' אלפים תתקפ"ז [ליט'] על ל"ט

Category	Comment	Link	Annotated text
fraction/fraction of fraction	⅜·⅖	ספר_חשבון#8Loc	If one says: how much are 3-eighths of 2-fifths? : $\scriptstyle\frac{3}{8}\sdot\frac{2}{5}$ ואם יאמ' כמה הם ג' שמיניות מב' חמישיות
fraction/fraction of fraction	⅜·⅖	ספר_יסודי_התבונה_ומגדל_האמונה#Vq4n	As the one who says: how much are 3 eighths of 2 fifths : $\scriptstyle\frac{3}{8}\sdot\frac{2}{5}$ כגון האומר כמה הם ג' שמיניות מב' חמישיות

Category	Comment	Link	Annotated text
simple fraction/division of fractions	¾÷⅖	ספר_המספר_/_אליהו_מזרחי#n2Fz	Example: if you wish to divide 3-quarters by 2-fifths. המשל אם רצית לחלק ג' רביעיות על ב' חמשיות
simple fraction/division of fractions	¾÷⅖	קצת_מענייני_חכמת_המספר#GaRS	When you wish to divide 3-quarters by two-fifths. :: $\scriptstyle{\color{blue}{\frac{3}{4}\div\frac{2}{5}}}$ רצית לחלק ג' רביעיות על שני חמישיות
simple fraction/division of fractions	¾÷⅖	ספר_המספר_/_אליהו_מזרחי#B4Qv	Example: if you wish to divide 3-quarters by 2-fifths. המשל בזה אם רצית לחלק הג' רביעיות על הב' חמישיות

Category	Comment	Link	Annotated text
simple fraction/division of fractions	⁶/₇÷⅖	בר_נותן_טעם#KqBF	Example: if you are told in our first example: we wish to divide 6-sevenths by 2-fifths. : $\scriptstyle\frac{6}{7}\div\frac{2}{5}$ המשל אם אמרו לך במשלינו הראשון רצינו לחלק ב' [ו'] שביעיות על ב' חמישיות
simple fraction/division of fractions	⁶/₇÷⅖	בר_נותן_טעם#WIah	Example: we wish to divide 6-sevenths by 2-fifths. : $\scriptstyle\frac{6}{7}\div\frac{2}{5}$ המשל רצינו לחלק ו' שביעיות על ב' חמישיות

Difference between revisions of "Mathematical formula"

Latest revision as of 09:18, 13 July 2024

Contents

Addition

Subtraction

Doubling

Halving

Multiplication

units by units

units by tens

units by hundreds

tens by tens

tens by hundreds

tens by thousands

hundreds by hundreds

hundreds by thousands

units by tens and hundreds

tens by tens and hundreds

tens and hundreds by tens and hundreds

tens by units and tens

units by units and tens

units and tens by units and tens

tens and hundreds by units and tens

units by units, tens and hundreds

units and tens by units, tens and hundreds

units, tens and hundreds by units, tens and hundreds

Word Problems

Permutations

Squaring

Cubing

Division

Smaller by greater

Greater by smaller

Fractions

fraction of fraction

division of fractions

fractions by fractions

fractions by integers

fractions by integers and fractions

fractions of fractions by fractions

fractions of fractions by integers

fractions of fractions by integers and fractions

integers by fractions

integers by integers and fractions

integers and fractions by fractions

integers and fractions by integers

integers and fractions by integers and fractions

Combined Division

multiplication of fractions

multiplication of integer and fraction by integer and fraction

multiplication of fraction by fraction

multiplication of fraction of fraction by fraction of fraction

multiplication of fraction of integer and fraction by fraction of integer and fraction

multiplication of fraction by integer or integer by fraction

multiplication of fraction of fraction of integer by fraction of fraction of integer

multiplication of sexagesimal fractions

addition of fractions

subtraction of fractions

Rule of Three

Roots

Extraction of Roots

Extraction of Cube Roots

Approximation

Addition of Roots

Subtraction of Roots

Multiplication of Roots

Division of Roots

Multiplication of Algebraic Species

Linear Equation

Quadratic Equation

ax²=bx

ax²=c

bx=c

ax²+bx=c

ax²+c=bx

bx+c=ax²

Compound Quadratic Equations

quadratic equation in two variables

quadratic equation in three variables

Cubic Equation

Category	Comment	Link	Annotated text
simple fraction/division of fractions	⅜÷⅖	ספר_יסודי_התבונה_ומגדל_האמונה#rzHD	As the one who says to you: divide 3 eighths by 2 fifths : $\scriptstyle\frac{3}{8}\div\frac{2}{5}$ כגון האומר לך חלק ג' שמיניות על ב' חמישיות
simple fraction/division of fractions	⅜÷⅖	ספר_חשבון#gX0D	If one says: divide 3-eighths by 2-fifths : $\scriptstyle\frac{3}{8}\div\frac{2}{5}$ ואם יאמ' חלק ג' שמיניות על ב' חמשיות

Category	Comment	Link	Annotated text
simple fraction/division of fractions	(⅔+¾)÷(²/₇+⅙)	קצת_מענייני_חכמת_המספר#8XVT	If you wish to divide the sum of 2-thirds and 3-quarters by the sum of two-sevenths and one-sixth, without having to sum up first, then to divide. :: $\scriptstyle{\color{blue}{\left(\frac{2}{3}+\frac{3}{4}\right)\div\left(\frac{2}{7}+\frac{1}{6}\right)}}$ ואולם אם רצית לחלק קבוץ הב' שלישיות וג' רביעיו' על קבוץ שני שביעיות וששית מבלתי שתצטרך לקבץ תחלה ואח"כ לחל‫'
simple fraction/division of fractions	(⅔+¾)÷(²/₇+⅙)	ספר_המספר_/_אליהו_מזרחי#Yb86	If you wish to divide, for instance, [the sum of] 2-thirds and 3-quarters by [the sum of] 2-sevenths and one-sixth, without having to sum up first, then to divide, but you get the right result all at once. :: $\scriptstyle{\color{blue}{\left(\frac{2}{3}+\frac{3}{4}\right)\div\left(\frac{2}{7}+\frac{1}{6}\right)}}$ הנה אם רצית לחלק עד"מ הב' שלישיות והג' רביעיות יחד על הב' שביעיות וששית מבלתי שתצטרך לקבץ תחלה ואח"כ לחלק אבל הכל יצא לך מתוקן בדרך א‫'

Category	Comment	Link	Annotated text
simple fraction/division of fractions	(¾÷⅔)+(²/₇÷⅙)	קצת_מענייני_חכמת_המספר#Vj8S	If you wish to sum up the result of division of 3-quarters by 2-thirds with the result of division of 2-sevenths by [one-sixth], without having to divide first, then sum up. :: $\scriptstyle{\color{blue}{\left(\frac{3}{4}\div\frac{2}{3}\right)+\left(\frac{2}{7}\div\frac{1}{6}\right)}}$ ואולם אם רצית לקבץ היוצא מחלוק הג' רביעיות על הב' שלישיות עם היוצא מחלוק הב' שביעיו' על הו' מבלתי שתצטרך לחלק תחלה ואח"כ לקבץ
simple fraction/division of fractions	(¾÷⅔)+(²/₇÷⅙)	ספר_המספר_/_אליהו_מזרחי#69mH	If you wish to sum up the result of division of 3-quarters by 2-thirds with the result of division of 2-sevenths by one-sixth, without having to divide first, then sum up, but you get the right result all at once. :: $\scriptstyle{\color{blue}{\left(\frac{3}{4}\div\frac{2}{3}\right)+\left(\frac{2}{7}\div\frac{1}{6}\right)}}$ ואולם אם רצית לקבץ היוצא מחלוק הג' רביעיות על הב' שלישיות עם היוצא מחלוק הב' שביעיות על הששית מבלתי שתצטרך לחלק תחלה ואח"כ לקבץ אבל יצא לך הכל מתוקן בדרך אחת

Category	Comment	Link	Annotated text
simple fraction/division of fractions	(⅔×¾)÷(²/₇×⅙)	קצת_מענייני_חכמת_המספר#TQ16	If you wish to divide the product of 2-thirds by 3-quarters by the product of 2-sevenths by one-sixth, without having to multiply first, then divide. :: $\scriptstyle{\color{blue}{\left(\frac{2}{3}\times\frac{3}{4}\right)\div\left(\frac{2}{7}\times\frac{1}{6}\right)}}$ ואולם אם רצית לחלק העולה מהכאת הב' שלישיו' עם הג' רביעי' על העולה מהכאת הב' שביעיו' עם הששי' מבלתי שתצטר' להכות תחלה ואח"כ לחלק
simple fraction/division of fractions	(⅔×¾)÷(²/₇×⅙)	ספר_המספר_/_אליהו_מזרחי#SxBq	If you wish to divide the product of 2-thirds by 3-quarters by the product of 2-sevenths by one-sixth, without having to multiply first, then divide, but you get the right result all at once. :: $\scriptstyle{\color{blue}{\left(\frac{2}{3}\times\frac{3}{4}\right)\div\left(\frac{2}{7}\times\frac{1}{6}\right)}}$ ואולם אם רצית לחלק העולה מהכאת הב' שלישיות עם הג' רביעיות על העולה מהכאת הב' שביעיות עם הששית מבלתי שתצטרך להכות תחלה ואח"כ לחלק אבל יצא לך הכל מתוקן בדרך אחת

Category	Comment	Link	Annotated text
simple fraction/division of fractions	(¾÷⅔)×(²/₇÷⅙)	קצת_מענייני_חכמת_המספר#Rr95	If you wish to multiply the result of division of 3-quarters by 2-thirds by the result of division of 2-sevenths by [one-sixth], without having to divide first, then multiply. :: $\scriptstyle{\color{blue}{\left(\frac{3}{4}\div\frac{2}{3}\right)\times\left(\frac{2}{7}\div\frac{1}{6}\right)}}$ ואולם אם רצית להכות היוצא מחלוק הג' רביעיות על הב' שלישיות עם היוצא מחלוק הב' שביעית על הו' מבלתי שתצטרך לחלק תחלה ואח"כ להכות
simple fraction/division of fractions	(¾÷⅔)×(²/₇÷⅙)	ספר_המספר_/_אליהו_מזרחי#S6kH	If you wish to multiply the result of division of 3-quarters by 2-thirds by the result of division of 2-sevenths by one-sixth, without having to divide first, then multiply, but you get the right result all at once. * $\scriptstyle{\color{blue}{\left(\frac{3}{4}\div\frac{2}{3}\right)\times\left(\frac{2}{7}\div\frac{1}{6}\right)}}$ ואולם אם רצית להכות היוצא מחלוק הג' רביעיות על הב' שלישיות עם היוצא מחלוק הב' שביעיות על הששית מבלתי שתצטרך לחלק תחלה ואח"כ להכות אבל יצא לך הכל מתוקן בדרך אחת

Category	Comment	Link	Annotated text
simple fraction/division of fractions	(¾-⅔)÷(²/₇-⅙)	ספר_המספר_/_אליהו_מזרחי#rxLg	If you wish to divide the remainder from subtraction of 2-thirds from 3-quarters by the remainder from subtraction of one-sixth from 2-sevenths, without having to subtract first then divide, but you get the right result all at once. * $\scriptstyle{\color{blue}{\left(\frac{3}{4}-\frac{2}{3}\right)\div\left(\frac{2}{7}-\frac{1}{6}\right)}}$ ואולם אם רצית לחלק הנשאר מחסור הב' שלישיות מהג' רביעיות על הנשאר מחסור הששית מהב' שביעיות מבלתי שתצטרך לחסר ראשונה ואח"כ לחלק אבל יצא לך הכל מתוקן בדרך אחת
simple fraction/division of fractions	(¾-⅔)÷(²/₇-⅙)	קצת_מענייני_חכמת_המספר#C9my	If you wish to divide the remainder from subtraction of 2-thirds from 3-quarters by the remainder from subtraction of one-sixth from 2-sevenths, without having to subtract first then divide. :: $\scriptstyle{\color{blue}{\left(\frac{3}{4}-\frac{2}{3}\right)\div\left(\frac{2}{7}-\frac{1}{6}\right)}}$ ואולם אם רצית לחלק הנשאר מחצור הב' שלישיות מג' רביעיות על הנשאר מחסור הששית מהב' שביעיות מבלתי שתצטרך לחסר ראשונה ואח"כ לחלק

Category	Comment	Link	Annotated text
simple fraction/division of fractions	(²/₇÷⅙)-(¾÷⅔)	ספר_המספר_/_אליהו_מזרחי#xuFi	If you wish to subtract the result of division of 3-quarters by 2-thirds from the result of division of 2-sevenths by one-sixth, without having to divide first, then subtract, but you get the right result all at once. :: $\scriptstyle{\color{blue}{\left(\frac{2}{7}\div\frac{1}{6}\right)-\left(\frac{3}{4}\div\frac{2}{3}\right)}}$ ואולם אם רצית לחסר היוצא מחלוק הג' רביעיות על הב' שלישיות מהיוצא מחלוק הב' שביעיות על הששית מבלתי שתצטרך לחלק ראשונה ואח"כ לחסרם אבל יצא לך הכל מתוקן בדרך א‫'
simple fraction/division of fractions	(²/₇÷⅙)-(¾÷⅔)	קצת_מענייני_חכמת_המספר#8JsO	If you wish to subtract the result of division of 3-quarters by 2-thirds from the result of division of 2-sevenths by [one-sixth], without having to divide first, then subtract. :: $\scriptstyle{\color{blue}{\left(\frac{2}{7}\div\frac{1}{6}\right)-\left(\frac{3}{4}\div\frac{2}{3}\right)}}$ ואולם אם רצית לחסר היוצא מחלוק הג' רביעיו' על הב' שלישיו' מהיוצא מחילוק הב' שביעיו' על הו' מבלתי שתצטרך לחלק ראשונה ואח"כ לחסרם

Category	Comment	Link	Annotated text
simple fraction/multiplication of fractions	⅖×⅒×⅔	ספר_חשבון#q3mD	If one says: how much are 2 parts of 5 by one-tenth by 2-thirds? : $\scriptstyle\frac{2}{5}\times\frac{1}{10}\times\frac{2}{3}$ ואם יאמ' ב' חלקים מה' בעשירית האחד בב' שלישי האחר כמה הם
simple fraction/multiplication of fractions	⅖×⅒×⅔	Anonymous#j0VS	If you multiply 2 parts of five by a tenth by 2 thirds. : $\scriptstyle\frac{2}{5}\times\frac{1}{10}\times\frac{2}{3}$ \|style="width: 50%; text-align:right;"\|ואם תערוך ב' חלקים מחמשה בעשירית האחד בב' שלישי אחד
simple fraction/multiplication of fractions	⅖×⅒×⅔	ספר_יסודי_התבונה_ומגדל_האמונה#yQcZ	If you are told: multiply two parts of 5 by one tenth by two thirds? : $\scriptstyle\frac{2}{5}\times\frac{1}{10}\times\frac{2}{3}$ ואם יאמר לך חשוב ב' חלקים מה' בעשירית האחד בב' שלישי האחד

Category	Comment	Link	Annotated text
simple fraction/multiplication of fractions	4½×⅔	מלאכת_המספר#V2wD	As the one who wants to multiply 4 and a half by two-thirds. : $\scriptstyle\left(4+\frac{1}{2}\right)\times\frac{2}{3}$ כמי שרוצה לרבע ד' וחצי עם שני שלישיות
simple fraction/multiplication of fractions	4½×⅔	ספר_דיני_ממונות#zVQ8	$\scriptstyle\left(4+\frac{1}{2}\right)\times\frac{2}{3}$ דמיון זה נרצה לכפול ד' וחצי על ב' שלישיות

Category	Comment	Link	Annotated text
simple fraction/multiplication of fractions	¼×¼	ספר_חשבון#q5Pm	The quarter by a quarter is a quarter of a quarter. : $\scriptstyle\frac{1}{4}\times\frac{1}{4}=\frac{1}{4}\sdot\frac{1}{4}$ והרביע ברביע הוא רביע הרביע
simple fraction/multiplication of fractions	¼×¼	קצור_המספר#dX0i	As if you say: we multiply once a quarter of 1 by a quarter. : $\scriptstyle\frac{1}{4}\times\frac{1}{4}$ כאלו תאמ' נכפול רביע א' ברביע פעם