history of mathematics - National STEM Centre

7 7 Towards a rigorous approach
The operations of + and x on the real numbers are said to be commutative. With
the operation x on vectors, you have found an operation which is not commutative;
that is, the order in which you combine the vectors does affect the result.
Consequently, the rules which govern the combination of objects by x would not
look the same as the rules governing the combination of real numbers under either +
or x. So De Morgan was wrong; not all systems need necessarily obey the same
rules. You can invent a new system, an extension of the number system, with
different rules from the number system.
Activity 11.13 The full two-dimensional Clifford algebra
Activity 11.12, question 2, shows that the turning operator / which turns one step
into another is a product of the two basic vectors i and j in Clifford's model of two
dimensions. It is called the basic bivector in the model.
1 a Investigate combinations of any number of vectors in two dimensions using
Clifford's operation x.
b Also investigate combinations of vectors and bivectors, bivectors and bivectors,
numbers and bivectors. What types of objects are you able to produce? Can you
come up with any rules to indicate the type of object that you will obtain from a
product of particular objects?
c What are the basic building blocks in Clifford's two-dimensional model?
Clifford's three kinds of elements in two dimensions,
• scalars (that is, numbers or the size and direction of steps)
• vectors i and j (that is, steps)
• the bivector j x i
form the basic building blocks of the most general algebraic system in two
dimensions, which is called a Clifford algebra. In a Clifford algebra you are allowed
to combine together any of these elements.
You have met the idea of extension on several occasions now. Wessel extended the
real number system to two dimensions in order to define the system of complex
numbers with similar rules of addition and multiplication to those of the real
numbers. His extension was based on a particular model of the number line and
operations on it. Clifford had a different model of the real numbers which, when
extended, produced his algebra of geometrical objects in two dimensions which
have different rules for addition and multiplication from the real numbers.
Activity 11.14 Extending to other dimensions
For further details see
Chapter 18 of the Space
and vectors unit of
BookS.
Clifford's approach can be further extended to three dimensions. Steps are built up
from three basic vectors i, j and k.
In three dimensions you can define three turning operators /, J and K as shown in
Figure 11.5.
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