history of mathematics - National STEM Centre

10 'Two minuses make a plus'
And, to use an awkward comparison which M. Chabert's pronounced
Grenoblois drawl made even more clumsy, let us suppose that the negative
quantities are a man's debts; how, by multiplying a debt of 10,000 francs by
500 francs, can this man have, or hope to have, a fortune of 5,000,000
francs?
Are M. Dupuy and M. Chabert hypocrits like the priests who come to say
Mass at my grandfather's, and can my beloved mathematics be a fraud?
Oh, how eagerly I would have listened then to one word about logic, or the
art of finding out the truthl
Reflecting on Chapter 10
What you should know
• negative numbers were initially rejected by the Chinese, Greeks and
Babylonians, among others, as not being practical
• subtracted numbers on the other hand were acceptable to these people and,
likewise, subtraction as an operation was considered acceptable
• rules for addition and subtraction of positive and negative numbers appeared as
early as the 3rd century BC in China
• rules for multiplying subtracted numbers appeared later in the 3rd century AD in
geometric form and in algebraic form in the 9th century (al-Khwarizmi)
• rules for multiplying and dividing negative numbers were first proposed by
Brahmagupta in the 7th century AD although they can be derived by
extrapolation from al-Khwarizmi's results
• much later in the 17th century in Europe, negative numbers started to be
accepted as solutions of equations
• negative numbers were modelled as directions and debts as a means of justifying
the rules for multiplying and dividing negative numbers. However, these rules
were not universally accepted because of apparent contradictions.
Preparing for your next review
• You should be prepared to explain how the operations of combining both
positive and negative numbers by addition and multiplication can be modelled as
movements along the number line.
• You should be aware of the difference between negative and subtracted numbers
and be able to discuss why the latter were more acceptable to early users of
numbers.
• You should be able to explain how algebraic formulae, involving multiplication
of subtracted numbers, may be derived from areas of geometrical figures.
• You should be able to discuss some of the objections to negative numbers that
were raised throughout the centuries.
• You should be able to explain some of the limitations in the derivation of rules
of multiplication of numbers in the work of, say, Euler and Saunderson.
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