Difference between revisions of "ספר אגריס"

Introduction: The Positional Decimal System

0 before any numeral indicates tens

• 20

• 30

• 40

• 90

• 100

• 200

• 300

• 400

• 900

• 1000

• 2000

• 3000

• 4000

• 9000

• 10000

• 20000

• 30000

• 40000

• 90000

• 100000, etc.

Addition

• Example: you wish to add 44 to 55.

Write as follows:

4 4 5 5

• Sum 4 with 5 in the bottom [row], it is 9.

• Sum the second 4 with the second 5, it is 9.

Thus, we found that their sum is 99, which is ninty-nine.

• Another example:

9 9 1 1

• Add 9 to 1 that is on the bottom [row], which is 11, [it is] ten.

Write 0 and you are left with 1, which is in the second rank.

• Add it to the second 1 that is also in the [second] rank, it is 2. Add the 2 to the 9 in the second rank, it is 11.

And 0 first, it is 110.

• Another example:

1 4 2 9 6 8

• Add 2 to 8, it is ten.

Write 0.

• Sum the remaining 1 with 6, it is 7. Add it to 4, it is 11.

Write 1 in the second rank.

• Add 1 in the third rank to 9 and 1, it is 11.

So it is 1110.

Subtraction

• Example: to subtract 24 from 46.

Write as follows:

4 6 2 4

• Subtract 4 from 6, 2 remains.

• Subtract 2 from 4, 2 remains, which is in the second rank.

It is 22.

• Example:

2 ‫0 ‫0 ‫0 1 9 9 9

• Subtract 9 from 10, one remains.

• Subtract 9 from 9, since we have shifted one for our matter, 0 remains.

• Subtract 9 from the third 9, 0 remains.

• Subtract 1 from 1, 0 remains.

Although only one [is borrowed], from it all the applied tens are generated:

• For, we have shifted the 1 to the preceding 9, so it is 10, from which 9 is subtracted and 1 remains.

• We shifted this 1 to the preceding 0, from which 9 is subtracted and 1 remains.

• We shifted this 1 to the first 0, so it is 10. Subtract the first 9 from the 10 and 1 remains on 1999, which is one.

Multiplication

If you multiply the last bottom digit by the last upper digit, write the result above the last bottom digit.

• interim product is of the units

If the result is less than 9, write the units above the last bottom digit (and shift the tens one rank forwards).

• interim product is of the tens

If there are no units, only tens, write 0 above the last bottom digit and shift the tens [forwards].

Then, multiply also the digit that precedes the last digit in the bottom row by the last digit in the top row and write the product above the bottom digit, by which you have multiplied.

Do the same for all the [bottom] digits - multiply them by the last [upper] digit.

After you have multiplied all the bottom row by the last upper digit, shift back the digits of the bottom row.

Multiply the last bottom digit by the digit that precedes the [last digit] in the top row, then the bottom digit that precedes the last [digit] by the same [upper digit], by which you have multiplied the last [digit], and likewise for all by the same [digit].

If there are more than two ranks in the top row, shift back [the bottom row] again as I explained to you.

• Example: multiply 23 by 23.

To know how much is the result, write like this:

2 3 2 3

• Multiply the bottom 2 by the upper 2, it is 4.

Write the 4 in the third rank after the upper 2, like this:

4 2 3 2 3

• Multiply also the 3 by 2 that is above it, it is 6.

Erase the upper 2 and write the 6 instead, such as:

4 6 3 2 3

• Write the [bottom] 23 beneath the second 3, it is:

4 6 3   2 3

• Multiply also the 2 by the 3, it is 12.

Erase the 6 and write the 2 instead.

Add the 1, which is in the third rank, to the 4, it is 5, such as:

5 2 3   2 3

• Multiply the 3 by the 3, it is 9.

[Write] it is also above 23, it is [5]29.

Multiply the bottom [digits] by the second rank, then shift back and multiply them also by the first rank. Multiply it by the bottom digit, then erase what is above it.

Division

• Here is an example: we want to know how many times 70 there are in 654?

• Write 654 in this row.

• Then, write 70 in the bottom row, like this:

6 5 4   7 0

If we would write the last 7 beneath the 6, which is also last [digit], we would not find 7 in it.

• Therefore, we have to write the 7 corresponding to the 5 of 654.

• Say: how many times 7 there are in 65?

We answer: 9 times.

• Write this 9 in a third row beneath the 0.

2 remains instead of the 5 of 654, and with the 4 of 654, which we did not divide yet, they are 24.

Hence we find that 70 is in 654 9 times and 24 remains.

• Another example: we want to know: how many times 19 there are in 5086?

• Write the greater number according to agorima, then write the 19 beneath is, like this:

5 0 8 6 1 9

• Say: how many times the bottom 1 is in the upper 5?

• If you would take 5 times, from what would you take 5 times the 9?

1 2 8 6 1 9       2

1 2 8 6   1 9     2

6 8 6

1 4 6 1 9   2

1 4 6 1 9   2 6

1 4 6   1 9 2 6

7 6   1 9 2 6

1 3   1 9 2 6 7

Proportions

• When one asks: you know that 11 roosters are sold for 21 pešiṭim. Now, you ask: what is the price of 8 roosters?

21 11   8

• Example: I bought 12 cubits of cloth for 28 dinar. You wish to know: what is the price of 7 cubits?

28 12   8

• Example: you hired a worker for 30 days for 41 dinar, but he worked with you only 17 days. You wish to know: how much should his payment for these days be?

41 30   17

Gelosia

the result: