history of mathematics - National STEM Centre

10 'Two minuses make a plus'
equally signed are to be added (in absolute values), if a positive quantity
has no opponent, make it positive; and if a negative has no opponent, make
it negative.
1 Translate the rules in the extract for the treatment of positive and negative
numbers into rules which your fellow students would understand. Use whatever
means you like to explain these rules - words, pictures, tables, and so on - but keep
the idea of the counting board and the way that the Chinese used it.
Although the Chinese probably provided the earliest known example of the use of
negative numbers, and as early as the 3rd century BC had rules for adding and
subtracting them, they were not the first to generate the rules for multiplying and
dividing negative numbers. In China, these rules did not appear until 1299, in a
work called the Suan-hsiao chi-meng.
The first known setting out of the rules for multiplying and dividing negative
numbers was in the 7th century AD. The Indian Brahmagupta wrote down these rules
which are instantly recognisable.
Positive divided by positive, or negative by negative is affirmative ....
Positive divided by negative is negative. Negative divided by affirmative is
negative.
Unlike those before him, Brahmagupta was prepared to consider negative numbers
as 'proper', valid solutions of equations.
Activity 10.5 A geometrical argument
You first learned about
al-Khwarizmi in The
Arabs unit in Chapter 5.
Figure 10.1
The rules presented by Brahmagupta had already been considered by the Greeks
some four centuries earlier. Theirs was a geometrical argument, so differences in
distances and areas were used and were represented as subtracted numbers. The
geometrical setting is presented here in a slightly different form from the way it was
set out originally in Book II of Euclid's Elements. It is as argued by al-Khwarizmi in
the 9th century, and it should look familiar to you from your previous work on
Chapter 2 of the Investigating and proving unit in Book 4.
1 Explain carefully in words how Figure 10.1 enables you to conjecture the
algebraic formula
(a - b)(c -d) = ac -bc — ad + bd.
In the algebraic expression that you obtained in Activity 10.5, the values of a, b, c
and d are all positive and are such that a > b and c > d, so you used positive
numbers throughout. There are no negative numbers present, only subtracted
numbers. Even if you subtracted some of these positive numbers, you made no use
of negative numbers standing on their own. Also, although you can very easily use
this expression to get a rule for multiplying negative numbers, this was not done at
the time, since negative numbers were assumed not to exist.
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