history of mathematics - National STEM Centre

10 'Two minuses make a plus'
This is an interesting case of how a result is generated first by considering a
diagram, where the result is apparent from the picture, and then, sometime later, is
turned into a symbolic formula. Viete's symbolic explanation of al-Khwarizmi's
geometric result was probably the first algebraic formulation of a geometric
argument. It is algebraic because it uses symbols which are not necessarily assumed
to represent distances.
Activity 10.7 The equivalence of the different cases
One dictionary gives the meaning of extrapolate as 'to infer, conjecture, from what
is known'.
1 How would you use al-Khwarizmi's equation, which you obtained in Activity
10.5, to extrapolate rules for multiplying two negative numbers together?
2 Even though al-Khwarizmi did not express it in quite the same way, his equation
is equivalent to Brahmagupta's rules as set out on page 127. This is because
al-Khwarizmi's equation enables you to derive the rules. Explain how you could
obtain the rules from al-Khwarizmi's equation by extrapolation.
As you can see from Activity 10.7, it is a simple extension of al-Khwarizmi's
equation to deduce the result of multiplying two negative numbers together.
However this was not done by any of the mathematicians mentioned so far, because
it required them to admit the idea of a negative number.
It appears to have been the Flemish mathematician, Albert Girard (1590-1633), who
first openly acknowledged the validity of negative roots of equations and thus
opened up the debate about the existence of negative numbers. He interpreted
negative numbers as having an opposite direction to positive ones - he used the
words 'retrogression' and 'advance' for negative and positive numbers respectively
- which might remind you of where this chapter began with the number line.
Activity 10.8 Why a lapse in development?
This activity is optional.
From Girard's recognition of negative numbers in 1629 there is a lag in time to the
mid- 18th century before the next significant developments regarding rules for
combining negative numbers.
1 From your knowledge of what was going on in the mathematics community
during that time, can you propose any reason which might account for this lapse?
Although the rules for combining negative numbers were not being developed
between Girard's time and the mid-18th century, negative numbers themselves were
not totally neglected, since further progress was being made on ordering of numbers
along the number line.
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