Now, addition and subtraction: now, the rule of operation for addition and subtraction, in half a verse.
12. The sum or difference of the numerals according to their places is to be made, in order or in reverse order.
Now, an example:
13. Oh Lilavati, intelligent girl, if you understand addition and subtraction, tell me the sum of the amounts 2, 5, 32, 193, 18, 10, and 100, as well as [the remainder of] those when subtracted from 10000.
Statement: 2, 5, 32, 193, 18, 10, 100. Result from addition: 360. Result when subtracted from ayuta (10000): 9640.
That is addition and subtraction.
Now, the method for multiplication: the rule of operation for multiplication, in two and a half verses.
14. One should multiply the last [i.e., most significant] digit in the multiplicand by the multiplier, [and then the other digits], beginning with the next to last, by the same [multiplier]. Or, the multiplicand is [set] down repeatedly [and multiplied] by the [separate] parts of the multiplier; the product is the sum [of those separate products].
15. Or, the multiplier is divided by some [number], and the multiplicand is multiplied by that [number] and by the quotient: [that is] the result. (There are two ways of dividing up the number in this manner [i.e., splitting either the multiplier or the multiplicand]). Or [when the multiplicand is] multiplied by [the multiplier's] separate digits, [the product] is added up [from those].
16. Or, [if the multiplicand is] multiplied by the multiplier decreased or increased by a given [number], [the product should be] increased or decreased by the multiplicand times the given [number].
Now, an example:
17. Fawn-eyed child Lilavati, tell me, how much is the number [resulting from] 135 multiplied by 12, if you understand multiplication by separate parts and by separate digits. And tell [me], beautiful one, how much is that product divided by the same multiplier?
Statement: Multiplicand 135, multiplier 12.
Multiply the last digit of the multiplicand by the multiplier; when it is done in the same way [for all the digits], the result is 1620. Or, when the multiplier is divided into two separate parts (4, 8), when the multiplicand is multiplied separately by both [of those] and [the products] combined, the result is the same, 1620. Or the multiplier is divided by three, [and] the quotient is 4. When the multiplicand is multiplied by that and by three, the result is the same, 1620. Or when [the multiplier] is divided into its digits (1, 2), when the multiplicand is multiplied separately by those and [the products] added up according to their place-value, the result is the same, 1620. Or when the multiplicand is multiplied separately by the multiplier minus two (10) and by two, and [those products] are added, the result is the same, 1620. Or when the multiplicand is multiplied by the multiplier plus eight (20), and diminished by the multiplicand times eight, the result is the same, 1620.
That is the method for multiplication.
Now division: the rule of operation for division, in one verse.
18. In division, the divisor [multiplied by some number] is subtracted from the last [digit(s) of the] dividend; that multiplier is the result. But when possible, one should divide after having reduced the divisor and dividend by a common [factor].
Now for the case of division, the statement of the digits of the product and the divisor ([formerly] its multiplier) in the previous example: Dividend 1620. Divisor 12. The quotient from the division is the multiplicand, 135.
Or the dividend and divisor are reduced by three (540/4), or by four (405/3). When [those dividends are] divided by their respective divisors, the result is the same, 135.
That is division.
Now squaring: the rule of operation for the square, in two verses.
19. The product of two equal [quantities] is called the "square." Now the square of the last [digit] is set down, and so are the subsequent digits multiplied by the last [digit] times two, each [product] above [the place] of its own [digit]. [Then] when one has moved [to a fresh location] the quantity [to be squared], disregarding the last [digit], [the procedure is performed] again. [I.e., the square of the n-digit number an an-1 ... a2 a1 is found as follows: after writing an2 above an, 2an an-1 above an-1, ..., 2an a1 above a1, write down again the first n-1 digits an-1 ... a2 a1 and repeat the process. The square of the entire number is the sum of all the resulting products, according to their place values.]
20. Or, the square of two [separate] parts [of a given number] is their product multiplied by two, added to the sum of the squares of those parts. Or, the square is the product of [two equal] numbers [separately] increased and decreased by a given [quantity], added to the square of that given [number].
Here is an example:
21. Friend, tell [me] the square of nine, of fourteen, of three hundred minus three, and of ayuta [10000] plus five, if you know the way to produce squares.
Statement: 9, 14, 297, 10005. The squares of those, produced according to the stated method, are 81, 196, 88209, 100100025.
Or, [there are] two parts of nine---4, 5. The product of those (20), times two (40), is added to the sum of the squares of those parts (41); the result is the same square, 81.
Or, [there are] two parts of fourteen---6, 8. The product of those (48) times two [is] 96. The squares of those parts [are] 36, 64. [96] is added to the sum of those (100); the result is the same square, 196.
Or, the two parts [are] 4, 10; and in the same way the same square is 196.
Or, the number [is] 297. This is decreased and separately increased by three: 294, 300. The product of those (88200) is added to the square of three (9); the result is that same square, 88209. It is always [to be done] in this way. That is the square.
Now, the square-root: the rule of operation for the square-root, in one verse.
22. Having subtracted the [largest possible] square from the last odd [decimal] place [i.e., from the one- or two-digit multiple of the highest even power of 10 contained in the given number], multiply the square-root [of the subtracted quantity] by two. When the [next] even place is divided by that [here, to operate upon the "next place" means to use the remainder from the previous operation with an additional least significant digit brought down from the subsequent decimal place]: after subtracting the square of the [integer] quotient from the next odd place after that, set down two times the quotient in the "row" [of successive digits of the answer]. When the [next] even place is divided by the "row," after subtracting the square of the quotient from the next odd place, set down that result multiplied by two in the "row"; [do] this repeatedly. Half of the "row" is the [desired] square-root.
Here is an example:
23. Friend, [you should] know the square-root of four, and similarly of nine, and of the four squares previously given, respectively, if your understanding of this [topic] has been increased.
Statement: 4, 9, 81, 196, 88209, 100100025. [Their] square-roots obtained in order [are] 2, 3, 9, 14, 297, 10005.
That is the square-root.
Now, the cube: the rule of operation for cubing, in three verses.
24. And the product of three equal [quantities] is defined [as] the cube. The cube of the last [digit] is set down, and then the square of the last multiplied by the first and by three, and then the square of the first multiplied by three and by the last, and also the cube of the first. All [those]
25. are added up according to their different place-values; [that] is the cube [of a two-digit number]. [Or if one] considers that [quantity] as [split into] two parts, the last [part may be divided] in the same way repeatedly. Or, in finding squares and cubes, the procedure may be performed [starting with] the first digit.
26. Or the quantity is multiplied by [each of its] two parts, multiplied by three, and added to the sum of the cubes of the parts. The cube of the square-root, multiplied by itself, is the cube of the square number.
Here is an example:
27. Friend, tell me the cube of nine, and the cube of the cube of three, and also the cube of the cube of five, and then the cube-root from the cube, if you [have] a solid understanding of cubes.
Statement: 9, 27, 125. The resulting cubes, in order, are 729, 19683, 1953125.
Or, the quantity is 9, [and] its two parts 4, 5. The quantity is multiplied by both of those (180), multiplied by three (540), and added to the sum of the cubes of the parts (189); the resulting cube is 729.
Or, the quantity is 27, [and] its two parts 20, 7. [It] is multiplied by both of those and multiplied by three (11340), [and] added to the sum of the cubes of the parts (8343); the resulting cube is 19683.
Or, the quantity is 4; its square-root is 2; the cube of that is 8. That, multiplied by itself, is the cube of four, 64.
Or, the quantity is 9; its square-root is 3; the cube of that is 27. The square of that is the cube of 9, 729. The cube of a square number is just the square of the cube of the square-root.
That is the cube.
Now, the rule of operation for the cube-root, in two verses.
28. The first [decimal place] is a cube place; then there are two non-cube [places], and so forth. When one has subtracted from the highest cube [place] the [greatest possible] cube, the [cube] root is put down separately. Divide the next [place] by the square of that [cube-root] multiplied by three;
29. set the result in the "row". Subtract the square of that [quotient] times three, multiplied by the last [digit of the root], from the next [place]; subtract the cube of the quotient from the next [place] after that. [Proceeding] in the same way repeatedly, the "row" becomes the cube-root.
Now the statement of the cubes [given] previously, in order [to find] the [cube-]roots: 729, 19683, 1953125. The roots obtained in order: 9, 27, 125.
That is the cube-root.
These are the eight operations.
Now, the eight operations for fractions.
Now, the rule of operation for reducing fractions to the same denominator, in one verse.
30. The numerator and denominator [of each fraction] are multiplied by the other's denominator: in this way [they are] reduced to the same denominator. Or, both numerator and denominator may be multiplied by [each other's] reduced denominators, by the intelligent [calculator].
Here is an example:
31. Three, one-fifth, one-third: tell [me], friend, [the values of] those [reduced to] a common denominator, in order to add [them]; and also one sixty-third and one-fourteenth, in order to subtract [them].
Statement: 3/1, 1/5, 1/3. Reduced to a common denominator: 45/15, 3/15, 5/15. The sum is 53/15.
Now, the statement in the second example: 1/63, 1/14. [The numerators] are multiplied by [each other's] denominators reduced by seven: 7, 2. Reduced to a common denominator: 2/126, 9/126. The result after subtraction is 7/126, and when reduced by seven, 1/18.
That is reduction to a common denominator.
Now, the rule of operation for fractions of fractions, in half a verse.
32. Numerators are multiplied by numerators, denominators by denominators: [that] is the procedure for simplifying fractions of fractions.
Here is an example:
33. One-fourth of one-sixteenth of one-fifth of three-fourths of two-thirds of one-half of a dramma, wise one, was given to a beggar by the one [he] begged from. Tell me, dear child, how many varatakas were offered by that stingy [one], if you know the reduction procedure in arithmetic for fractions of fractions.
Statement: 1/1, 1/2, 2/3, 3/4, 1/5, 1/16, 1/4. When simplified, the result is 6/7680; reduced by six, the result is 1/1280. So one varataka was given.
That is the reduction of fractions of fractions.
Now, the rule of operation for quantities increased or decreased by a fraction, in one and a half verses.
34. The numerator is added to or subtracted from an integer multiplied by the denominator, [depending] whether the fractional part is positive or negative. If the quantity is to be increased or decreased by a part of itself, multiply [its] denominator by the denominator [of the next fraction] underneath [it], and [multiply] the numerator by the same increased or decreased by [the fraction's] own numerator.
Here is an example:
35. Tell [me], how much is two plus one-fourth, and three minus one-fourth, when simplified, if [you] understand increase and decrease by fractions.
Statement:
| 2 | 3 |
| 1/4 | .1/4 |
Here is an example:
36. How much is one-fourth plus its third part, plus one-fourth of the sum? and how much is two-thirds minus one-eighth of that, minus three-sevenths of the difference? And tell [me], friend, how much is one-half minus one-eighth of it, plus nine-sevenths of the difference, if you know increase and decrease by fractions?
Statement:
| 1/4 | 2/3 | 1/2 |
| 1/3 | .1/8 | .1/8 |
| 1/2 | .3/7 | 9/7 |
Those are the four [rules] of simplification.
Now, the rule of operation for addition and subtraction of fractions, in half a verse.
37. The sum or difference of numerators with a common denominator [produces the sum or difference of the fractions]. The denominator of an [integer] quantity with no denominator is considered [to be] one.
Here is an example:
38. Friend, tell [me] how much one-fifth, one-quarter, one-third, one-half, and one-sixth make when added together. Tell [me] quickly the remainder [from] three minus those fractions.
Statement: 1/5, 1/4, 1/3, 1/2, 1/6. After addition, the result is 29/20.
Now the remainder [from] three diminished by those: 31/20.
That is [the procedure] in addition and subtraction of fractions.
Now, the rule of operation in multiplication of fractions, in half a verse.
39. The product of the numerators is divided by the product of the denominators; the quotient is the result in multiplication of fractions.
Here is an example:
40. What is two and one-seventh multiplied by two and one-third? and tell [me] [how much is] one-half multiplied by one-third, if you are skilled in the procedure for multiplying fractions.
Statement:
| 2 | 2 |
| 1/3 | 1/7 |
Statement: 1/2, 1/3. After multiplication, the result is 1/6.
That is multiplication of fractions.
Now, the rule of operation for division of fractions, in half a verse.
41. In division, after one has inverted the denominator and numerator of the divisor, the rest is to be done [according to] the rule for multiplication.
Here is an example:
42. Tell me [how much is] five divided by two plus one-third, and one-sixth [divided] by one-third, if your understanding, sharp as the point of the sheath of darbha-grass, is adequate for division of fractions.
Statement:
| 2 |
| 1/3 |
That is division of fractions.
Now the rule of operation for squares etc. of fractions, in half a verse.
43. For squaring, the two squares of numerator and denominator---and for cubing, [their] two cubes---are to be given. For determining the root, [find] the roots [of numerator and denominator].
Here is an example:
44. Friend, tell [me] quickly the square of three and one-half, and then the square-root of the square, and the cube [of it], and then the root of the cube, if you know squares and cubes of fractions.
Statement:
| 3 |
| 1/2 |
Those are the eight operations for fractions.
Now, the rule of operation for operations with zero, in two verses.
45. In addition, zero [produces a result] equal to the added [quantity], in squaring and so forth [it produces] zero. A quantity divided by zero has zero as a denominator; [a quantity] multiplied by zero is zero, and [that] latter [result] is [considered] "[that] times zero" in subsequent operations.
46. A [finite] quantity is is understood to be unchanged when zero is [its] multiplier if zero is subsequently [its] divisor, and similarly [if it is] diminished or increased by zero.
Here is an example:
47. Tell [me], what is zero plus five, [and] zero's square, square-root, cube, and cube-root, and five multiplied by zero, and ten minus zero? And what [number], multiplied by zero, added to its own one-half, multiplied by three, and divided by zero, [gives] sixty-three?
Statement: 0. That, added to five, [gives] the result 5. The square of zero is 0; the square-root, 0, the cube, 0; the cube-root, 0.
Statement: 5. That, multiplied by zero, [gives] the result 0.
Statement: 10. That, divided by zero, is 10/0.
[There is] an unknown number whose multiplier is 0. Its own half is added: 1/2. [Its] multiplier is 3, [its] divisor 0. The given [number] is 63. Then, by means of the method of inversion or assumption [of some arbitrary quantity], [which] will be explained [later], the [desired] number is obtained: 14. This calculation is very useful in astronomy.
Those are the eight operations involving zero.