history of mathematics - National STEM Centre

y i
Activity 5.7
Figure 5.6a
y'
Figure 5.6b
X\ XT.
5 Arab mathematics
3 Write notes on the accuracy of this method. For example, when is the
approximation good, and when is it bad?
The rule of 'double false'
In the Chinese classic, Chui-chang suan-shu (Nine chapters on the mathematical
arts), from the beginning of the Christian era, is the following problem.
A tub of full capacity lOtou contains a certain quantity of coarse, that is,
husked, rice. Grains, that is, unhusked rice, are added to fill the tub. When
the grains are husked it is found that the tub contains 7 tou of coarse rice
altogether. Find the original amount of rice in the tub.
Assume that 1 tou of grains yields 6 sheng of coarse rice, where 1 tou is equal to 10
sheng and 1 sheng is approximately 0.2 litres.
The suggested solution is as follows.
If the original amount of rice in the tub is 2 tou, a shortage of 2 sheng
occurs; if the original amount of rice is 3 tou, there is an excess of 2 sheng.
Cross-multiply 2 tou by the surplus 2 sheng, and then cross-multiply 3 tou
by the deficiency of 2 sheng, and add the two products to give 10 tou.
Divide this sum, that is 10, by the sum of the surplus and deficiency to get
the answer: 2 tou and 5 sheng.
1 Take x { and x2 to be the preliminary, incorrect, guesses of the answer, x. Take
e\ and e2 to be the errors: that is, the shortage and surplus, arising from these
guesses. Express in symbolic form the algorithm used in the solution quoted.
Questions 2 and 3 interpret this solution geometrically, in linear and other forms.
2 The linear case: In Figure 5.6a, x t and x 2 are incorrect guesses for x, and e } and
ei are the consequent errors. Derive the expression obtained for.v in question 1.
3 The non-linear case: In Figure 5.6b, let jc, and x2 be numbers which lie close to
and on each side of a root x of the equation f (x) = 0. You can find an approximate
solution for the root x in two stages.
a Apply the formula that you obtained in question 1 to find x$.
b Repeat the process, choosing the appropriate pair, x } , x 3 or x3 , x 2 so that one is
on either side of the solution of the equation. Use this method to find the
approximate root which lies in the range between 1 and 2, correct to two places of
decimals, of the equation x' - x — 1 = 0.
This method was called the rule of double false. In the linear case the solution for x
is unique and independent of the choice of x { and x 2 if jc, and x2 are distinct. In the
non-linear case, the solution is approximate. If you apply the method iteratively, as
in the example above, a closer and closer solution may be found. The number of
iterations needed depends on the accuracy of your initial approximations!
The rule of double false provides one of the oldest methods of approximating to the
real roots of an equation. Variations of it are found in Babylonian mathematics, in
Greek mathematics in Heron's method for the extraction of a square root, and in an
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