Number Theory: An Introduction to Mathematics

42 I The Expanding Universe of Numbers

|z+w|^2 ≤|z|^2 + 2 |z||w|+|w|^2 =(|z|+|w|)^2 ,

and (iii) follows by taking square roots.
Several other properties are consequences of these three, although they may also
be verified directly. By takingz=w=1 in (ii) and using (i), we obtain| 1 |=1. By
takingz=w=−1 in (ii) and using (i), we now obtain|− 1 |=1. Takingw=− 1
andw=z−^1 in (ii), we further obtain

|−z|=|z|, |z−^1 |=|z|−^1 ifz= 0.

Again, by replacingzbyz−win (iii), we obtain

||z|−|w|| ≤ |z−w|.

This shows that|z|is a continuous function ofz. In factCis a metric space, with the
metric d(z,w)=|z−w|. By considering real and imaginary parts separately, one
verifies that this metric space is complete, i.e. every fundamental sequence is conver-
gent, and that the Bolzano–Weierstrass property continues to hold, i.e. any bounded
sequence of complex numbers has a convergent subsequence.
It will now be shown that any complex number has a square root. Ifw=u+iv
andz=x+iy,thenz^2 =wis equivalent to

x^2 −y^2 =u, 2 xy=v.

Since

(x^2 +y^2 )^2 =(x^2 −y^2 )^2 +( 2 xy)^2 ,

these equations imply

x^2 +y^2 =

√

(u^2 +v^2 ).

Hence

x^2 =

{

u+

√

(u^2 +v^2 )

}/

2.

Since the right side is positive ifv=0,xis then uniquely determined apart from sign
andy=v/ 2 xis uniquely determined byx.Ifv=0, thenx=±

√

uandy=0when
u>0;x=0andy=±

√

(−u)whenu<0, andx=y=0whenu=0.

It follows that any quadratic polynomial

q(z)=az^2 +bz+c,

wherea,b,c∈Canda=0, has two complex roots, given by the well-known formula

z=

{

−b±

√

(b^2 − 4 ac)

}/

2 a.

However, much more is true. The so-calledfundamental theorem of algebraasserts
that any polynomial