86 complex numbers

Multiplying by –1 and multiplying by i

Polar coordinates of a complex number

i

Also, noting that , we see that

it is also possible to raise a complex number to a com-

plex power to obtain a real result:

(Technically, since ican also be expressed as

for any whole number k, there are infinitely different

(real) values for the quantity ii.)

The Geometry of Complex Numbers

Multiplying the entries of the real

NUMBER LINE

by –1

has the effect of rotating the line about the point 0

through an angle of 180°. It is natural to ask: multipli-

cation by which number creates a 90°rotation about

the point zero?

Call the desired number x. Multiplication by x

twice, that is, multiplication by x×x= x2, would have

the effect of performing two 90°rotations, namely, a

rotation by 180°. Thus multiplication by x2has the

same effect as multiplication by –1, and the desired

number xmust therefore satisfy the equation x2= –1.

This shows that x= iand that it is natural to interpret

purely imaginary numbers of the form ib as members

of a vertical number line.

This model provides a natural correspondence

between complex numbers and points in the plane. The

horizontal axis is called the real axis, the vertical axis

the imaginary axis, and an arbitrary complex number

z= a+ ib appears as the point with coordinates (a,b)

on the plane.

This representation of complex numbers as points

on a plane is called an Argand diagram in honor of

J

EAN

R

OBERT

A

RGAND

(1768–1822) who, along with

surveyor Casper Wessel (1745–1818), first conceived of

depicting complex numbers in this way. The plane of

all complex numbers is also called the complex plane.

The angle θthat a complex number z= a+ ib

makes with the positive real axis is called the argument

of the complex number, and the distance rof the com-

plex number from the origin is called its modulus (or,

simply, absolute value), denoted |z|. P

YTHAGORAS

’

S

THEOREM

and the

DISTANCE FORMULA

show that:

Using trigonometry we see that the values aand bcan

be expressed in terms of rand θin a manner akin to

POLAR COORDINATES

. We have:

a= r cos θ

b= r sin θ

With the aid of Euler’s formula, this shows that any

complex number zcan be expressed in polar form:

z= r cos θ+ ir sin θ= reiθ

From this it follows, for instance, that the product of

two complex numbers z1= r1eiθ1and z2= r2eiθ2has

modulus r1r2and argument θ1+ θ2:

z1z2= r1eiθ1r2eiθ2= r1r2ei(θ1+ θ2)

It is convenient to define the conjugate of a complex

number z= a+ ib to be the number –

z= a– ib. We have:

z · –

z= a2+ b2= |z|2

|z| = √a2+ b2