Number Theory: An Introduction to Mathematics

5 Complex Numbers 47

Supposeeix

′

=1forsomex′∈Rand choosen∈Zso thatn≤x′/ 2 π<n+1.

Ifx=x′− 2 nπ,theneix=1and0≤x< 2 π.Ifx=0, then 0<x/ 4 <π/2and

hence 0<sinx/ 4 <1. Thuseix/^4 =± 1 ,±i. But this is a contradiction, since

(eix/^4 )^4 =eix= 1.

We show next that the mapx→eixmaps the interval 0≤x< 2 πbijectively onto

theunit circle, i.e. the set of all complex numberswsuch that|w|=1. We already

know that|eix|=1ifx∈R.Ifeix=eix

′

,where0≤x≤x′< 2 π,thenei(x

′−x)

=1.

Since 0≤x′−x< 2 π, this impliesx′=x.

It remains to show that ifu,v∈Randu^2 +v^2 =1, then

u=cosx,v=sinx

for somexsuch that 0≤x< 2 π.Ifu,v>0, then alsou,v<1. Henceu=cosxfor

somexsuch that 0<x<π/2. It follows thatv=sinx,sincesin^2 x= 1 −u^2 =v^2

and sinx>0. The other possible sign combinations foru,vmay be reduced to the

caseu,v>0 by means of the relations

sin(x+π/ 2 )=cosx, cos(x+π/ 2 )=−sinx.

Ifzis any nonzero complex number, thenr=|z|>0and|z/r|=1. It follows

that any nonzero complex numberzcan be uniquely expressed in the form

z=reiθ,

wherer,θare real numbers such thatr>0and0≤θ< 2 π. We call theser,θthe
polar coordinatesofzandθtheargumentofz.Ifz=x+iy,wherex,y∈R,then
r=

√

(x^2 +y^2 )and

x=rcosθ, y=rsinθ.

Hence, in the geometrical representation of complex numbers by points of a plane,r

is the distance ofzfromOandθmeasures the angle between the positivex-axis and

the ray

−→

Oz.

We now show that the exponential function assumes every nonzero complex

valuew.Since|w|>0, we have|w|=exfor somex∈R.Ifw′=w/|w|,then

|w′|=1andsow′=eiyfor somey∈R. Consequently,

w=|w|w′=exeiy=ex+iy.

It follows that, for any positive integern, a nonzero complex numberwhasn

distinctn-th roots. In fact, ifw=ez,thenwhas the distinctn-th roots

ζk=ζωk(k= 0 , 1 ,...,n− 1 ),

whereζ=ez/nandω=e^2 πi/n. In the geometrical representation of complex numbers

by points of a plane, then-th roots ofware the vertices of ann-sided regular polygon.