history of mathematics - National STEM Centre

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For example, if an equation contains the expression a - 2b, you could consider it as
a + (-2)b, where -2, the coefficient of b, is a negative number.
However the Babylonian, Chinese and Indian mathematicians viewed a-2b as 2b
being subtracted from a. If their equations contained a-2b, they were only
prepared to consider those solutions in which a-2b was positive.
Activity 10.3 A Chinese livestock problem
Examples of equations with subtracted numbers are to be found in a Chinese text
dating back to about the 3rd century BC, called the Chiu-chang suan-shu, or the
'Nine chapters on the mathematical arts'.
One example is from a problem involving payment for livestock. Selling 2 sheep
and 5 pigs while buying 13 cows leaves a debt of 580 pieces of money; selling 3
cows and 3 pigs while buying 9 sheep leaves no money at all; and buying 5 cows
but selling 6 sheep and 8 pigs leaves 290 pieces.
For solving problems that involved what you would call simultaneous equations, the
Chinese used algorithms carried out on a counting board with coloured rods: red for
added numbers and black for subtracted ones. On the counting board the problem
would be set out as
-5 3 -13 cows
6-92 sheep
835 pigs
290 -580 pieces
Activity 10.4 The Chinese rules
1 Set out the Chinese livestock problem as a system of simultaneous equations and
solve it using the method you developed in Chapter 5, in the Equations and
inequalities unit in Book 3.
Ways of combining negative numbers
The Chiu-chang suan-shu gives rules for adding and subtracting positive and
negative numbers. However, the rules are somewhat indirect because signs for the
numbers were not used.
This activity is optional.
Here is an extract from the Chiu-chang suan-shu.
When the equally signed quantities are to be subtracted and the different
signed are to be added (in their absolute values), if a positive quantity has
no opponent, make it negative; and if a negative has no opponent make it
positive. When the different signed quantities are to be subtracted and the