12.1 What are complex numbers?
Numbers such as integers, rational numbers, etc. that we have been using so far are called
real numbers– they can be used to count with, or measure distances, time, etc. From the
rules of signs for combining negative numbers (11
➤
) we know that if we square two
such real numbers, the result will always be positive:
22 =+ 4 (− 2 )^2 =+ 4An obvious question is, is thereanysort of mathematical object that results in a nega-
tive number when squared? Why should such an object be of interest anyway? In real
life we don’t need such quantities – no one ever measured the length of a line to be,
for example
√
−1 metres. True, we don’t need such ‘numbers’ for measuring and repre-
senting real physical quantities. However, it does turn out that such ‘numbers’ are very
useful as a mathematical tool in representing physical objects which have two parame-
ters – for example the amplitude and phase of a signal waveform may be conveniently
combined in ‘complex’ form. Also, such ‘numbers’ provide nice tools for calculational
purposes – as for example in differential equations. So, let’s accept that it’s useful to
look at the properties of such ‘numbers’ as
√
−1 and develop the tools necessary to
use them.
Start by considering the quadratic equation
x^2 + 1 =0orx^2 =− 1This has no ‘real’ solution, but by introducing the symbol
j=√
− 1 (alternative notationi)we can write
x=±jThe objectjis sometimes called ‘imaginary’, and is an example of a ‘complex number’.
There is, however, nothing imaginary about it. While it certainly does not represent a
‘quantity’ in the normal sense that a ‘real number’ such as 2 does, it is still a perfectly
proper symbol that serves a very useful purpose in mathematics. In particular, having
defined it as above, it now allows us to write down the ‘solution’ of any quadratic
equation.
Problem 12.1
Write the solution of the equationx^2 Y 2 xY 2 =0, as given by the formula
(66➤
), in terms ofj=√
−1.By the formula we have
x=− 2 ±√
− 4
2=− 2 ±√
− 1
2=− 1 ±j