complex numbers 85

complex numbers There is no real number xwith

the property that x2= –1. By introducing an “imagi-

nary” number ias a solution to this equation we obtain

a whole host of new numbers of the form a+ ib with a

and breal numbers. These new numbers form the sys-

tem of complex numbers. It is customary to use the

variable zto denote an arbitrary complex number:

z=a+ ib. If b= 0, then zis a real number. Thus the set

of complex numbers includes the set of real numbers. If

a= 0 so that zis of the form z= ib, then zis said to be

purely imaginary. In general, if z= a+ ib, then ais

called the real part of zand bthe imaginary part of z.

We write: Re(z) = aand Im(z) = b.

The number iis usually regarded as the square

root of negative one: i= √

–

–1. (One must be careful as

there are, in fact, two square roots of this quantity,

namely iand –i.) The roots of other negative quanti-

ties follow: √

–

–9 = √

–

–1 .√

–

9 = ±3iand √

–

–30 = ±i√

–

30,

for instance.

The set of all complex numbers is denoted C.

Arithmetic can be performed on the complex numbers

by following the usual rules of algebra and replacing i2

by –1 whenever it appears. For example, we have:

(2 + 3i) + (4 – i) = 6 + 2i

(2 + 3i) – (4 – i) = –2 + 4i

(2 + 3i)(4 – i) = 8 + 12i– 2i– 3i2

= 8 + 10i+ 3 = 11 + 10i

The

QUOTIENT

of two complex numbers can be

computed by the process of

RATIONALIZING THE

DENOMINATOR

:

One can show that with these arithmetic properties,

the set of complex numbers constitutes a mathemati-

cal

FIELD

.

In the early 18th century, French mathematician

A

BRAHAM

D

E

M

OIVRE

noticed a striking similarity

between complex multiplication and the

ADDITION

for-

mulae of the sine and cosine functions from

TRIGONOM

-

ETRY

. Given that:

(a+ ib) · (c+ id) = (ac – bd) + i(ad + bc)

and:

cos(x+ y) = cos(x)cos(y) – sin(x)sin(y)

sin(x+ y) = sin(x)cos(y) + cos(x)sin(y)

we obtain the compact formula:

(cos(x) + isin(x))(cos(y) + isin(y)) = cos(x+ y) + isin(x+ y)

This observation formed the basis for the famous for-

mula that now bears his name:

(cos(x) + isin(x))n= cos(nx) + isin(nx)

A few years later L

EONHARD

E

ULER

(1707–83) took

matters one step further and used the techniques of cal-

culus to establish his extraordinary formula:

eix = cos(x) + isin(x)

from which D

E

M

OIVRE

’

S FORMULA

follows easily.

(Use (eix)n= ei(nx).) Moreover, this result shows that

de Moivre’s formula also holds for noninteger values

of n.

That the cosine and sine functions appear as the

real and imaginary parts of a simple

EXPONENTIAL

FUNCTION

shows that all of trigonometry can be

greatly simplified by rephrasing matters in terms of

complex numbers. Although some might argue that

complex numbers do not exist in the real world, the

mathematics of the complex number system has proved

to be very powerful and has offered deep insights into

the workings of the physical world. Engineers and

physicists phrase a great deal of their work in terms of

complex number theory. (Engineers prefer to use the

symbol jinstead of i.)

It is a surprise to learn that the introduction of a

single new number ias a solution to the equation x2+

1 = 0 provides all that is needed to completely solve

any

POLYNOMIAL

equation anxn+ an–1xn–1 + … + a1x+

a0= 0.

The

FUNDAMENTAL THEOREM OF ALGEBRA

asserts

that a polynomial equation of degree nhas precisely n

roots (counted with multiplicity) in the complex num-

ber system.

It is possible to raise a real number to a complex

power to obtain a real result. For example, by E

ULER

’

S

FORMULA

, we have:

eiπ= cos(π) + isin(π) = –1 + i · 0 = –1