Graphics Reference
In-Depth Information
Chapter 2
Complex Numbers
2.1 Introduction
Complex numbers have been described as the 'king' of numbers, probably because
they resolve all sorts of mathematical problems where ordinary real numbers fail.
For example, the rather innocent looking equation
x
2
1
+
=
0
has no real solution, which seems amazing when one considers the equation's sim-
plicity. But one does not need a long equation to show that the algebra of real num-
bers is unable to cope with objects such as
√
x
=
−
1
.
However, this did not prevent mathematicians from finding a way around such an
inconvenience, and fortuitously the solution turned out to be an incredible idea that
is used everywhere from electrical engineering to cosmology. The simple idea of
declaring the existence of a quantity
i
, such that
i
2
=−
1, permits us to express the
solution to the above equation as
x
=±
i.
All very well, you might say, but what is
i
? What is mathematics? One could also
ask, and spend an eternity searching for an answer!
i
is simply a mathematical object
whose square is
1. Let us continue with this strange object and see how it leads us
into the world of rotations.
−
2.2 Complex Numbers
A complex number has two parts: a
real
part and an
imaginary
part. The real part
is just an ordinary number that may be zero, positive or negative, and the imaginary
part is another real number multiplied by
i
. For example, 2
+
3
i
is a complex number
