The Chemistry Maths Book, Second Edition

8 Complex numbers

8.1 Concepts

We saw in Section 1.7 and in Chapter 2 that the solutions of algebraic equations are

not always real numbers; in particular, the solutions of the equation x

2

1 = 1 − 1 are

, and the square root of a negative number is not a real number.

1

Such

numbers are incorporated into the system of numbers by defining the square root of

− 1 as a new number which is usually denoted by the symbol i(or jin engineering

mathematics) with the property

i

2

1 = 1 − 1 (8.1)

A number containing is called a complex number; examples of complex

numbers are 2 i, − 3 i, and 21 + 15 i. The general complex number has the form (the letter

zis usually used to denote a complex number)

z 1 = 1 x 1 + 1 iy (8.2)

where xand yare real numbers. The number xis called the real partof z, and yis called

the imaginary part of z:

x 1 = 1 Re (z), y 1 = 1 Im (z) (8.3)

Ifx 1 = 10 thenz 1 = 1 iyis called pure imaginary. Ify 1 = 10 thenz 1 = 1 xis real, so that the

set of complex numbers includes the real numbers as subset.

Powers of i

Every integer power of iis one of the numbersi,−i, 1 ,− 1 ; for example

In general, for integersn 1 = 1 0, ±1, ± 2 , =,

i

4 n

1 = 1 +1, i

4 n+ 1

1 = 1 +i, i

4 n+ 2

1 = 1 −1, i

4 n+ 3

1 = 1 −i (8.4)

iii i i i i

i

i

i

i

i

32 4 22 1

2

1

1

1

=×=−, = =, == =

−

=−

−

()

i=− 1

x=± − 1

1

Square roots of negative numbers are mentioned in Cardano’s Ars Magnaof 1545 in connection with the

solution of quadratic and cubic equations. Cardano called such a result ‘as subtle as it is useless’. Rafael Bombelli

(1526–1572), Italian engineer, called ‘più di meno’ (plus of minus) and ‘meno di meno’ (minus of minus),

and presented the arithmetic of complex numbers in his Algebraof 1572. The first serious consideration of

complex numbers was by Albert Girard (1595–1632) who in his L’invention nouvelle en l’algèbre(New discovery

in algebra), 1629, published the first statement of the fundamental theorem of algebra. He called the complex

solutions of equations ‘impossible’ but ‘good for three things: for the certainty of the general rule (the theorem),

for the fact that there are no other solutions, and for their use’. Descartes used the words ‘real’ and ‘imaginary’.

Leibniz factorized Euler proposed the symbol ifor in

1777, and this was adopted by Gauss in his Disquisitiones arithmeticaeof 1801.

x a xaixaixa ixa i − 1

44

+=+−+−−−()()( )( ).

±− 1