Chinese Extraction of Cube Roots 1

Paul Yiu

December, 1996

As I have been enjoying for the most part of the year many of the enlighten- ing discussions in this list on various topics in the history of mathematics, I would like to respond to several recent postings on ancient Chinese methods of extraction of square and cube roots. I am not a historian of mathematics, nor a Chinese classicist, but a student of mathematics keenly interested in the pedagogical insights in the Chinese classics Jiuzhang Suanshu, especially its commentary by LIU Hui in the third century. So, please take this series of postings as an amateur’s ﬁnding prompted by interesting questions posed recently in this list. I am sure most of these are well known among historians of mathematics.

In his posting on December 6, Professor J. Gonzalez Cabillon quoted two passages in the works of Professor LAM Lay Yong of Singapore:

The Chinese also developed a similar method to find the cube root of a number and in so doing were able to solve a cubic equation of the form x3 + ax2 + bx = c,wherea, b and c are positive. Through the method of finding a side of a square or cube with the rod numeral system, the Chinese invented a notation to represent quadratic and cubic equations and were able to solve numerically aparticular2 type of such equations. One could say that at this stage there were indications that arithmetic was forging into the yet to be known field of algebra. Arch. Hist. Exact Sci., 47 (1) 1994, 1-51.

The ancient Chinese performed the four fundamental operations on fractions and the extractions of square and cube roots as

1Minor revision of 5 postings to the Math-History-List, Mathematical Association of America, December 14, 15, 1996. 2Underscore mine.

1 YIU: Chinese extraction of cube roots 2

easily as we perform these operations using our numeral system. More important, in spite of the two thousand years’ gap and the diﬀerence in media, the procedures that they used bear parallel resemblance to the ones we use. [...] Arch. Hist. Exact Sci., 37 (4) 1987, 365 - 392.

My library here does not have Arch. Hist. Exact Sci., nor have I seen Professor LAM’s book (with ANG Tian Se) Fleeting Footsteps, Tracing the Conception of Arithmetic and Algebra in Ancient China. 3 But I happened to have a copy of her 1994 paper on an overview of Jiuzhang Suanshu (Nine Chapters of the Mathematical Art). To me, these paragraphs are terse summaries of a long tradition of Chinese mathematics. In the first passage I underline the words numerically and particular. It should not be taken to mean that ancient Chinese knew how to solve cubic equations in radicals. It means that the principles and methods of extraction of square roots and cube roots as explained in Jiuzhang Suanshu have gradually evolved into a method of solving polynomial equations, a method known as zeng cheng kai fang (extraction of roots by addition and multiplication) in the Song period (9th – 11th century), finally capable of dealing with equations of arbitrarily high degrees and coefficients positive, negative, and zero. Such numerical methods are indeed spigot algorithms which give a positive root of a polynomial equation digit by digit. Here, I am borrowing the term ‘spigot algorithm’ from an article of Stan Rabinowitz and Stan Wagon in the American Mathematical Monthly. 4 A spigot algorithm “pumps out digits one at a time and does not use the digits after they are computed”. Professor SIU Man Keung has explained the extraction of square root in an early posting 5 In practice, each of the digits is determined by deliberation. In Jiuzhang Suanshu and other Chinese texts, the digits obtained in this way are called ‘shang’, the same character for the word ‘quotient’. This character shang (deliberation) captures the essence of the division process, (unless one insists on performing division by repeated subtraction). In a posting on November 28, SIU wrote

If by ”trial and error” one means to ﬁrst guess an answer and see how good or bad it is and then proceed to the next step, then I agree that the ancient Chinese method for calculating square roots, cube roots, etc. is of that category. But if by ”trial

3World Scientiﬁc, Singapore, 1992. 4vol. 102 (1995) 195 – 203. 5November . YIU: Chinese extraction of cube roots 3

and error” one means that it is just empirical guesswork without theory behind it, then I cannot agree that the ancient Chinese method is of that category.

Then he went on to explain the geometry given in LIU Hui’s commentary of the method.

In a paper (written in Chinese) on the constructive and mechanistic aspects of traditional Chinese mathematics, 6 Professor WU Wentsun has incorporated the method of zeng cheng kai fang into a computer programme for the determination of the real roots of a cubic equation of the form

3 2 x + a1x + a2x = a3 with positive a1, a2,anda3. WU remarks that “the principle of such a method [like the extraction of square root] is self evident from the procedures, and that it is precise to any desired degree of accuracy”.

Perhaps it is appropriate to read a passage by Joseph Needham on this subject. In his Science and Civilisation in China, 7 Needham wrote: 8

The solution of numerical higher equations for approximate val- ues of roots begins, as far as we know, in China. It has been called the most characteristic Chinese mathematical contribu- tion. That it was well developed in the work of the Sung al- gebraists has long been known, but it is possible to show that if the text of the Han Jiuzhang [Suanshu], (Nine Chapter of the Mathematical Art) is very carefully followed, the essentials of the method are already there at a time which may be dated as of the −1 century. ...

Among the problems in the Jiuzhang, one, the ﬁnding of a cube root, may be written x3 =1, 860, 867, and another, involving thesquareandthesimpleterm,x2 +34x +71, 000 = 0. Natu- rally, these equations were not written as such, but set up with counting rods on a board. The uppermost line in the table is to contain the radical solution sought, the second line (shi)is

6QIN Jiushao and Shushu Jiuzhang, pp.73 – 87, Beijing, 1987. 7volume III (1959), Chapter 19 on Mathematics, pp. 126 ﬀ. 8I have replaced Needham’s transliterations of Chinese characters by Pinyin. Again, emphasis mine. YIU: Chinese extraction of cube roots 4

the constant term before the operation, the third (fa,orlaterin the operation, ting fa) is for numbers obtained in its successive stages, the fourth (zhong) receives another number found at an intermediate state, and the ﬁfth (jie suan) is the coeﬃcient of x3 before the operation.

unnamed line for the root 1860867 shi (dividend) fa (divisor) zhong (middle) jie suan (borrowed rod)

The process starts by raising the 1 in the jie suan line to 106 so that 3 1, 000, 000x1 − 1, 860, 867 = 0. After this the text, not at all explicit, uses the word yi, ‘discussion’, which must mean the choosing of a ﬁrst root ﬁgure a (e.g. 1). This is then inserted in its proper place on the top line. Then follows a set of operations by which the numbers on the counting - board are converted into a pattern corresponding to Horner’s transformed equation:

3 2 1000x2 +30, 000x2 + 300, 000x2 − 860, 867 = 0,

Here x2 = 10(x1 − a). With the transformed equation, ‘discussion’ (yi) again takes place, and the next digit of the solution is inserted. The same process is repeated (with modiﬁcation) to ﬁnd the third digit, at which point the Jiuzhang procedure ends. In this particular case, the answer was 123. LIU Hui in the +3rd century, however, pointed out that the process could be continued past the decimal point (as we should say) as long as required. The Jiuzhang author himself seems to have carried square-root extraction to one decimal place.

The system can be found in all the later mathematical books with terminological variations, and minor changes in the steps, gen- erally not improvements. Numerical equations of degrees higher than the third occur ﬁrst in the work of QIN Jiushao around +1245. He deals very clearly with equations such as YIU: Chinese extraction of cube roots 5

−x4 + 763, 200x2 − 40, 642, 560, 000 = 0.

... In considering the long lead of Chinese mathematics in this ﬁeld, one may suggest that the counting - board, with its lines repre- senting increasing powers, was particularly convenient for the purpose, though of course no attempt was made, even in the Song, to produce a generalized theory of these equations.

This last paragraph in Needham’s naturally brings me back to the second passage of LAM’s cited by Professor Gonzalez Cabillon. 9 But I would like to ﬁrst address Professor Domingo Gomez Morin’s question on December 12:

[A]part from trivial examples on perfect cubes, I think it would be equally easy to show here many numerical examples on ancient Chinese approximations to the solution of cube roots. In this way, could anyone bring me any ancient Chinese approximation to the solution of cube roots, as for example, the cube root of 2?”

My resources in Chinese literature is extremely limited. I am sure other scholars more knowledgeable and resourceful in Chinese mathematics can help. But here I shall report on my findings. Ididnot expect to be able to find informations on the cube root of 2. Apparently, the ancient Chinese have not been particularly interested in the duplication of the cube. However, I was surprised to find that all examples on cube roots in Jiuzhang Suanshu are perfect cubes, (which Professor Gomez has ruled out), or round off to a simple fraction “after the decimal point”! The methods in the texts of Jiuzhang tells of rounding off square roots (respectively cube roots) of numbers which are not perfect squares (respectively cubes) in the form of fractions. But in his commentary on the extraction of square roots, LIU Hui indicated that in such cases, the process could be continued as long as required. (See Needham’s passage quoted above).

9See §1. YIU: Chinese extraction of cube roots 6

Then I looked up Shushu Jiuzhang and Ze Yuan Hai Jing (texts in the Song and Yuan periods containing an abundant number of problems involving polynomial equations), a few modern textbooks and collections of essays on the history of Chinese mathematics. On the topics on square roots, cube roots, and the numerical solution of polynomial equations using the method of zeng cheng kai fang, I could only find examples with simple, integer answers, or roots round off to a simple fraction “after the decimal point”. In thecaseofZe Yuan Hai Jing, a text on some 170 problems on right triangles with an inscribed circle where cubic and higher degree equations frequently turn up, the author has designed the problems to admit simple integer so- lutions. It may not be relevant to the present discussion on the extraction of cube root 2. But it is good pedagogical principle to write exercises (and especially examinations!) to illustrate mathematical methods with calcula- tions leading to simple answers. It is along this line, for example, that the question of finding Pythagorean triples naturally turned up. And Chinese authors from ancient times have consistently adhered to this sound principle. Sometimes it may not be at all easy, or even impossible, to do this; such endeavours may lead to nontrivial “Diophantine” problems).

Let us agree for a moment that the methods of extraction of square roots and of cube roots are essentially the same. 10 Here is evidence that ancient Chinese mathematicians have carried out the extraction of square roots “long after the decimal point”. In his commentary on the method of computing the area of a circular field, 11 LIU Hui explains his famous method ge yuan, (dissecting the circle) of computing π by approximating the area of a circle successively with regular polygons of 6, 12, 24, 48, 96, and 192 sides. Beginning with a regular hexagon in a circle of radius 1 che (= 10 cun), each side of which has length 10 cun, he expressed the length of a side of an inscribed regular 2n−gon in termsofthatofaninscribedregularn−gon. In this way, LIU obtained π 64 169 correct to 2 places after the decimal point, as between 3.14 625 and 3.14 625 . Here, LIU had to extract a large number of square roots to obtain the sides of the regular polygon recursively.√ The simplest square root LIU encountered 2 in the first step was 75, which he gave as 8.66025 5 . √ This means that he computed 75 by starting with the digit 8, and apply the ‘standard’ procedure 5 times, and then round off with a fraction 2 5 . In this calculation, LIU calculated all numbers involved to 6 significant

10See §4 on the extraction of cube root using counting rods. 11Jiuzhang Suanshu, Chapter 1, following Problem 32. YIU: Chinese extraction of cube roots 7

ﬁgures, round oﬀ at the end with a simple fraction. This is an illustration of what LIU commented on the method of extracting square roots, that the process could be continued as long as required”. 12

A translation (by Professors K. Chemla and GUO Shuchen) of the method of extraction of square roots can be found in Professor Fowler’s posting on December 5. As Needham, LAM, and others have pointed out, this is essentially the same as what we are using today. See Professor SIU’s posting on November 21. The same remarks applies to the extraction of cube roots.

In an attempt to answer Professor Domingo Gomez Morin’s question on “ancient Chinese approximation to the solution of cube roots, as for example, the cube root of 2”, my digression in §2 on LIU Hui’s determination of π leads to something unexpected to me. (To historians or more knowledgeable scholars, such may have been well known). The ancient Chinese, including 9 the author of Jiuzhang, took the volume of a sphere to be 16 times the cube of diameter. In Chapter 4 (Shaoguang)ofJiuzhang Suanshu, there were two problems on the volume of a sphere.

Problem IV.23: There is a sphere of volume 4500 [cubic] feet. What is the diameter of the sphere? Answer: 20 feet. Problem IV.24: There is another sphere of volume 1,644,866,437,500 [cubic] feet. What is the diameter of the sphere? Answer: 14300 feet.

As to the method, the text explains:

Multiply the volume by 16, and take one nineth. Extract the cube root to ﬁnd the diameter of the sphere.

Note that here the author of Jiuzhang has designed the problems to admit simple answers. While the first one was really trivial, the second one required two steps of the cube root process to get the answer. LIU Hui, however, was not satisfied with this formula for the volume of a sphere. In his brilliant commentary on this method, he explained the flaw

12See the passage of Needham’s at the end §1. YIU: Chinese extraction of cube roots 8 of the traditional argument for to this formula: for a sphere is inscribed in a cylinder, which in turn is inscribed in a cube,

(1) volume of cube : volume of cylinder = 4 : π; (2) volume of cylinder : volume of sphere = 4 : π.

From these it would follow that

π2 (3) volume of sphere = 16 times cube of diameter.

The method in Jiuzhang corresponds to taking π = 3 in this formula. Immediately following this brief review, LIU Hui considered the solid mu he fang gai and demonstrated ingeniously that instead of (2), it should be

volume of mu he fang gai : volume of sphere = 4 : π.

But he was not able to determine precisely the volume of mu he fang gai. It was left to ZU Chongzhi (5th century) to complete the task to obtain the precise formula

π (4) volume of sphere = 6 times cube of diameter.

Now, following the answers to Problems IV.23, 24 are the following commentaries. IV.23: With the ﬁne ratio (mi lu), a sphere of diameter 20 has volume 10 4190 21 [cubic] feet.

But the next one is much more interesting:

3 IV.24: With the ﬁne ratio (mi lu), the diameter is 14643 4 feet.

I have two questions here. (i) Whose commentaries are these? (ii) Which formula is being used for the volume of a sphere here?

The commentary on IV.23 gives the volume of a sphere of diameter 20 10 using a more accurate value for π. From 4190 21 it is easy to see that this came from the precise formula (4) (not obtained by LIU Hui) with the familar 22 approximation π = 7 (not used by LIU Hui). (If this commentary were 157 given by LIU Hui with his “Hui lu” 3.14 or 50 using the formula (3) (known 1 to him to be incorrect), the answer would have been 4743 5 ). Furthermore, YIU: Chinese extraction of cube roots 9 in his commentaries on earlier Problems IV.17, 18 on areas of circles, LIU Hui made corrections begining with the saying “with Hui’s method”. In each of these items, following LIU Hui’s commentary, we find annotations by LI Chunfeng (in the Tang Dynasty) beginning the words “with mi lu” (the fine ratio). We conclude that the corrections to the answers of IV.23,24 above were made by LI using the precise formula (4) obtained by ZU Chongzhi in 22 the fifth century. Incidently, the approximation 7 is referred to here as mi lu (the fine ratio). It is well known, however, that ZU gave this as shu lu 355 (the crude ratio), and 113 as mi lu (the fine ratio).

22 Now, with π = 7 and V = 1,644,866,437,500, and using the correct formula (4) for the volume of a sphere, the commentary to Problem IV.24 3 gave 14643 4 for the diameter of the sphere. This calculation made use of the cube root method, beginning with the ﬁrst digit 1, applying the process four times, and rounding oﬀ with a simple fraction.

This, I believe, is an example of ancient Chinese approximation of cube roots for numbers which are not perfect cubes. It√ did not go beyond the decimal point. But together with the example on 75 in §2, it is clear that LIU and his contemporaries had no trouble going beyond the decimal point “as long as required”. This, of course, presumes a strong resemblence of the square root and cube root process on the counting board.

In this section we consider the extraction of cube roots on the counting board. Professor SIU has described an algorithm in an earlier posting: 13

[...] A similar algorithm goes for the cube root. Start with say, 1860867. You group the digits by threes from right to left — 1 860 867. Make an estimate of some A such that A3 does not exceed 1, in this case take A = 1. Subtract 1 × 106 from the number to obtain 860 867. Make the next estimate of some B such that (3A2 × 102 +3A × B × 10 + B2)B does not exceed 860, in this case take B = 2 ( because 364 × 2 = 728 while 369 × 3= 1107). Continue in a similar fashion. In this particular case, the third estimate gives C=3 and the procedure ends, yielding the answer 123. In general, you can calculate with an accuracy

13November 21, 1996. YIU: Chinese extraction of cube roots 10

to within any decimal places. The mathematical basis is the identity

(10A + B)3 = 1000A3 + (300A2 +30AB + B2)B.

I review this for a comparison of the method on the counting board (given in Jiuzhang Suanshu) described below. We can then see that how remarkably similar they are.

Allow me to make a personal recollection here. I learned the same method as a middle school student in Hong Kong in the 60’s. In those days, we just began to hear of calculators but hardly saw one; we used four-ﬁgure tables and, in more senior years, also the slide rule. But I still remember the sheer joy of being taught how to calculate cube roots by hand, precisely, to as many digits as we like. It was something superior to using logarithm tables. I still have my arithmetic textbook with me, the famous (then in our society) C.V.Durell’s General Arithmetic for Schools. 14 To my surprise, I cannot ﬁnd the cube root method there! It must have been out of my teacher’s enthusiasm to teach us this method.

Now, a method of extracting cube roots is explained in Jiuzhang Suan- shu, following Problems IV.21,22. I am aware of the diﬃculties involved in the exegesis of this passage. The description may have been imprecise; but from LIU Hui’s commentary one can ﬁgure out the method. My translation below is only tentative; but I believe that the method described below is faithful to Jiuzhang.

The extraction of cube root is carried out on a counting board, in five rows. The first row is for the root. (See the arrangement in Needham’s passage, quoted in §1. Whether it is nearest to or farthest from the ‘cal- culator’ I do not know, but perhaps unimportant. In the second row one puts down the digits of the dividend (the number whose cube root is to be extracted). The third and fourth rows are left blank in the beginning. They are respectively referred to as the divisor and the middle number. In the fifth (and bottom) row, under the units digit of the dividend is a borrowed rod (jie suan).

The method begins by moving the borrowed rod to the left, jumping two places at a time. This has the eﬀect of dividing the digits of the dividend into blocks of threes. 14Bell, London, 1936. YIU: Chinese extraction of cube roots 11

Suppose the number is not the cube of one of the digits 1, 2, ... 9. (The problem would then be trivial). Figure out the ﬁrst quotient (digit of the root).

Multiply it twice to the number represented by the borrowed rod; use this number as divisor (fa) to perform division in the ﬁrst block. Treble this and leave it as “ﬁxed divisor” (ding fa,inthe third row). Once more, divide this divisor by the quotient and leave the result in the next row, treble it as the middle number; again, put a borrowed rod in the bottom row.

All these have the effect of (i) removing the cube of the quotient from the first block, (this first quotient is so chosen that its cube is just no greater than the three - digit number in the leftmost block), (ii) putting 3 times the first quotient in the fourth row as “the middle number”, (iii) putting 3 times the square of the first quotient in the third row as divisor, (iv) leaving the borrowed rod in the bottom row. 15

Now, shift the ﬁxed divisor one place to the right, the middle number two places, and the borrowed row in the bottom row three places.

17 Let us illustrate this with the cube root of 1, 937, 541 27 ,asinProblem IV.22 of Jiuzhang. Ignoring the fractional part, we should have 1 for the ﬁrst quotient, and following the method up to now,

1quotient 937541 dividend 3 ﬁxed divisor 3 middle number 1 bottom number

Figure out the quotient again; multiply it to the middle number; multiply it twice to the bottom number; add these (two) to the ﬁxed divisor. 15I cannot ﬁgure out how the original borrowed rod may have disappeared. But in the end, there can only be one borrowed row in the bottom row. YIU: Chinese extraction of cube roots 12

12 quotient 937541 dividend 3 6 4 ﬁxed divisor 6 middle number 4 bottom number

Perform division with the ﬁxed divisor. After the division, dou- ble the bottom number, add this, together with the middle number, to the ﬁxed divisor.

12 quotient 209541 dividend 4 3 2 ﬁxed divisor 6 middle number 4 bottom number

Divide again (the ﬁxed divisor) by the quotient (not just the digit, but the number in the top row), and put the answer in the next row.

12 quotient 209541 dividend 4 3 2 ﬁxed divisor 3 6 middle number 1 bottom number

Shift the ﬁxed divisor one place to the right, the middle number two places, and the bottom number three places.

12 quotient 209541 dividend 4 3 2 ﬁxed divisor 3 6 middle number 1 bottom number

At this point, the text indicates that “if the division does not terminate, then the cube root cannot be extracted precisely. It is intended, of course, that the above process (of choosing a digit and adjusting the bottom, middle YIU: Chinese extraction of cube roots 13 numbers and the ﬁxed divisors) be carried out up to the block containing the units digit. Then it deals the case when the dividend has a fractional part, as Professor SIU has explained in his posting yesterday. 16 As pointed out before, LIU Hui has remarked (in his commentary on the square root process) that this can be continued as long as required.

In the present example, the next quotient is 4:

124 quotient 209541 dividend 4 3 3 fixed divisor 1 4 4 middle number 1 6 bottom number 124 quotient 209541 dividend 4 4 6 5 6 fixed divisor 1 4 4 middle number 1 6 bottom number 124 quotient 30917 dividend 46128 fixeddivisor 3 7 2 middle number 1 bottom number To work beyond the decimal point, one would shift the numbers in the three bottom rows as follows, and choose 6 for the next quotient:

12 4 quotient 30917, 0 0 0 dividend 4612 8 ﬁxeddivisor 3 7 2 middle number 1 bottom number But perhaps this is enough for an illustration of the process.

In his posting, Professor SIU remarked

It is quite diﬃcult for us to imagine how to calculate “easily” usingcountingrods,forwearenowsousedtocalculatingwith numerals. 16December 12, 1996. YIU: Chinese extraction of cube roots 14

Andwithcalculatorsandcomputers!

A Chinese saying goes shu neng sheng qiao (skill comes from practice). I remember as school students, my peers and I were often amazed at the speed and accuracy our (mostly) illiterate fathers performed multiplications and long divisions with the abacus. The method of extracting cube root in Jiuzhang is indeed quite tedious. In the next posting, I shall outline a modifification of the same process by YANG Hui in the 13th century, which basically consists of only one type of operation performed somewhat “symmetrically” to the bottom three rows. It is a lot easier to master. With this, I suggest one works out a few examples with coins or chips (in place of counting rods). It is then easier to appreciate that the counting board is at least as efficient as paper and pencil. Of course, no comparison with calculators or computers is intended!

Here is a method of ﬁnding cube roots explained by YANG Hui in the 13th century. It is called zeng cheng kai li fang fa (method of extraction of cube roots by additions and multiplications), and is a modiﬁcation of the method in Jiuzhang Suanshu.

The ﬁve rows of the counting board are named slightly diﬀerently from the text of Jiuzhang, (but in line with the terms used by LIU Hui):

(1) the top row shang (quotient), (2) the second row shi (dividend), (3) the third row fang (square) (4) the fourth row lian (rod) (5) the bottom row xia (bottom).

One begins by shifting a jie suan (borrowed rod) in the bottom as in the Jiuzhang method. The borrowed rod ends up in the same column as the units digit in the leftmost block of the dividend. At this point, the fang and lian are empty.

Choose a ﬁrst quotient whose cube does not exceed the 3-digit number in the ﬁrst block. Multiply the quotient to the bottom number (which is always 1) and add to lian (the 4th row); YIU: Chinese extraction of cube roots 15

Multiply the quotient to lian and add to fang (the 3rd row)...... Multiply the quotient to fang (3rd row) and subtract it from the dividend. (If this diﬀerence is zero, the process stops). Multiply the quotient to the bottom number and add to lian (4th row); multiply the quotient to the lian (4th row) and add to fang (3rd row); again, multiply the quotient to the bottom number and add to lian (4th row)...... Shift fang (3rd row) one place to the right, lian (4th row) two places, and the bottom number three places. Repeat the process by choosing the next quotient.

Let’s illustrate this with the cube root of 3,140,199,562,500, the one in LI Chenfeng’s emendation of Problem IV.24 (diameter of a sphere of given volume) mentioned in §3.

With the ﬁrst quotient 1, we follow the steps and get to

1 2 140199562500 3 3 1 The second quotient should be 4:

14 2 140199562500 436 34 1 300 + 4 × 34 = 436; 2140 − 4 × 436 = 396.

14 396199562500 588 42 1 YIU: Chinese extraction of cube roots 16

436 + 4 × 38 = 588.

14 396199562500 588 4 2 1 ************

14 6 396199562500 61356 426 1 58, 800 + 6 × 426 = 61, 356; 396, 199 − 6 × 61, 356 = 28, 063.

146 28063562500 63948 438 1 61, 356 + 6 × 432 = 63, 948.

146 28063562500 6 3948 4 38 1 ************

1464 28063562500 6 412336 4 384 1 63, 94800 + 4 × 4384 = 6, 412, 336; 28, 063, 562 − 4 × 6, 412, 336 = 2, 414, 218. YIU: Chinese extraction of cube roots 17

1464 2 414218500 6 429888 4 392 1 6, 412, 336 + 4 × 4, 388 = 6, 429, 888.

1464 2 414218500 6429888 4392 1 ************

14643 2 414218500 643120569 43923 1 642, 988, 800 + 3 × 43, 923 = 643, 120, 569 2, 414, 218, 500 − 3 × 643, 120, 569 = 484, 856, 793.

14643 484856793 643252347 43929 1 If we want to go beyond the decimal point, the next digit should be 7:

1464 3 484856793. 000 6 43252 347 43 929 1 ******************* This method is easier because, (i) the bottom number always remains 1; YIU: Chinese extraction of cube roots 18

(ii) apart from shifting, all operations are of the same kind: multiplying the quotient to either the bottom number or lian (4th row) and adding to the previous row.

Here is a simple scheme to memorize the steps in each cycle:

choose quotient, (5)-(4), (4)-(3), divide, (5)-(4), (4)-(3), (5)-(4), shift.

Here “divide” means subtracting (quotient times fang (3rd line)) from the dividend (2nd line). The process terminates when division is without remainder.

Since the multiplications involved are only by single digit numbers, these operations can be done much more quickly with counting rods than we might have imagined. As suggested in the previous section, one can play this game with chips or coins. Perhaps it is a good exercise to try it on the cube root of 2. Then, it is easier to appreciate how eﬃcient this was in those days when Chinese people wrote with brushes, or even compared with paper and pencil (that we were obliged to use not long ago)! Today, such techniques of extracting square and cube roots properly belong to history. However, it is the algorithmic nature, and the simple geometry behind the method makes us marvel at the genius of mathematicians in the distant past.

Department of Mathematics Florida Atlantic University Boca Raton, FL 33431-0991

Email: [email protected]