history of mathematics - National STEM Centre

Activity 11.4 Two counter-examples
Activity 11.5 Looking at axioms
/ / Towards a rigorous approach
1 Find two counter-examples to the 'principle of the permanence of equivalent
forms'. That is, find two examples of results which are true for positive numbers but
which would not be true if you applied them to negative numbers.
The rules of the number game
The first attempt by Peacock to put negative numbers on a firmer footing did not
lead to much success since the same comments could be made of Peacock's work as
of Euler's and Saunderson's: he was using extrapolation and analogy. The work of
all of them assumed that results held for the negative numbers by analogy with what
held for the positives. It was necessary to be even more formal and systematic and
to produce rules for combining numbers. The axioms as a whole form a set of 'self-
evident' rules from which all the other properties of the numbers can be deduced.
The axioms are not proved; rather they are part of the definition of the system which
includes ways of combining the numbers. The set of axioms is minimal; this means
that properties are not included as axioms if they can be deduced from the other
axioms in the set. Basically the axioms are the bare minimum with which you can
get by and still be able to deduce all of the properties of the numbers under the usual
rules of combination.
The German mathematician Hermann Hankel was the first to produce a logical
theory of rational numbers in 1867 but, he had fundamental objections to some
aspects of the irrational numbers. The axioms of both the rational and the irrational
numbers are attributed to another German, David Hilbert, as late as 1899.
The axioms assume that there are two operations which you can use to combine
pairs of numbers. These operations are labelled by + and x. The effect of each of
these operations is to combine two numbers to produce a third number. The third
number is not necessarily different from each of the other two. The result of the
operation is still a number; it is not some new kind of mathematical object.
1 a The axioms require that there is a particular number which, when combined
with any other number using the operation of +, leaves that number unchanged.
There is also a number, a different one, which has a similar effect when x is
considered instead of +. What are these two numbers?
b If a number is chosen at random, then the axioms require that it is always
possible to find another number which, when combined with it using the operation
+, gives the number 0. If 2 is the number selected at random, then what is the
required number?
c Suppose that you take 2 again and ask for the number which, when combined
with 2 using the operation x, gives 1. What is that number? It you had chosen any
other number instead of 2, would you still have been able to satisfy the request?
Justify your answer.
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