history of mathematics - National STEM Centre

7 / Towards a rigorous approach
although these operations may not be labelled by + and x. Write down the rules for
combining these objects. Translate these rules into a new language by consistently
replacing the symbols used for your two operations by the symbols + and x. If your
objects were represented by numbers, would these translated rules look the same as
the rules for combining real numbers by addition and multiplication? If so, then
your collection of objects satisfies the same axioms as the real numbers. If not, then
the axioms for the two systems are different; in which case, how do they differ?
Charting the progress and status of negative numbers through history highlights
some interesting points concerning the development of any piece of mathematics. It
has shown how many results are first generated by analogy, intuition and
extrapolation, and do not necessarily develop a rigorous foundation until later.
However, once the rigour has been established, extensions and generalisations can
be produced in which the rules governing the combination of numbers by + and x
are used to represent the combination of new objects. These rules then acquire the
status of the rules governing an algebraic system, since they no longer just apply to
the original numbers. And, potentially, this algebraic system allows the possibility
of generalisation to an abstract structure, which exists in spaces which you cannot
necessarily visualise.
Reflecting on Chapter 11
What you should know
• that mathematical results are conjectured using various methods: extrapolation,
analogy, intuition; and that none of these methods is necessarily rigorous
• that complex numbers can be described as an extension of the real number
system, and that they may be modelled in the plane
• that the properties of negative numbers do not follow the 'principle of
permanence of equivalent forms'
• that it is possible to model a positive number as a step along the number line and
that multiplication of the number by -1 is analogous to reversing its direction
• that it is possible to extend this model of the real numbers into the plane to
produce a different algebraic system from the complex numbers (this is
Clifford's model)
• that the unit vectors can be modelled as basic steps at right angles to one another
in both two and three dimensions, and that it is possible to define an operator
which turns one basic step into the other
• that it is possible to combine numbers, steps and turning operators, using an
operation analogous to multiplication, but which has different properties from
multiplication of numbers
• that systems, which can be visualised and represented by geometrical objects,
can be extended and generalised to more abstract situations, for example, to
higher dimensions, which can no longer be represented pictorially but which are
equally valid mathematically.
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