history of mathematics - National STEM Centre

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d Do you believe that the following statement is true?
Activity 11.6 Another look at axioms
SBayley was one of the s*
first people to study
matrices - see the
Modelling with matrices
unit in Book 4.
742
'If a number is chosen at random, then the axioms require that it is always possible
to find another number which, when combined with the original number using the
operation x, gives 1.'
If you believe the statement is false, then how can you modify it to make it true?
You will not be introduced here to the axioms of the number system in any further
detail or be required to deduce from them the familiar and well-used properties of
the real numbers. Deductions of this kind can seem tedious since you already use
the properties without thinking in your everyday mathematics.
It is also unnecessary, once the axioms have been established and the properties
deduced. However it is necessary that someone actually sets up these foundations.
Without them, there is a danger that a whole mathematical edifice could crumble
due to an overlooked inconsistency.
Axioms are often presented in a very general form. Even though they are describing,
for example, the number system with addition and multiplication as the rules of
combination, the language may not refer to numbers or to addition and
multiplication. It may refer to elements which you might choose to think of as
numbers, and the operations for combining the elements might be labelled by using
function notation rather than the symbols + and x.
So the elements may be a, b, ... and an operation to combine them may be labelled
by f. In a particular case, you may want to think of f applied to a and b as producing
the sum of the numbers a and b, that is, f(a,b) = a + b. However this interpretation
is not part of the axioms, which are quite abstract.
This activity is optional.
1 a Write out the requirements of the axioms as given in Activity 11.5 using
function notation where f (a, b) = a + b and g(a, b) = a x b.
b Is it true that f(a,b) = f(b,a) and g(a,b) = g(b,a)l Justify your answers.
Other developments in the 19th century
The 19th century was an active period for European mathematics, in England and
Ireland in particular. You have already read about Peacock; other active
mathematicians in England at the time were Augustus De Morgan (1806-1871),
Arthur Cayley (1821-1895); William Rowan Hamilton (1805-1865) was from
Trinity College in Dublin. You will be hearing about some of their attempts to
extend the number system.