history of mathematics - National STEM Centre

Figure 11.3
Activity 1 /. 10 Clifford's model again
Figure 114
step in
'"•^ the plane
/ / Towards a rigorous approach
Describe other ways in which you can start with the same step and achieve the same
outcome. By analogy, what can you deduce about the rules for combining numbers
using the operations of x and +?
2 You can think of the basic building block of a step of length 1 as being a step
corresponding to some standard size, like a metre length. You can make a smaller
step than the basic one by multiplying it by a positive number a, where a < 1.
a Describe how you would obtain a step to the left which has a length of one-half
of the standard step.
b If you performed this step three successive times and then reversed its direction,
what would be the outcome? How might you express this process in symbols?
You can extend Clifford's picture to two dimensions. The basic element is still a
step but it can come in different varieties because there are many more directions in
which the step can be taken in the plane compared with the line. However, all steps
in the plane can be built up from two basic vectors together with the operations of +
and - as described above. The two basic vectors i and j are those with which you are
already familiar. Numbers, called scalars, are the size of the steps. Then, for
example, you can interpret multiplying a vector by a positive number a as changing
the size of the step while leaving its direction unchanged. If a > 1 then the size
increases, whereas if a < I then the size decreases.
1 Referring to Figure 11.3, draw diagrams to represent in Clifford's model the
outcome of the following combinations of steps with other steps and with scalars.
a -2xi b i + j A c -(ixi)-(f xj)
By multiplying i by the scalar -1 you can turn it into - i; similarly you can turn j
into - j. Thus, multiplication of a step by a negative number allows you to move up
and down a line in the plane, but it does not allow you to convert one line into
another. For example, to turn i into j you need a turning operator; call this turning
operator /. Figure 11.4 shows the way / operates.
When you apply the operator / to a step, it turns through 90° in an anti-clockwise
direction. In particular, it will turn i into j and j into - i. Using symbols you can
write this as
/ x i = j and 7 x j = -i
Here the multiplication symbol x is being used to represent the application of the
turning operator to the step.
Activity 11.11 More on Clifford's model
1 Use Clifford's interpretation of /, i and j to show that the following statements
are true.
a /x/xi = -i b /x/xj = -j c 7 2 =-l
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