Chinese Approximation of Square Roots
Historical Context:
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When: ca. 100 A.D.
Where: China
What: Chiu Chang or Jiuzhang Suanshu (i.e. Nine Chapters on the
•
Mathematical Art)
Mathematics focus: Demonstrate use of iteration to approximate square roots.
•
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Suggested Readings:
History of square root algorithms:
http://www.speedylook.com/Square_root_(history).html
• Chinese mathematics:
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Chinese_overview.html
or
http://aleph0.clarku.edu/~djoyce/mathhist/china.html
• The Nine Chapters on the Mathematical Art:
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Nine_chapters.html
•
•
•
NCTM’s Historical Topics for the Mathematics Classroom (1969):
“Incommensurables and irrational numbers” (pp. 70-72), “Radical symbol” (pp.
147-148), and “Chinese-Japanese numeration system” (pp. 43-44)
Key search words/phrases: Chinese mathematics, square roots, iteration,
approximation, Chiu Chang, Jiuzhang Suanshu, Nine chapters
Problem to Explore:
Find the square root of a positive number m.
Why This Problem is Important:
•
•
Using a method similar to that of the Greeks, the Chinese algorithmic
approach to extracting approximations to square roots is visual, somewhat
intuitive, and powerful.
The Chinese extraction of square roots is geometrical, yet is based on the
binomial expansion in algebra.
Problem Solving Experiences:
“Shao kuang,” or Chapter Four in the Chiu Chang, includes 24 problems dealing with
areas and side lengths of square tracts of land. The chapter title is appropriate since
shao means “how much” and kuang means “width.” Within this chapter, one finds the
first discussion of Chinese techniques for determining square roots and cube roots.
However, the discussion is quite vague, with the current meaning (or interpretation)
gleaned from the subsequent work of Liu Hui (3rd century) and Yang Hui (13th
century), two Chinese commentators on the Chiu Chang.
For example, one of the chapter’s problems asks: There is a [square] field of area
71824 [square] pu [or paces]. What is the side of the square? Answer: 268 pu.
The text Chiu Chang included only the answer, with no discussion of how it was
determined. Again, our thanks to Liu Hui and Yang Hui, who worked out both the
numerical and geometrical aspects of the calculations involved. [Note: the discussion
that follows is a variation on Joseph’s interpretation (2000).]
To determine 71824 using the Chinese approach, two things must be established
at the start. First, the method (algebraic and geometrical aspects) is based on a
binomial expansion. Given n, suppose we are to find n , where the latter is a 3-digit
number abc. To represent place-values, let α =100a, β = 10b, and γ =c. Using
expanded notation, n = (100a + 10b + c)2 = ( α + β + γ )2.
1. Show that n = α 2 + (2 α + β ) β + [2( α + β ) + γ ] γ , an identity that becomes
both the driving force and the goal for the Chinese extraction of square roots.
Second, counting-rods were laid out in four rows, labeled Shang (result), Shih (given
number), Fang Fa (square element), and Chieh Suan (carrying rod, which fixes the
position of digits during calculations). To determine 71824 , the original lay-out is:
Result
Given number
Square element
Carrying rod
n
1
7
1
8
2
4
1
Step 1: Use approximations to determine the value of α , or the hundred’s
component of the root. We find that α = 200 (or a = 2) works and α = 300 would be
too great since (300)2 = 90000 > 71824. Enter α = 200 in the first “result” row, but
leave the columns blank when a zero is involved. Though non-intuitive, move the
“carrying rod” to “ten-thousands position,” multiply it by a = 2, placing the resulting 2
in the “square element” row above the “carrying rod” to represent the number 20000.
Subtract the product of the “square element” entry and a=2 from the “given number,”
resulting in 71824 – 2(20000) = 31824. The revised lay-out now looks like this:
Result
Given number
Square element
Carrying rod
α
n-α2
α
3
2
1
1
2
8
1
2
4
Step 2: The next action may seem strange at first. Double α to get 2 α = 400, then
enter 2 α as the new “square element” but shifted one cell to the right to represent
4000 (i.e. backwards in place value). Shift the “carrying rod” value of 1 two cells to
the right, basically reflecting the “doubling” effect of squaring on place value (i.e.
102=100). To determine β (or b, the ten’s value), ask what value of b will lead to a
“best” estimate so the product (b)(4b00) ≤ 31824. Try β =60 (and b=6), entering it in
the appropriate cell in the “result” row. The revised lay-out now looks like this:
Result
Given number
Square element
Carrying rod
α +β
n-α2
2α
3
2
8
1
4
1
6
2
4
1
Step 3: Repeat the previous process done with α originally. Multiply b=6 and the 1
in the “carrying rod,” placing it visually in the cell above the “carrying rod,” which
essentially is adding the product to the 2 α in the “square element” row, making it
now 2 α + β . Finally, subtract the product of b and the new “square element” (2 α + β )
from the “given number, or 31824 – (6)(4600) = 4224. The revised lay-out now looks
like this:
Result
Given number
Square element
Carrying rod
α +β
n - α - β (2 α + β )
2α + β
2
1
4
4
2
2
6
1
6
2
4
Step 4: Double α + β to get 2( α + β ) = 5200, then enter 2 α as the new “square
element” but shifted one cell to the right to represent 520. Shift the “carrying rod”
value of 1 two cells to the right. To determine γ (or c, the one’s value), ask what
value of c will lead to a “best” estimate so the product (c)(52c) ≤ 4224. Try c = γ = 8,
entering it in the appropriate cell in the “result” row. The revised lay-out now looks like
this:
Result
α + β +γ
2
6
8
Given number
Square element
Carrying rod
n - ( α + β )2
2( α + β )
1
4
2
5
2
2
4
1
Step 5: Multiply c=8 and the 1 in the “carrying rod,” placing it visually in the cell above
the “carrying rod,” which essentially is adding the product to the 2( α + β ) in the
“square element” row, making it now 2( α + β )+ γ . Finally, subtract the product of c
and the new “square element” 2( α + β )+ γ from the “given number, or 4224 –
(8)(528) = 0. The final lay-out is:
Result
Given number
Square element
Carrying rod
α + β +γ
n - ( α + β )2-[2( α + β )+ γ ] γ
2( α + β )+ γ
2
6
8
5
2
8
1
1
A single visual can represent the Chinese extraction of square roots. For the given
example to extract 71824 , the associated visual is:
60
200
200
40000
8
12000
2080
60
8
12000
3600
2080
64
2. Carefully study the visual, comparing it to actions and computational results in
each of the five steps in the extraction process. What are the visual
interpretations of:
• n - α 2?
• (2 α + β ) β ?
• n - α 2- β (2 α + β )?
• n - ( α + β )2?
• [2( α + β )+ γ ] γ ?
• ( α + β + γ )2?
Extension and Reflection Questions:
Extension 1: Use the Chinese approach to extract the root 315844 . Draw the
associated visual as well.
Extension 2: Ch'in Chiu-shao
(ca.1202–1261) was a Chinese mathematician who approximated square roots
using a method called “adding a fraction to the whole part of the root.” For nonsquare numbers n, the first step is to determine the largest possible square a2 < n,
then write n = a2+r for some “remainder” term r. Then, Chiu-shao used these
inequalities to approximate n :
r
r
≤ a2 + r ≤ a +
a+
2a + 1
2a
Use this method to approximate these roots: 15 , 17 , 200 , 71800 , and
315800 . Does the method seem to work better for small or large numbers? Also,
what happens if you choose a square a2 that is not the “largest possible,” such as
rewriting 15 = 22 + 11 or 17 = 32 + 8 = 12 + 16?
Open-ended Exploration: The examples of the Chinese algorithm for extracting
square roots always involved perfect squares (i.e. the last step had a remainder of 0).
Will it work to extract approximations of square roots for non-perfect square
numbers? Also, can you adjust it to extract cube roots of positive numbers? Nth roots
of positive numbers?
Joseph, G.C. (2000). The Crest of the Peacock: Non-European Roots of
Mathematics. Princeton University Press