Graphics Reference
In-Depth Information
3
2
1
0
1
2
3
Fig. 2.1. The number line.
never terminate and are always subject to a small error when stored within a
computer.
2.6 Real Numbers
Rational and irrational numbers together comprise the real numbers .
2.7 The Number Line
It is convenient to organize numbers in the form of an axis to give them a
spatial significance. Figure 2.1 shows such a number line , which forms an axis
as used in graphs and coordinate systems. The number line also helps us
understand complex numbers, which are the 'king' of all numbers.
2.8 Complex Numbers
Leonhard Euler (1707-1783) (whose name rhymes with boiler ) played a sig-
nificant role in putting complex numbers on the map. His ideas on rotations
are also used in computer graphics to locate objects and virtual cameras in
space, as we shall see later on.
Complex numbers resolve some awkward problems that arise when we
attempt to solve certain types of equations. For example, x 2
4=0has
2. But x 2 +4 = 0 has no obvious solutions using real or integer
numbers. However, the number line provides a graphical interpretation for a
new type of number, the complex number. The name is rather misleading: it
is not complex, it is rather simple.
Consider the scenario depicted in Figure 2.2. Any number on the number
line is related to the same number with the opposite sign via an anti-clockwise
rotation of 180 . For example, if 3 is rotated 180 about zero it becomes
solutions x =
±
3,
2 is rotated 180 about zero it becomes 2.
We can now write
and if
1 is effectively
a rotation through 180 . But a rotation of 180 can be interpreted as two
consecutive rotations of 90 , and the question now arises: What does a rotation
of 90 signify? Well, let's assume that we don't know what the answer is going
to be - even though some of you do - we can at least give a name to the
operation, and what better name to use than i.
3=(
1)
×
3, or 2 = (
1)
×−
2, where
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