Page 16 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 1] NUMBERS 7

From the point of view of an axiomatic foundation of complex numbers, it is desirable to treat a
complex number as an ordered pair ða; bÞ of real numbers a and b subject to certain operational rules
which turn out to be equivalent to those above. For example, we define ða; bÞþðc; dÞ¼ða þ c; b þ dÞ,
ða; bÞðc; dÞ¼ðac bd; ad þ bcÞ, mða; bÞ¼ ðma; mbÞ, and so on. We then find that ða; bÞ¼ að1; 0Þþ
bð0; 1Þ and we associate this with a þ bi, where i is the symbol for ð0; 1Þ.

POLAR FORM OF COMPLEX NUMBERS
If real scales are chosen on two mutually perpendicular axes X OX and Y OY (the x and y axes) as
0
0
in Fig. 1-2 below, we can locate any point in the plane determined by these lines by the ordered pair of
numbers ðx; yÞ called rectangular coordinates of the point. Examples of the location of such points are
indicated by P, Q, R, S, and T in Fig. 1-2.
Y
4 Y
P(3, 4)
_
Q( 3, 3) 3
P(x, y)
2
ρ
1 y
T(2.5, 0)
φ
X _ 4 _ 3 _ 2 _ 1 O 1 2 3 4 X
X′ O x X

_ 1
_ _
R( 2.5, 1.5) _ 2
_
S(2, 2)
_
3
Y Y′
Fig. 1-2 Fig. 1-3
Since a complex number x þ iy can be considered as an ordered pair ðx; yÞ,we can represent such
numbers by points in an xy plane called the complex plane or Argand diagram. Referring to Fig. 1-3
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
x þ y ¼jx þ iyj and , called the amplitude or
above we see that x ¼ cos , y ¼ sin where ¼
argument,is the angle which line OP makes with the positive x axis OX.It follows that
z ¼ x þ iy ¼ ðcos þ i sin Þ ð2Þ
called the polar form of the complex number, where and are called polar coordintes.Itis sometimes
convenient to write cis instead of cos þ i sin .
and by using the
If z 1 ¼ x 1 þ iy i ¼ 1 ðcos 1 þ i sin 1 Þ and z 2 ¼ x 2 þ iy 2 ¼ 2 ðcos 2 þ i sin 2 Þ
addition formulas for sine and cosine, we can show that
z 1 z 2 ¼ 1 2 fcosð 1 þ 2 Þþ i sinð 1 þ 2 Þg ð3Þ
z 1 1
¼ fcosð 1 2 Þþ i sinð 1 2 Þg ð4Þ
z 2 2
n n n
z ¼f ðcos þ i sin Þg ¼ ðcos n þ i sin n Þ ð5Þ
where n is any real number. Equation (5)is sometimes called De Moivre’s theorem.We can use this to
determine roots of complex numbers. For example, if n is a positive integer,

1=n 1=n
z ¼f ðcos þ i sin Þg ð6Þ
þ 2k þ 2k
¼ 1=n cos þ i sin k ¼ 0; 1; 2; 3; .. . ; n 1
n n