12 Complex Numbers
Complex numbers extend the notion of ordinary ‘real numbers’ to include a new kind of
‘number’,j=
√
−1. Such ‘numbers’ can be combined by the usual rules of algebra, except
that we replacej^2 by−1 wherever it occurs. Complex numbers do not serve the usual
role of numbers – we don’t measure physical quantities with them – but they provide a
very useful shorthand tool for dealing with certain combinations of real numbers. There is
not much that is conceptually new in this chapter, but we have to become more practised
at using basic, elementary mathematics.
Prerequisites
It will be helpful if you know something about:- solving quadratic equations (65
➤
)- elementary algebra (Chapter 2
➤
)- polar coordinates (206
➤
)- elementary trig, including compound angle formulae (187
➤
)- the exponential function (124
➤
)Objectives
In this chapter you will find:- definition and representation of a complex number
- addition, subtraction, multiplication and division of complex numbers
- the Argand plane
- modulus and argument of a complex number
- polar form of a complex number
- multiplication and division in polar form
- the exponential form of a complex number
- powers and roots – De Moivre’s theorem
Motivation
You may need the material of this chapter for:- solving polynomial equations
- solving differential equations
- describing voltages, currents, etc. in AC electricity theory