history of mathematics - National STEM Centre

Being
critical
11.2
Arguing
by analogy
11.3
Wessel's
extension
11.4
Two
counter-examples
11.5
Looking
at axioms
11.7
Do all systems
have the same
axioms?
11.9
Clifford's
model
11.10
Clifford's
model again
11.11
More on
Clifford's model
Towards a rigorous approach
11.6
Another
look at axioms
11.8
Boolean
algebra
11.12
Calculations in
Clifford algebra
11.13
The full two-
dimensional
Clifford algebra
11.14
Extending to
other dimensions
You have, by now, read a substantial section on the historical development of
addition, subtraction and multiplication of negative numbers. You have seen the
rules for combining numbers, both positive and negative, in the work of many
people from different cultures and across the centuries. All the work is different,
with different motives and viewpoints. Yet the work all has one thing in common:
none of it is what mathematicians would call rigorous. It is based on intuition,
analogy or extrapolation.
This chapter shows why mathematicians needed to adopt a more rigorous approach
to mathematics, and how they have gone about it.
In Activities 11.1 and 11.4 you are asked to criticise 'proof by
extrapolation'.
In Activity 11.2 you are asked to use argument by analogy to justify the
rules for multiplying negative numbers evolved in the previous chapter.
In Activity 11.3 you will see how Wessel extended the notion of a
number line to include complex numbers.
Then, in Activities 11.5 to 11.7, you will learn more about an axiomatic
approach to numbers.
Activity 11.8 extends the idea to Boolean algebra.
In Activities 11.9 to 11.14, you will meet Clifford's development of the
rigorous approach to number and finally in Activity 11.15 you will see
how Clifford's ideas extend beyond number.
All the activities are suitable for working in a small grouf
Work through the activities in sequence.
Activity 11.6 is optional.
11.15
Reflecting
on algebra
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