A course of pure mathematics, by G.H. Hardy.

36, 37]

COMPLEX NUMBERS

85

THEOREM 1. If R (x + yi) is a rational function of x + yi, it
can be reduced to the form X + Yi, where X and Y are rational
functions of x and y with real coefficients.
In the first place it is evident that any polynomial P (x + yi)
can be reduced, in virtue of the definitions of addition and multiplication, to the form A + Bi, where A and B are polynomials
in x and y with real coefficients. Similarly Q (x + yi) can be
reduced to the form C +Di. Hence
R (x + yi) = P (x + yi)/Q (x + yi)
can be expressed in the form
(A + B) + i)( Di) = (A + Bi) (C - Di)l(C +Di) (C - Di)
AC+BD BC-AD.
C2~+ D2 2+ C~+D2
which proves the theorem.
THEOREM 2. If R (x + yi) = X + i, R denoting a rational
function as before, but with real coefficients, then R(x - yi) = X -Yi.
In the first place this is easily verified for a power (x + yi)n
by actual expansion.
It follows by addition that the theorem is true for any polynomial with real coefficients. Hence, in the notation used above,
A-Bi AC+BD BC-AD.
R( - yi)= C- = C2+D2 - C2+D2 l,
the reduction being the same as before except that the sign of i
is changed throughout. It is evident that results similar to those
of Theorems 1 and 2 hold for functions of any number of complex
variables.
THEOREM 3. The roots of an equation
aozn + alzn-1 +... + an = 0,
whose coefficients are real, may, in so far as they are not themselves
real, be arranged in conjugate pairs.
For it follows from Theorem 2 that if x + yi is a root, so is
- yi. A particular case of this theorem is the result (~ 34)
that the roots of a quadratic equation with real coefficients are
either real or conjugate.
This theorem is sometimes stated as follows-in an equation