Difference between revisions of "עיר סיחון"

== Chapter Nine – On Knowing the Fractions of Integer, whether in Multiplication, in Division, in Addition, or in Subtraction ==

== Chapter Nine – On Knowing the Fractions of Integer, whether in Multiplication, in Division, in Addition, or in Subtraction ==

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|This rule of fractions is opposite to the rule of integers, since [the integers] increase when they are multiplied by themselves, or by others, more than they are increased by their summing.

|This rule of fractions is opposite to the rule of integers, since [the integers] increase when they are multiplied by themselves, or by others, more than they are increased by their summing.

:<math>\scriptstyle{\color{OliveGreen}{n+m<n\times m}}</math>

:<math>\scriptstyle{\color{OliveGreen}{n+m<n\times m}}</math>

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|On the other hand, this [rule of fractions] is somewhat similar to the rule of the unit, that if we multiply it by itself the result is only one, while its sum with itself is two.

|On the other hand, this [rule of fractions] is somewhat similar to the rule of the unit, that if we multiply it by itself the result is only one, while its sum with itself is two.

|Therefore, the number two is mean between all the other [integers] and [one and the fractions], since the sum of two with itself and its product by itself are both equal to four.

|Therefore, the number two is mean between all the other [integers] and [one and the fractions], since the sum of two with itself and its product by itself are both equal to four.

:<math>\scriptstyle{\color{blue}{2+2=2\times2=4}}</math>

:<math>\scriptstyle{\color{blue}{2+2=2\times2=4}}</math>

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|In order that the practice in this chapter will be clear and known for all by elaborate explanation, I write various calculations of operations with fractions, from which the procedure in other cases can be known and learned.  

|In order that the practice in this chapter will be clear and known for all by elaborate explanation, I write various calculations of operations with fractions, from which the procedure in other cases can be known and learned.  

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|The method of finding the common denominator:

|The method of finding the common denominator:

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::The denominator of the quarter is four and the denominator of the fifth is five; the product of the one by the other is twenty.

::The denominator of the quarter is four and the denominator of the fifth is five; the product of the one by the other is twenty.

::<math>\scriptstyle{\color{blue}{4\sdot5=20}}</math>

::<math>\scriptstyle{\color{blue}{4\sdot5=20}}</math>

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::We multiply also the denominator by itself; the result is sixty-five thousand and twenty-five.

::We multiply also the denominator by itself; the result is sixty-five thousand and twenty-five.

::<math>\scriptstyle{\color{blue}{255\sdot255=65025}}</math>

::<math>\scriptstyle{\color{blue}{255\sdot255=65025}}</math>

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|This sought-after is more complicated to be extracted than the others mentioned, because its fractions are numerous. I shall present you the method to find it easily:

|This sought-after is more complicated to be extracted than the others mentioned, because its fractions are numerous. I shall present you the method to find it easily:

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::When we take three, for the third that is derived from it, and multiply it by four, since the quarter is derived from it, it is twelve. We multiply it by five, for the fifth; it is sixty and this is the denominator, in which you find a third, a quarte and a fifth.

::When we take three, for the third that is derived from it, and multiply it by four, since the quarter is derived from it, it is twelve. We multiply it by five, for the fifth; it is sixty and this is the denominator, in which you find a third, a quarte and a fifth.

::<math>\scriptstyle{\color{blue}{3\sdot4\sdot5=12\sdot5=60}}</math>

::<math>\scriptstyle{\color{blue}{3\sdot4\sdot5=12\sdot5=60}}</math>

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::Its fifth is twelve; its quarter is three; and its two-thirds two. We see and know clearly that two-thirds of a quarter of a fifth is two parts of sixty.

::Its fifth is twelve; its quarter is three; and its two-thirds two. We see and know clearly that two-thirds of a quarter of a fifth is two parts of sixty.

::<math>\scriptstyle{\color{blue}{\frac{2}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}=\frac{\frac{2}{3}\sdot\left[\frac{1}{4}\sdot\left(\frac{1}{5}\sdot60\right)\right]}{60}=\frac{\frac{2}{3}\sdot\left(\frac{1}{4}\sdot12\right)}{60}=\frac{\frac{2}{3}\sdot3}{60}=\frac{2}{60}}}</math>

::<math>\scriptstyle{\color{blue}{\frac{2}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}=\frac{\frac{2}{3}\sdot\left[\frac{1}{4}\sdot\left(\frac{1}{5}\sdot60\right)\right]}{60}=\frac{\frac{2}{3}\sdot\left(\frac{1}{4}\sdot12\right)}{60}=\frac{\frac{2}{3}\sdot3}{60}=\frac{2}{60}}}</math>

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::We take also a denominator that has a seventh and an eighth; it is fifty-six.

::We take also a denominator that has a seventh and an eighth; it is fifty-six.

::<math>\scriptstyle{\color{blue}{7\sdot8=56}}</math>

::<math>\scriptstyle{\color{blue}{7\sdot8=56}}</math>

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::Its eighth is seven; and its six-sevenths is six. This six is parts of fifty-six.

::Its eighth is seven; and its six-sevenths is six. This six is parts of fifty-six.

::<math>\scriptstyle{\color{blue}{\frac{6}{7}\sdot\frac{1}{8}=\frac{\frac{6}{7}\sdot\left(\frac{1}{8}\sdot56\right)}{56}=\frac{\frac{6}{7}\sdot7}{56}=\frac{6}{56}}}</math>

::<math>\scriptstyle{\color{blue}{\frac{6}{7}\sdot\frac{1}{8}=\frac{\frac{6}{7}\sdot\left(\frac{1}{8}\sdot56\right)}{56}=\frac{\frac{6}{7}\sdot7}{56}=\frac{6}{56}}}</math>

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::What we are asking for is as if we say to multiply two parts of sixty by six parts of fifty-six.

::What we are asking for is as if we say to multiply two parts of sixty by six parts of fifty-six.

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::So, we multiply one denominator by the other; it is three thousand, three hundred and sixty.

::So, we multiply one denominator by the other; it is three thousand, three hundred and sixty.

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::When we multiply the two parts by the six, it is twelve.

::When we multiply the two parts by the six, it is twelve.

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::Its ratio to the product of one denominator by the other is a fifth of an eighth of its seventh.

::Its ratio to the product of one denominator by the other is a fifth of an eighth of its seventh.

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::<span style=color:green>'''Check:'''</span> Because a seventh of the product is four hundred and eighty; its eighth is sixty, and twelve is a fifth of sixty.

::<span style=color:green>'''Check:'''</span> Because a seventh of the product is four hundred and eighty; its eighth is sixty, and twelve is a fifth of sixty.

::<math>\scriptstyle{\color{blue}{\frac{1}{5}\sdot\left[\frac{1}{8}\sdot\left(\frac{1}{7}\sdot3360\right)\right]=\frac{1}{5}\sdot\left(\frac{1}{8}\sdot480\right)=\frac{1}{5}\sdot60=12}}</math>

::<math>\scriptstyle{\color{blue}{\frac{1}{5}\sdot\left[\frac{1}{8}\sdot\left(\frac{1}{7}\sdot3360\right)\right]=\frac{1}{5}\sdot\left(\frac{1}{8}\sdot480\right)=\frac{1}{5}\sdot60=12}}</math>

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|You will understand [what] to do through your wisdom, as you see in the previous examples.

|You will understand [what] to do through your wisdom, as you see in the previous examples.

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|Up to this point I have explained the method of multiplying the fractions by themselves or by other fractions.

|Up to this point I have explained the method of multiplying the fractions by themselves or by other fractions.

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*{{#annot:5×⁴/₆|17|chfZ}}Example of multiplying integers by fractions: we wish to multiply five integers by four sixths.

*{{#annot:5×⁴/₆|17|chfZ}}Example of multiplying integers by fractions: we wish to multiply five integers by four sixths.

:<math>\scriptstyle5\times\frac{4}{6}</math>

:<math>\scriptstyle5\times\frac{4}{6}</math>

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::The denominator of the sixths is six.

::The denominator of the sixths is six.

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::We multiply the five integers by the numerator of the four-sixths; it is twenty.

::We multiply the five integers by the numerator of the four-sixths; it is twenty.

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::We divide it by the denominator; the result is three integers and two-sixths of one, which is its third.

::We divide it by the denominator; the result is three integers and two-sixths of one, which is its third.

::<math>\scriptstyle{\color{blue}{5\times\frac{4}{6}=\frac{5\sdot4}{6}=\frac{20}{6}=3+\frac{2}{6}=3+\frac{1}{3}}}</math>

::<math>\scriptstyle{\color{blue}{5\times\frac{4}{6}=\frac{5\sdot4}{6}=\frac{20}{6}=3+\frac{2}{6}=3+\frac{1}{3}}}</math>

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|We can find it another way:

|We can find it another way:

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::We multiply it by four-sixths; they are one hundred and twenty sixths of a sixth

::We multiply it by four-sixths; they are one hundred and twenty sixths of a sixth

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::The hundred and eight of them are three integers and the remaining twelve are two-sixths

::The hundred and eight of them are three integers and the remaining twelve are two-sixths

::<math>\scriptstyle{\color{blue}{5\times\frac{4}{6}=\frac{\left(5\sdot6\right)\sdot4}{6\sdot6}=\frac{30\sdot4}{6\sdot6}=\frac{120}{6}\sdot\frac{1}{6}=\frac{108+12}{6}\sdot\frac{1}{6}=3+\frac{2}{6}}}</math>

::<math>\scriptstyle{\color{blue}{5\times\frac{4}{6}=\frac{\left(5\sdot6\right)\sdot4}{6\sdot6}=\frac{30\sdot4}{6\sdot6}=\frac{120}{6}\sdot\frac{1}{6}=\frac{108+12}{6}\sdot\frac{1}{6}=3+\frac{2}{6}}}</math>

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|We find that both ways lead to the same result.

|We find that both ways lead to the same result.

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*{{#annot:(3+⅘)×⅗|17|TGUm}}Example of multiplying integers and fractions by fractions of one type alone: we wish to multiply three integers and four-fifths by three-fifths.

*{{#annot:(3+⅘)×⅗|17|TGUm}}Example of multiplying integers and fractions by fractions of one type alone: we wish to multiply three integers and four-fifths by three-fifths.

:<math>\scriptstyle\left(3+\frac{4}{5}\right)\times\frac{3}{5}</math>

:<math>\scriptstyle\left(3+\frac{4}{5}\right)\times\frac{3}{5}</math>

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::We take the denominator of the fifths; it is five.

::We take the denominator of the fifths; it is five.

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::We multiply it by the three integers; they are fifteen-fifths.

::We multiply it by the three integers; they are fifteen-fifths.

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::We multiply them by the three-fifths; they are fifty-seven fifths of a fifth.

::We multiply them by the three-fifths; they are fifty-seven fifths of a fifth.

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::The fifty of them are two integers and the remaining seven-fifths of a fifth are one fifth and two-fifths of a fifth.

::The fifty of them are two integers and the remaining seven-fifths of a fifth are one fifth and two-fifths of a fifth.

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::We multiply the three integers by the numerator of the three-fifths; they are nine-fifths.

::We multiply the three integers by the numerator of the three-fifths; they are nine-fifths.

::<math>\scriptstyle{\color{blue}{3\times\frac{3}{5}=\frac{3\sdot3}{5}=\frac{9}{5}}}</math>

::<math>\scriptstyle{\color{blue}{3\times\frac{3}{5}=\frac{3\sdot3}{5}=\frac{9}{5}}}</math>

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::When we multiply the four-fifths also by the three-fifths, they are twelve-fifths of a fifth, which are two-fifths and two-fifths of a fifth.

::When we multiply the four-fifths also by the three-fifths, they are twelve-fifths of a fifth, which are two-fifths and two-fifths of a fifth.

::<math>\scriptstyle{\color{blue}{\frac{4}{5}\times\frac{3}{5}=\frac{4\sdot3}{5\sdot5}=\frac{12}{5}\sdot\frac{1}{5}=\frac{2}{5}+\left(\frac{2}{5}\sdot\frac{1}{5}\right)}}</math>

::<math>\scriptstyle{\color{blue}{\frac{4}{5}\times\frac{3}{5}=\frac{4\sdot3}{5\sdot5}=\frac{12}{5}\sdot\frac{1}{5}=\frac{2}{5}+\left(\frac{2}{5}\sdot\frac{1}{5}\right)}}</math>

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*{{#annot:(4+⅖)×¾|17|IoIr}}Example of multiplying integers and fractions by fractions that are not of one type: we wish to multiply four integers and two fifths by three quarters.

*{{#annot:(4+⅖)×¾|17|IoIr}}Example of multiplying integers and fractions by fractions that are not of one type: we wish to multiply four integers and two fifths by three quarters.

:<math>\scriptstyle\left(4+\frac{2}{5}\right)\times\frac{3}{4}</math>

:<math>\scriptstyle\left(4+\frac{2}{5}\right)\times\frac{3}{4}</math>

רצינו לכפול ארבעה שלמים ושתי חמשיות על שלש רביעיות{{#annotend:IoIr}}

רצינו לכפול ארבעה שלמים ושתי חמשיות על שלש רביעיות{{#annotend:IoIr}}

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::We take the denominator of the fifths; it is five.

::We take the denominator of the fifths; it is five.

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::We multiply it by the four integers; it is twenty-fifths.

::We multiply it by the four integers; it is twenty-fifths.

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::We add to them the two-fifths; they are twenty-two fifths.

::We add to them the two-fifths; they are twenty-two fifths.

::<math>\scriptstyle{\color{blue}{4+\frac{2}{5}=\frac{\left(4\sdot5\right)+2}{5}=\frac{20+2}{5}=\frac{22}{5}}}</math>

::<math>\scriptstyle{\color{blue}{4+\frac{2}{5}=\frac{\left(4\sdot5\right)+2}{5}=\frac{20+2}{5}=\frac{22}{5}}}</math>

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::We multiply them by the three-quarters; they are sixty-six quarters of a fifth.

::We multiply them by the three-quarters; they are sixty-six quarters of a fifth.

::<math>\scriptstyle{\color{blue}{\frac{22}{5}\sdot\frac{3}{4}=\frac{22\sdot3}{4}\sdot\frac{1}{5}=\frac{66}{4}\sdot\frac{1}{5}}}</math>

::<math>\scriptstyle{\color{blue}{\frac{22}{5}\sdot\frac{3}{4}=\frac{22\sdot3}{4}\sdot\frac{1}{5}=\frac{66}{4}\sdot\frac{1}{5}}}</math>

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::We multiply it by the denominator of the fifths; it is twenty.

::We multiply it by the denominator of the fifths; it is twenty.

::<math>\scriptstyle{\color{blue}{4\sdot5=20}}</math>

::<math>\scriptstyle{\color{blue}{4\sdot5=20}}</math>

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::We divide the sixty-six quarters by it; we find them three times in it, which are three integers and six-quarters of a fifth still remain indivisible that are one-fifth and two-quarters of a fifth.

::We divide the sixty-six quarters by it; we find them three times in it, which are three integers and six-quarters of a fifth still remain indivisible that are one-fifth and two-quarters of a fifth.

::<math>\scriptstyle{\color{blue}{\left(4+\frac{2}{5}\right)\times\frac{3}{4}=\frac{66}{20}=3+\left(\frac{6}{4}\sdot\frac{1}{5}\right)=3+\frac{1}{5}+\left(\frac{2}{4}\sdot\frac{1}{5}\right)}}</math>

::<math>\scriptstyle{\color{blue}{\left(4+\frac{2}{5}\right)\times\frac{3}{4}=\frac{66}{20}=3+\left(\frac{6}{4}\sdot\frac{1}{5}\right)=3+\frac{1}{5}+\left(\frac{2}{4}\sdot\frac{1}{5}\right)}}</math>

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|In another way:

|In another way:

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|<math>\scriptstyle{\color{OliveGreen}{\left(n+\frac{a}{b}\right)\times\frac{c}{d}=\frac{n\sdot c}{d}+\frac{a\sdot c}{b\sdot d}}}</math>

|<math>\scriptstyle{\color{OliveGreen}{\left(n+\frac{a}{b}\right)\times\frac{c}{d}=\frac{n\sdot c}{d}+\frac{a\sdot c}{b\sdot d}}}</math>

::We multiply the four integers by three-quarters; they are twelve-quarters that are three integers.

::We multiply the four integers by three-quarters; they are twelve-quarters that are three integers.

::<math>\scriptstyle{\color{blue}{4\sdot\frac{3}{4}=\frac{4\sdot3}{4}=\frac{12}{4}=3}}</math>

::<math>\scriptstyle{\color{blue}{4\sdot\frac{3}{4}=\frac{4\sdot3}{4}=\frac{12}{4}=3}}</math>

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::Then, we multiply the two-fifths by the three-quarters; they are six-quarters of a fifth that are one-fifth and two-quarters of a fifth, as we found at first.

::Then, we multiply the two-fifths by the three-quarters; they are six-quarters of a fifth that are one-fifth and two-quarters of a fifth, as we found at first.

::<math>\scriptstyle{\color{blue}{\frac{2}{5}\sdot\frac{3}{4}=\frac{2\sdot3}{4}\sdot\frac{1}{5}=\frac{6}{4}\sdot\frac{1}{5}=\frac{1}{5}+\left(\frac{2}{4}\sdot\frac{1}{5}\right)}}</math>

::<math>\scriptstyle{\color{blue}{\frac{2}{5}\sdot\frac{3}{4}=\frac{2\sdot3}{4}\sdot\frac{1}{5}=\frac{6}{4}\sdot\frac{1}{5}=\frac{1}{5}+\left(\frac{2}{4}\sdot\frac{1}{5}\right)}}</math>

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*{{#annot:(2+¾)×(3+²/₄)|17|lTDC}}Example of multiplying integers and fractions by integers and fractions of the same type: we wish to multiply two integers and three quarters by three integers and two quarters.

*{{#annot:(2+¾)×(3+²/₄)|17|lTDC}}Example of multiplying integers and fractions by integers and fractions of the same type: we wish to multiply two integers and three quarters by three integers and two quarters.

:<math>\scriptstyle\left(2+\frac{3}{4}\right)\times\left(3+\frac{2}{4}\right)</math>

:<math>\scriptstyle\left(2+\frac{3}{4}\right)\times\left(3+\frac{2}{4}\right)</math>

רצינו לכפול שני שלמים ושלש רביעיות על שלשה שלמים ושתי רביעיות{{#annotend:lTDC}}

רצינו לכפול שני שלמים ושלש רביעיות על שלשה שלמים ושתי רביעיות{{#annotend:lTDC}}

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::So, we multiply it by the two integers; they are eight-quarters.

::So, we multiply it by the two integers; they are eight-quarters.

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::We add the three-quarters to them; they are eleven.

::We add the three-quarters to them; they are eleven.

::<math>\scriptstyle{\color{blue}{\left(2\sdot4\right)+3=8+3=11}}</math>

::<math>\scriptstyle{\color{blue}{\left(2\sdot4\right)+3=8+3=11}}</math>

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::We add the two-quarters to them; their number is fourteen.

::We add the two-quarters to them; their number is fourteen.

::<math>\scriptstyle{\color{blue}{\left(3\sdot4\right)+2=12+2=14}}</math>

::<math>\scriptstyle{\color{blue}{\left(3\sdot4\right)+2=12+2=14}}</math>

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::We divide them by the product of the denominator by itself; the product is sixteen.

::We divide them by the product of the denominator by itself; the product is sixteen.

::<math>\scriptstyle{\color{blue}{4^2=16}}</math>

::<math>\scriptstyle{\color{blue}{4^2=16}}</math>

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::We find them nine times in it that are nine integers; and ten-quarters of a quarter remain indivisible that are two-quarters and two-quarters of a quarter and this is the result of the multiplication.

::We find them nine times in it that are nine integers; and ten-quarters of a quarter remain indivisible that are two-quarters and two-quarters of a quarter and this is the result of the multiplication.

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::We multiply the integers by the integers; they are six integers.

::We multiply the integers by the integers; they are six integers.

::<math>\scriptstyle{\color{blue}{2\sdot3=6}}</math>

::<math>\scriptstyle{\color{blue}{2\sdot3=6}}</math>

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::We multiply also the two integers by the two-quarters; they are four-quarters.

::We multiply also the two integers by the two-quarters; they are four-quarters.

::<math>\scriptstyle{\color{blue}{2\sdot\frac{2}{4}=\frac{4}{4}}}</math>

::<math>\scriptstyle{\color{blue}{2\sdot\frac{2}{4}=\frac{4}{4}}}</math>

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::Then, we multiply the three-quarters by the three integers; they are nine-quarters.

::Then, we multiply the three-quarters by the three integers; they are nine-quarters.

::<math>\scriptstyle{\color{blue}{\frac{3}{4}\sdot3=\frac{9}{4}}}</math>

::<math>\scriptstyle{\color{blue}{\frac{3}{4}\sdot3=\frac{9}{4}}}</math>

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::We add them to the four-quarters we have, because they are all of the same type; they are thirteen-quarters.

::We add them to the four-quarters we have, because they are all of the same type; they are thirteen-quarters.

::<math>\scriptstyle{\color{blue}{\frac{4}{4}+\frac{9}{4}=\frac{13}{4}}}</math>

::<math>\scriptstyle{\color{blue}{\frac{4}{4}+\frac{9}{4}=\frac{13}{4}}}</math>

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::Then, we multiply the three-quarters by the two; they are six-quarters of a quarter that are one quarter and two-quarters of a quarter.

::Then, we multiply the three-quarters by the two; they are six-quarters of a quarter that are one quarter and two-quarters of a quarter.

::<math>\scriptstyle{\color{blue}{\frac{3}{4}\sdot\frac{2}{4}=\frac{6}{4}\sdot\frac{1}{4}=\frac{1}{4}+\left(\frac{2}{4}\sdot\frac{1}{4}\right)}}</math>

::<math>\scriptstyle{\color{blue}{\frac{3}{4}\sdot\frac{2}{4}=\frac{6}{4}\sdot\frac{1}{4}=\frac{1}{4}+\left(\frac{2}{4}\sdot\frac{1}{4}\right)}}</math>

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::We add them to the six; they are nine integers.

::We add them to the six; they are nine integers.

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::Two-quarters still remain indivisible, plus the two-quarters of a quarter we have.

::Two-quarters still remain indivisible, plus the two-quarters of a quarter we have.

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|We find that this way leads to the same as the first [way].

|We find that this way leads to the same as the first [way].

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*{{#annot:(5+⅔)×(2+³/₆)|17|NcZF}}Example of multiplying integers and fractions by integers and fractions that are not of the same type: we wish to multiply five integers and two thirds by two integers and three sixths.

*{{#annot:(5+⅔)×(2+³/₆)|17|NcZF}}Example of multiplying integers and fractions by integers and fractions that are not of the same type: we wish to multiply five integers and two thirds by two integers and three sixths.

:<math>\scriptstyle\left(5+\frac{2}{3}\right)\times\left(2+\frac{3}{6}\right)</math>

:<math>\scriptstyle\left(5+\frac{2}{3}\right)\times\left(2+\frac{3}{6}\right)</math>

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::It is known that the denominator of the third is three.

::It is known that the denominator of the third is three.

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::We multiply it by the five integers; they are fifteen-thirds.

::We multiply it by the five integers; they are fifteen-thirds.

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::We add the two-thirds to them; their number is seventeen.

::We add the two-thirds to them; their number is seventeen.

::<math>\scriptstyle{\color{blue}{\left(5\sdot3\right)+2=15+2=17}}</math>

::<math>\scriptstyle{\color{blue}{\left(5\sdot3\right)+2=15+2=17}}</math>

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::We add the three-sixths to them; their number is fifteen.

::We add the three-sixths to them; their number is fifteen.

::<math>\scriptstyle{\color{blue}{\left(2\sdot6\right)+3=12+3=15}}</math>

::<math>\scriptstyle{\color{blue}{\left(2\sdot6\right)+3=12+3=15}}</math>

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::We multiply one number by the other; they are two hundred and fifty-five.

::We multiply one number by the other; they are two hundred and fifty-five.

::<math>\scriptstyle{\color{blue}{17\sdot15=255}}</math>

::<math>\scriptstyle{\color{blue}{17\sdot15=255}}</math>

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::Then, we multiply one denominator by the other; the product is eighteen.

::Then, we multiply one denominator by the other; the product is eighteen.

::<math>\scriptstyle{\color{blue}{3\sdot6=18}}</math>

::<math>\scriptstyle{\color{blue}{3\sdot6=18}}</math>

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:<math>\scriptstyle{\color{blue}{\left(5+\frac{2}{3}\right)\times\left(2+\frac{3}{6}\right)=\frac{255}{18}=14+\left(\frac{3}{3}\sdot\frac{1}{6}\right)=14+\frac{1}{6}}}</math>

:<math>\scriptstyle{\color{blue}{\left(5+\frac{2}{3}\right)\times\left(2+\frac{3}{6}\right)=\frac{255}{18}=14+\left(\frac{3}{3}\sdot\frac{1}{6}\right)=14+\frac{1}{6}}}</math>

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|Another method:

|Another method:

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::We multiply this denominator by the five integers; they are ninety parts, every eighteen of which are one integer.

::We multiply this denominator by the five integers; they are ninety parts, every eighteen of which are one integer.

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::For the two-thirds, we add two-thirds of the denominator to them, which is twelve; the number of these parts is one hundred and two.

::For the two-thirds, we add two-thirds of the denominator to them, which is twelve; the number of these parts is one hundred and two.

::<math>\scriptstyle{\color{blue}{\left(5\sdot18\right)+\left(\frac{2}{3}\sdot18\right)=90+12=102}}</math>

::<math>\scriptstyle{\color{blue}{\left(5\sdot18\right)+\left(\frac{2}{3}\sdot18\right)=90+12=102}}</math>

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::We also multiply the denominator by the two integers; they are thirty-six parts.

::We also multiply the denominator by the two integers; they are thirty-six parts.

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::<math>\scriptstyle{\color{blue}{\left(2\sdot18\right)+\left(\frac{3}{6}\sdot18\right)=36+9=45}}</math>

::<math>\scriptstyle{\color{blue}{\left(2\sdot18\right)+\left(\frac{3}{6}\sdot18\right)=36+9=45}}</math>

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::We multiply one number by the other; the result is four thousand, five hundred and ninety.

::We multiply one number by the other; the result is four thousand, five hundred and ninety.

::<math>\scriptstyle{\color{blue}{102\sdot45=4590}}</math>

::<math>\scriptstyle{\color{blue}{102\sdot45=4590}}</math>

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::The result of division is fourteen, which are integers, and fifty-four remain indivisible, which is a sixth of the product of the denominator by itself. We find the result of multiplication is fourteen integers and a sixth, as [found] at first.

::The result of division is fourteen, which are integers, and fifty-four remain indivisible, which is a sixth of the product of the denominator by itself. We find the result of multiplication is fourteen integers and a sixth, as [found] at first.

:<math>\scriptstyle{\color{blue}{\left(5+\frac{2}{3}\right)\times\left(2+\frac{3}{6}\right)=\frac{4590}{324}=14+\frac{54}{324}=14+\frac{1}{6}}}</math>

:<math>\scriptstyle{\color{blue}{\left(5+\frac{2}{3}\right)\times\left(2+\frac{3}{6}\right)=\frac{4590}{324}=14+\frac{54}{324}=14+\frac{1}{6}}}</math>

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|All these ways are good and right for those who find knowledge.

|All these ways are good and right for those who find knowledge.

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|Now I will start talking about the way of dividing the fractions by each other, or integers and fractions by integers and fractions.

|Now I will start talking about the way of dividing the fractions by each other, or integers and fractions by integers and fractions.

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*{{#annot:⅔÷²/₇|552|VpDd}}Example for dividing fractions by fractions: we wish to divide two thirds by two sevenths.

*{{#annot:⅔÷²/₇|552|VpDd}}Example for dividing fractions by fractions: we wish to divide two thirds by two sevenths.

:<math>\scriptstyle\frac{2}{3}\div\frac{2}{7}</math>

:<math>\scriptstyle\frac{2}{3}\div\frac{2}{7}</math>

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::Its two-thirds is fourteen.

::Its two-thirds is fourteen.

::<math>\scriptstyle{\color{blue}{\frac{2}{3}\sdot21=14}}</math>

::<math>\scriptstyle{\color{blue}{\frac{2}{3}\sdot21=14}}</math>

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::Its two-sevenths is six.

::Its two-sevenths is six.

::<math>\scriptstyle{\color{blue}{\frac{2}{7}\sdot21=6}}</math>

::<math>\scriptstyle{\color{blue}{\frac{2}{7}\sdot21=6}}</math>

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===== <span style=color:green>Division of integers and fractions by integers and fractions</span> =====

===== <span style=color:green>Division of integers and fractions by integers and fractions</span> =====

*{{#annot:(4+⅔)÷(2+⅖)|552|eAW6}}Example for dividing integers and fractions by integers and fractions: we wish to divide four integers and two thirds by two integers and two fifths.

*{{#annot:(4+⅔)÷(2+⅖)|552|eAW6}}Example for dividing integers and fractions by integers and fractions: we wish to divide four integers and two thirds by two integers and two fifths.

:<math>\scriptstyle\left(4+\frac{2}{3}\right)\div\left(2+\frac{2}{5}\right)</math>

:<math>\scriptstyle\left(4+\frac{2}{3}\right)\div\left(2+\frac{2}{5}\right)</math>

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::So, we multiply it by the four integers; they are sixty parts.

::So, we multiply it by the four integers; they are sixty parts.

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::For the two-thirds, we add two-thirds of the denominator to them, which are ten; the number of the parts is seventy.

::For the two-thirds, we add two-thirds of the denominator to them, which are ten; the number of the parts is seventy.

::<math>\scriptstyle{\color{blue}{\left(4+\frac{2}{3}\right)\sdot15=\left(4\sdot15\right)+\left(\frac{2}{3}\sdot15\right)=60+10=70}}</math>

::<math>\scriptstyle{\color{blue}{\left(4+\frac{2}{3}\right)\sdot15=\left(4\sdot15\right)+\left(\frac{2}{3}\sdot15\right)=60+10=70}}</math>

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::We add six to them, which are two-fifths of the denominator; their number is thirty-six.

::We add six to them, which are two-fifths of the denominator; their number is thirty-six.

::<math>\scriptstyle{\color{blue}{\left(2+\frac{2}{5}\right)\sdot15=\left(2\sdot15\right)+\left(\frac{2}{5}\sdot15\right)=30+6=36}}</math>

::<math>\scriptstyle{\color{blue}{\left(2+\frac{2}{5}\right)\sdot15=\left(2\sdot15\right)+\left(\frac{2}{5}\sdot15\right)=30+6=36}}</math>

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::We divide the seventy by it; we find it two times in it, minus half a ninth of the divisor; or we say that the result of division is one time plus three-quarters and two-ninths minus a quarter of a ninth.

::We divide the seventy by it; we find it two times in it, minus half a ninth of the divisor; or we say that the result of division is one time plus three-quarters and two-ninths minus a quarter of a ninth.

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| colspan="2"|

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|After the ways of multiplying and dividing fractions have been clarified, it is also necessary to clarify the ways of adding them to each other and the way of subtracting smaller fractions from greater fractions

|After the ways of multiplying and dividing fractions have been clarified, it is also necessary to clarify the ways of adding them to each other and the way of subtracting smaller fractions from greater fractions

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|Know that when we wish to know how much is the sum of known fractions with known fractions:

|Know that when we wish to know how much is the sum of known fractions with known fractions:

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::We do it this way: we take the common denominator of both; it is twenty-four.

::We do it this way: we take the common denominator of both; it is twenty-four.

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::We take the twenty-four from it; fourteen remains from it, which is a whole half and half its sixth; or, if we wish the remainder is a half and a third of a quarter.

::We take the twenty-four from it; fourteen remains from it, which is a whole half and half its sixth; or, if we wish the remainder is a half and a third of a quarter.

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|When we wish to subtract from known fractions known fractions that are smaller than them and know the remainder.

|When we wish to subtract from known fractions known fractions that are smaller than them and know the remainder.

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|<math>\scriptstyle{\color{OliveGreen}{\frac{a}{b}-\frac{c}{d}=\frac{\left[\frac{a}{b}\sdot\left(b\sdot d\right)\right]-\left[\frac{c}{d}\sdot\left(b\sdot d\right)\right]}{b\sdot d}}}</math>

|<math>\scriptstyle{\color{OliveGreen}{\frac{a}{b}-\frac{c}{d}=\frac{\left[\frac{a}{b}\sdot\left(b\sdot d\right)\right]-\left[\frac{c}{d}\sdot\left(b\sdot d\right)\right]}{b\sdot d}}}</math>

*{{#annot:²/₄-⅕|678|tqJF}}As we say for example: we subtract one fifth from two quarters. How much is the remainder?

*{{#annot:²/₄-⅕|678|tqJF}}As we say for example: we subtract one fifth from two quarters. How much is the remainder?

:<math>\scriptstyle\frac{2}{4}-\frac{1}{5}</math>

:<math>\scriptstyle\frac{2}{4}-\frac{1}{5}</math>

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::We do as follows: we take a common denominator of both; it is 20.

::We do as follows: we take a common denominator of both; it is 20.

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::Its one-fifth is four.

::Its one-fifth is four.

::<math>\scriptstyle{\color{blue}{\frac{1}{5}\sdot20=4}}</math>

::<math>\scriptstyle{\color{blue}{\frac{1}{5}\sdot20=4}}</math>

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::We subtract four from the ten; six remains from it.

::We subtract four from the ten; six remains from it.

::<math>\scriptstyle{\color{blue}{10-4=6}}</math>

::<math>\scriptstyle{\color{blue}{10-4=6}}</math>

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|style="text-align:right;"|והוא רביעית המדומה וחומש רביעיתו וככה ערך הנשאר אל השלם

|style="text-align:right;"|והוא רביעית המדומה וחומש רביעיתו וככה ערך הנשאר אל השלם

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|Praise be to God, blessed be He.

|Praise be to God, blessed be He.

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|}

== Chapter Ten – Knowing the Square Roots of "Deaf" and "Mute" Numbers by Approximation ==

== Chapter Ten – Knowing the Square Roots of "Deaf" and "Mute" Numbers by Approximation ==

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|The reason that it is called a "wise number" is that its root, which is the measure of each of the four sides of the square, can be truly found.

|The reason that it is called a "wise number" is that its root, which is the measure of each of the four sides of the square, can be truly found.

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|In every scientific discipline there are many secrets hidden and concealed from the hearts of men to be found.

|In every scientific discipline there are many secrets hidden and concealed from the hearts of men to be found.

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|We also see this in the science of medicine, when physicians instruct to take a well-known herb root or stone to heal the sick person who carries them by their virtue, but no wise man can know the reason of the [healing] virtue of the stone, or the herb root, why it is so.

|We also see this in the science of medicine, when physicians instruct to take a well-known herb root or stone to heal the sick person who carries them by their virtue, but no wise man can know the reason of the [healing] virtue of the stone, or the herb root, why it is so.

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|Many things like these are known only to the Knower of all concealed things alone, blessed be He and blessed be His name.

|Many things like these are known only to the Knower of all concealed things alone, blessed be He and blessed be His name.

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|Now, we return to the premises and say: since most ancients extracted the approximate roots of "deaf" or "mute" numbers by using the method of the astronomers, some of their methods should be introduced and explained first, before I discuss the extraction of roots [of non-square numbers].

|Now, we return to the premises and say: since most ancients extracted the approximate roots of "deaf" or "mute" numbers by using the method of the astronomers, some of their methods should be introduced and explained first, before I discuss the extraction of roots [of non-square numbers].

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|The astronomers divided the zodiac into twelve shapes that are called zodiacal signs.

|The astronomers divided the zodiac into twelve shapes that are called zodiacal signs.

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|The reason for this division is that there is no number smaller than twelve that has as many whole parts [= divisors, in modern terminology] like it, for it has a half, a third, a quarter, a sixth, and half a sixth.

|The reason for this division is that there is no number smaller than twelve that has as many whole parts [= divisors, in modern terminology] like it, for it has a half, a third, a quarter, a sixth, and half a sixth.

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|They divided each zodiacal sign into thirty parts and called them degrees.

|They divided each zodiacal sign into thirty parts and called them degrees.

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|They divided it into this number, since there is no number smaller than it that has as many parts [= divisors] as it has, for it has a half, a third, a fifth, a sixth, and a tenth.

|They divided it into this number, since there is no number smaller than it that has as many parts [= divisors] as it has, for it has a half, a third, a fifth, a sixth, and a tenth.

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|They divided each degree into sixty parts and called them primes.

|They divided each degree into sixty parts and called them primes.

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|They divided each prime into sixty seconds; each second into sixty thirds; and each third into sixty fourths.

|They divided each prime into sixty seconds; each second into sixty thirds; and each third into sixty fourths.

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|And so on, they divided their fractions as needed, endlessly, each one is equal to sixty of the rank that follows it and sixty of each one are a unit of the preceding rank.

|And so on, they divided their fractions as needed, endlessly, each one is equal to sixty of the rank that follows it and sixty of each one are a unit of the preceding rank.

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|I tell you that the degrees [act] as the integers:

|I tell you that the degrees [act] as the integers:

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|When we multiply degrees by degrees, the result of multiplication are always degrees.

|When we multiply degrees by degrees, the result of multiplication are always degrees.

:<math>\scriptstyle{\color{OliveGreen}{\left(a\sdot60^0\right)\times\left(b\sdot60^0\right)=\left(a\sdot b\right)\sdot60^0}}</math>

:<math>\scriptstyle{\color{OliveGreen}{\left(a\sdot60^0\right)\times\left(b\sdot60^0\right)=\left(a\sdot b\right)\sdot60^0}}</math>

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|If we multiply them by other fractions, such as primes, seconds, or others, the type of these fractions always remains the same, their type does not change because of this multiplication.

|If we multiply them by other fractions, such as primes, seconds, or others, the type of these fractions always remains the same, their type does not change because of this multiplication.

:<math>\scriptstyle{\color{OliveGreen}{\left(a\sdot60^0\right)\times\left(b\sdot\frac{1}{60^n}\right)=\left(a\sdot b\right)\sdot\frac{1}{60^n}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{\left(a\sdot60^0\right)\times\left(b\sdot\frac{1}{60^n}\right)=\left(a\sdot b\right)\sdot\frac{1}{60^n}}}</math>

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|The primes, the seconds, and all other types, follow the rule of the [simple] fractions.

|The primes, the seconds, and all other types, follow the rule of the [simple] fractions.

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|So, if we multiply seconds by seconds, the product is fourths.

|So, if we multiply seconds by seconds, the product is fourths.

:<math>\scriptstyle{\color{OliveGreen}{a^{\prime\prime}\times b^{\prime\prime}=\left(a\sdot b\right)^{iv}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{a^{\prime\prime}\times b^{\prime\prime}=\left(a\sdot b\right)^{iv}}}</math>

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|The product of primes by primes is seconds.

|The product of primes by primes is seconds.

:<math>\scriptstyle{\color{OliveGreen}{a^\prime\times b^\prime=\left(a\sdot b\right)^{\prime\prime}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{a^\prime\times b^\prime=\left(a\sdot b\right)^{\prime\prime}}}</math>

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|The product of seconds by thirds is fifths.

|The product of seconds by thirds is fifths.

|The product of sixths by seconds, or fourths by fourths, is eighths.

|The product of sixths by seconds, or fourths by fourths, is eighths.

:<math>\scriptstyle{\color{OliveGreen}{a^{vi}\times b^{\prime\prime}=a^{iv}\times b^{iv}=\left(a\sdot b\right)^{viii}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{a^{vi}\times b^{\prime\prime}=a^{iv}\times b^{iv}=\left(a\sdot b\right)^{viii}}}</math>

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|The product of seconds by fifths is sevenths.

|The product of seconds by fifths is sevenths.

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|I will give you a method for multiplying these mentioned fractions:

|I will give you a method for multiplying these mentioned fractions:

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|When you wish to multiply a line of known fractions by a line of known fractions; in each line there are many higher and lower types, do as follows: take one of the two lines and convert all the higher types into the lowest type of that line. Add what you receive from the decomposing to the lowest type, because now they are all the same type. Apply the same procedure with the second line as well.

|When you wish to multiply a line of known fractions by a line of known fractions; in each line there are many higher and lower types, do as follows: take one of the two lines and convert all the higher types into the lowest type of that line. Add what you receive from the decomposing to the lowest type, because now they are all the same type. Apply the same procedure with the second line as well.

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|Then, divide them by sixty many times, until you reach a type which you cannot divide by sixty.

|Then, divide them by sixty many times, until you reach a type which you cannot divide by sixty.

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|This multiplication technique is long as well as short, but it is well-guided.

|This multiplication technique is long as well as short, but it is well-guided.

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|After clarifying and introducing what is appropriate for the art of this chapter, I will return to discuss the method of extracting the root a deaf square:

|After clarifying and introducing what is appropriate for the art of this chapter, I will return to discuss the method of extracting the root a deaf square:

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|style="text-align:right;"|ואם היה פחות מגדרו נעשה זאת

|style="text-align:right;"|ואם היה פחות מגדרו נעשה זאת

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*<math>\scriptstyle{\color{OliveGreen}{b<a}}</math>

*<math>\scriptstyle{\color{OliveGreen}{b<a}}</math>

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::<math>\scriptstyle\sqrt{5}</math>

::<math>\scriptstyle\sqrt{5}</math>

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