history of mathematics - National STEM Centre

Activity 10.7, page 129
13 Hints
1 Make a special choice of the numbers a and c which reduces the equation to one
which just has two negative numbers multiplied together on the left-hand side.
Having made this choice, substitute it in both sides of al-Khwarizmi's equation to
obtain the rule for multiplying negative numbers together.
2 Follow the same sort of argument here as applied in question 1 except that you
are not necessarily restricting yourself to choices of a and c. Remember that division
by a number e is the same as the operation of multiplication with — .
e
Activity 10.8, page 129
1 Think of any single significant mathematical development during that period
which might be occupying the minds of mathematicians. You could look through
the other units in this book. Does that give you any ideas?
Activity 10.9, page 130
1 Start by making a brief summary of the passage.
Activity 10.10, page 131
1 a Here is the translation of Arnauld's extract.
I do not understand why the square of -5 is the same as the square of + 5 , and that
they are both 25. Neither do I know how to reconcile that with the basis of the
multiplication of two numbers which requires that unity is to the one of the numbers
what the other is to the product. This is both for whole numbers and for fractions.
For 1 is to 3, what 4 is to 12. And 1 is to ^ what \ is to -^ . But I cannot reconcile
that with the multiplication of two negatives. For can one say that +1 is to -4 , as
- 5 is to +20? I do not see it. For +1 is more than -4 . And by contrast - 5 is less
than +20. Whereas in all other propositions if the first term is larger than the second,
the third must be larger than the fourth.
Activity 10.11, page 133
1 a Summarise both extracts before you try to list the assumptions which each
author makes. Here are notes on Euler's extract as an example:
• no doubt that positive multiplied by positive is positive, but separately examines
positive by negative and negative by negative;
• negative numbers modelled as a debt; multiplying a debt by a positive number
bigger than 1 gives a larger debt; thus -a times b and a times —b is —ab;
• -a times —b must be either —ab or ab. It cannot be —ab since -a times b gives
that and a times -b cannot give same result; so -a times -b must be ab .
By looking through your summary can you see the assumptions which Euler and
Saunderson make?
b You will first need to summarise Arnauld's objections to negative numbers and
the assumptions which he makes. You can get help to do this if you consult the hint
and also the answers to Activity 10.10.
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