history of mathematics - National STEM Centre

operation of
changing sign
number
line
You can take the study of
complex numbers further
in the option Complex
numbers and numerical
methods.
You can find a copy of the
paper in, for example, A
source book in
mathematics, D E Smith
1959 pp 55-56. ,,
Activity 11.3 Wessel's extension
-1 .+1
Figure 11.2
7 7 Towards a rigorous approach
experience of particular cases, Euler knows that multiplying one number, -a, by a
second number, —b, should not give the same result as multiplying -a by the
number b. Euler does not prove that these results are different, but the fact that 2x3
is different from 3x3 makes it seem reasonable.
None of the work to which you have been introduced so far is truly rigorous. If it
had been rigorous, then intuitive arguments, analogies and extrapolation would not
have been used. All the results would have been firmly established or proved;
nothing would have been taken for granted. Mathematics derived from intuition and
analogy is often valuable. Much of the mathematics which you use today would not
have developed if intuitive arguments had been ruled out. Some of the best
mathematics originates as intuitive arguments based on models that help
understanding. This is how you, and other mathematicians, are likely to be most
creative.
However, there comes a time when intuitively based results underpin so much other
mathematics that they need to be put on a firmer footing and to be made rigorous.
For example, in the mid-18th century new mathematical ideas which relied on the
number system were being developed, so it became necessary to establish a rigorous
foundation for the number system and of operations based on it.
One example of a new mathematical idea based on the number system is the system
of complex numbers. A paper written in 1799 by a Norwegian surveyor, Caspar
Wessel, shows clearly how the graphical representation of complex numbers is
modelled on an extension of the idea of the number line. The new idea is based on
the interpretation of a minus sign as producing a reversal of direction along a
number line. You can interpret this reversal of direction as a turn through an angle
of 180° (Figure 11.1).
It is useful if you already have some knowledge of complex numbers for this
activity. However, it is possible for you to try it without this knowledge. Take the
imaginary number i to be a line of unit length as shown in Figure 11.2.
Multiplying two negative numbers together involves putting two minus signs in
front of the equivalent positive numbers, so involves two turns, each of them
through 180°. Adding together the angles of the turning factors involved, namely
180°+180°, gives 360°, or a turn of the positive number through a complete circle.
This means that the result is the same as turning through no angle at all, so you
remain facing in the positive direction.
You can obtain the same result by thinking of -1 as being a line of length one unit
inclined at an angle of 180° to the unit length line +1. Incorporating imaginary
numbers into this picture involves extending into a plane. Wessel proposed that the
imaginary number i is a line of unit length inclined at 90° to the positive number +1.
and that -i is a line of unit length inclined at an angle of 270°.
1 a Multiplying these lines of unit length by adding together the angles that they
make with the number +1, show that you can interpret i as a line of unit length
inclined to the positive unit at an angle of 90° + 90° = 180°. How is this consistent
with the way the imaginary number i is defined?
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