Number Theory: An Introduction to Mathematics

5 Complex Numbers 41

andIzrespectively. Complex numbers of the formiy,wherey∈R, are said to be
pure imaginary.
It is worth noting thatCcannot be given the structure of anorderedfield, since in
an ordered field any nonzero square is positive, whereasi^2 + 12 =(− 1 )+ 1 =0.
It is often suggestive to regard complex numbers as points of a plane, the complex
numberz=x+iybeing the point with coordinates (x,y) in some chosen system of
rectangular coordinates.
Thecomplex conjugateof the complex numberz=x+iy,wherex,y∈R,isthe
complex number ̄z=x−iy. In the geometrical representation of complex numbers,
̄zis the reflection ofzin thex-axis. From the definition we at once obtain

Rz=(z+ ̄z)/ 2 , Iz=(z− ̄z)/ 2 i.

It is easily seen also that

z+w= ̄z+ ̄w, zw= ̄zw, ̄ ̄ ̄z=z.

Moreover,z ̄=zif and only ifz∈R. Thus the mapz→ ̄zis an ‘involutory auto-
morphism’ of the fieldC, with the subfieldRas its set of fixed points. It follows that
−z=− ̄z.
Ifz=x+iy,wherex,y∈R,then

zz ̄=(x+iy)(x−iy)=x^2 +y^2.

Hencez ̄zis a positive real number for any nonzeroz∈C.Theabsolute value|z|of
the complex numberzis defined by

| 0 |= 0 , |z|=

√

(z ̄z)ifz= 0 ,

(with the positive value for the square root). This agrees with the definition in§3if
z=xis a positive real number.
It follows at once from the definition that | ̄z|=|z|for everyz ∈ C,and
z−^1 = ̄z/|z|^2 ifz=0.
Absolute values have the following properties:for all z,w∈C,

(i)| 0 |= 0 ,|z|> 0 if z=0;
(ii)|zw|=|z||w|;
(iii)|z+w|≤|z|+|w|.

The first property follows at once from the definition. To prove (ii), observe that
both sides are non-negative and that

|zw|^2 =zwzw=zwz ̄w ̄=zz ̄ww ̄=|z|^2 |w|^2.

To prove (iii), we first evaluate|z+w|^2 :

|z+w|^2 =(z+w)(z ̄+ ̄w)=zz ̄+(zw ̄+wz ̄)+ww ̄=|z|^2 + 2 R(zw) ̄ +|w|^2.

SinceR(zw) ̄ ≤|zw ̄|=|z||w|, this yields