history of mathematics - National STEM Centre

10 'Two minuses make a plus'
numbers. The same is true of people across the world and across the centuries. But
why did people consider the possibility of negative numbers in the first place? What
practical use do you think negative numbers have?
2 You can visualise numbers by considering them as points on a number line. The
integers are equally spaced and increase from left to right. You can model an
operation combining numbers as a movement along the line; the number which is
the result of the operation corresponds to the point where the movement ends.
a How can you interpret the operations of addition and subtraction as movements
along the number line?
b What about multiplication? Does your picture of multiplication include the
multiplication of a positive number by a negative number and a negative by a
positive? Does it extend further to a negative by a negative? If not, why?
Before you read more about the changing views about negative numbers through
history, consult the answers for this activity to find out more about the use to which
negative numbers were put and how the number line can be used to model
operations on numbers.
Negative numbers have a chequered history. Although they were found as solutions
of equations as early as the 3rd century BC, they were usually rejected because they
were not considered to correspond to proper solutions of practical problems.
Negative numbers were called 'absurd' by Diophantus in the 3rd century and by
Michael Stifel, a German algebraist, in the first half of the 16th century, and
'fictitious' by Geronimo Cardano, again in the 16th century. Descartes called
negative solutions of equations 'racines fausses', literally 'false roots'. The Chinese
used the word 'fu' in front of the word for number when describing negatives. 'Fu'
is like the English prefix 'un' or 'dis'. This re-inforces the general impression that,
for a long time, negative numbers were not thought of as proper numbers.
Activity 10.2 Subtracted numbers are more acceptable
Historically, the concept of a 'subtracted number' was more common than that of a
negative number. A subtracted number is one such as a-b, where a > b.
1 People found subtracted numbers acceptable even when they rejected negative
numbers out of hand. Why do you think that this was so?
Subtracted numbers, and operations on them, can be traced back to civilisations that
you studied at the beginning of this book.
2 Find some examples of the use of subtracted numbers in The Babylonians and
The Greeks units. Can you propose some other uses to which they could be put?
Subtracted numbers appeared in equations which were considered, and solved, by
the Babylonians, Chinese and Indians.
If you look at such equations today, you might interpret them differently by
believing that they contain negative terms, that is terms with negative coefficients.
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