2 Chapter 1 Complex Numbers and the Complex Plane

1.1 Complex Numbers and Their Properties

No one person 1.1
“invented” complex numbers, but controversies surrounding the use of these
numbers existed in the sixteenth century. In their quest to solve polynomial equations by
formulas involving radicals, early dabblers in mathematics were forced to admit that there
were other kinds of numbers besides positive integers. Equations
 such asx2 + 2x + 2 = 0
√ √ √
and x3 = 6x + 4 that yielded “solutions” 1 + −1 and 2 + −2 + 3 2 − −2 caused
3

particular consternation within the community of √ fledgling mathematical

√ scholars because
everyone knew that there are no numbers such as −1 and −2, numbers whose square is
negative. Such “numbers” exist only in one’s imagination, or as one philosopher opined, “the
imaginary, (the) bosom child of complex mysticism.” Over time these “imaginary numbers”
did not go away, mainly because mathematicians as a group are tenacious and some are even
practical. A famous mathematician held that even though “they exist in our imagination
. . . nothing prevents us from . . . employing them in calculations.” Mathematicians
also hate to throw anything away. After all, a memory still lingered that negative numbers
at first were branded “fictitious.” The concept of number evolved over centuries; gradually
the set of numbers grew from just positive integers to include rational numbers, negative
numbers, and irrational numbers. But in the eighteenth century the number concept took a
gigantic evolutionary step forward when the German mathematician Carl Friedrich Gauss
put the so-called imaginary numbers—or complex numbers, as they were now beginning to
be called—on a logical and consistent footing by treating them as an extension of the real
number system.
Our goal in this first section is to examine some basic definitions and the arithmetic of
complex numbers.

The Imaginary Unit Even after gaining wide respectability, through

the seminal works of Karl Friedrich Gauss and the French mathematician Au-
gustin Louis Cauchy, the unfortunate name “imaginary” has survived down
the centuries. The
√ symbol i was originally used as a disguise for the embar-
rassing symbol −1. We now say that i is the imaginary unit and define
it by the property i2 = –1. Using the imaginary unit, we build a general
complex number out of two real numbers.

Definition 1.1 Complex Number

A complex number is any number of the form z = a + ib where a and

b are real numbers and i is the imaginary unit.

Terminology The notations a + ib and a + bi are used interchangeably.

The real number a in z = a+ ib is called the real part of z; the real number b
is called the imaginary part of z. The real and imaginary parts of a complex
number z are abbreviated Re(z) and Im(z), respectively. For example, if
Note: The imaginary part of ☞ z = 4 − 9i, then Re(z) = 4 and Im(z) = −9. A real constant multiple
z = 4 − 9i is −9 not −9i. of the imaginary unit is called a pure imaginary number. For example,
z = 6i is a pure imaginary number. Two complex numbers are equal if their
1.1 Complex Numbers and Their Properties 3

corresponding real and imaginary parts are equal. Since this simple concept
is sometimes useful, we formalize the last statement in the next definition.

Definition 1.2 Equality

Complex numbers z1 = a1 + ib 1 and z2 = a2 + ib 2 are equal, z1 = z2 , if

a1 = a2 and b1 = b2 .

In terms of the symbols Re(z) and Im(z), Definition 1.2 states that z1 = z2 if
Re(z1 ) = Re(z2 ) and Im(z1 ) = Im(z2 ).
The totality of complex numbers or the set of complex numbers is usually
denoted by the symbol C. Because any real number a can be written as
z = a + 0i, we see that the set R of real numbers is a subset of C.

Arithmetic Operations numbers can be added, subtracted,

multiplied, and divided.If z1 = a1 + ib1 and z2 = a2 + ib2, these operations are
defined as follows.
Addition: z1 + z2 = ( a1 + ib1) + ( a2 + ib2) = ( a1 + a2) + i(b1 + b2)

Subtraction: z1−z2 = ( a1 + ib1) − (a2 + ib2) = ( a1 − a2) + i(b1 − b2)

Multiplication: z1 · z2 = (a1 + ib1 )(a2 + ib2 )

= a1 a2 − b1 b2 + i(b1 a2 + a1 b2 )
z1 a1 + ib1
Division: = , a2 = 0, or b2 = 0
z2 a2 + ib2
a1 a2 + b1 b2 b1 a2 − a1 b2
= 2 2 +i
a2 + b2 a22 + b22
The familiar commutative, associative, and distributive laws hold for com-
plex numbers:

z +z = z +z
1 2 2 1
Commutative laws:
 z z =z z
1 2 2 1


 z + (z + z ) = (z + z ) + z
1 2 3 1 2 3
Associative laws:
 z (z z ) = (z z )z
1 2 3 1 2 3

Distributive law: z1 (z2 + z3 ) = z1 z2 + z1 z3

In view of these laws, there is no need to memorize the definitions of

addition, subtraction, and multiplication.
Take two complex numbers in general as z1 = a1+ib1 and z2 =a2+ib2 and prove these laws.
Just see the left hand side should be equal to right hand side.
4 Chapter 1 Complex Numbers and the Complex Plane

Addition, Subtraction, and Multiplication

(i ) To add (subtract ) two complex numbers, simply add (subtract ) the

corresponding real and imaginary parts.
(ii ) To multiply two complex numbers, use the distributive law and the
fact that i2 = −1.

The definition of division deserves further elaboration, and so we will discuss

that operation in more detail shortly.

EXAMPLE 1 Addition and Multiplication

If z1 = 2 + 4i and z2 = −3 + 8i, find (a) z1 + z2 and (b) z1 z2 .

Solution (a) By adding real and imaginary parts, the sum of the two complex
numbers z1 and z2 is

z1 + z2 = (2 + 4i) + (−3 + 8i) = (2 − 3) + (4 + 8)i = −1 + 12i.

(b) By the distributive law and i2 = −1, the product of z1 and z2 is

z1 z2 = (2 + 4i) (−3 + 8i) = (2 + 4i) (−3) + (2 + 4i) (8i)

= −6 − 12i + 16i + 32i2
= (−6 − 32) + (16 − 12)i = −38 + 4i.

Zero and Unity The zero in the complex number system is the num-
ber 0 + 0i and the unity is 1 + 0i. The zero and unity are denoted by 0 and
1, respectively. The zero is the additive identity in the complex number
system since, for any complex number z = a + ib, we have z + 0 = z. To see
this, we use the definition of addition:

z + 0 = (a + ib) + (0 + 0i) = a + 0 + i(b + 0) = a + ib = z.

Similarly, the unity is the multiplicative identity of the system since, for
any complex number z, we have z · 1 = z · (1 + 0i) = z.
There is also no need to memorize the definition of division, but before
discussing why this is so, we need to introduce another concept.

Conjugate If z is a complex number, the number obtained by changing

the sign of its imaginary part is called the complex conjugate, or simply
conjugate, of z and is denoted by the symbol z̄. In other words, if z = a + ib,
1.1 Complex Numbers and Their Properties 5

then its conjugate is z̄ = a − ib. For example, if z = 6 + 3i, then z̄ = 6 − 3i;

if z = −5 − i, then z̄ = −5 + i. If z is a real number, say, z = 7, then
z̄ = 7. From the definitions of addition and subtraction of complex numbers,
it is readily shown that the conjugate of a sum and difference of two complex
numbers is the sum and difference of the conjugates:

z1 + z2 = z̄1 + z̄2 , z1 − z2 = z̄1 − z̄2 . (1)

Moreover, we have the following three additional properties:

 
z1 z̄1
z1 z2 = z̄1 z̄2 , = , z̄¯ = z. (2)
z2 z̄2

Of course, the conjugate of any finite sum (product) of complex numbers is

the sum (product) of the conjugates.
The definitions of addition and multiplication show that the sum and
product of a complex number z with its conjugate z̄ is a real number:

z + z̄ = (a + ib) + (a − ib) = 2a (3)

z z̄ = (a + ib)(a − ib) = a − i b = a + b .
2 2 2 2 2
(4)

The difference of a complex number z with its conjugate z̄ is a pure imaginary

number:

z − z̄ = (a + ib) − (a − ib) = 2ib. (5)

Since a = Re(z) and b = Im(z), (3) and (5) yield two useful formulas:
z + z̄ z − z̄
Re(z) = and Im(z) = . (6)
2 2i
However, (4) is the important relationship in this discussion because it enables
us to approach division in a practical manner.

Division
To divide z 1 by z 2 , multiply the numerator and denominator of z 1 /z2 by
the conjugate of z 2 . That is,
z1 z1 z̄2 z1 z̄2
= · = (7)
z2 z2 z̄2 z2 z̄2
and then use the fact that z2 z̄2 is the sum of the squares of the real and
imaginary parts of z 2 .

The procedure described in (7) is illustrated in the next example.

EXAMPLE 2 Division
If z1 = 2 − 3i and z2 = 4 + 6i, find z1 /z2 .
6 Chapter 1 Complex Numbers and the Complex Plane

Solution We multiply numerator and denominator by the conjugate

z̄2 = 4 − 6i of the denominator z2 = 4 + 6i and then use (4):

z1 2 − 3i 2 − 3i 4 − 6i 8 − 12i − 12i + 18i2 −10 − 24i

= = = = .
z2 4 + 6i 4 + 6i 4 − 6i 42 + 62 52
Because we want an answer in the form a + bi, we rewrite the last result by
dividing the real and imaginary parts of the numerator −10 − 24i by 52 and
reducing to lowest terms:
z1 10 24 5 6
= − − i = − − i.
z2 52 52 26 13

Inverses In the complex number system, every number z has a unique

additive inverse. As in the real number system, the additive inverse of
z = a + ib is its negative, −z, where −z = −a − ib. For any complex number
z, we have z + (−z) = 0. Similarly, every nonzero complex number z has a
multiplicative inverse. In symbols, for z = 0 there exists one and only one
nonzero complex number z −1 such that zz −1 = 1. The multiplicative inverse
z −1 is the same as the reciprocal 1/z.

EXAMPLE 3 Reciprocal

Find the reciprocal of z = 2 − 3i.

Solution By the definition of division we obtain

1 1 1 2 + 3i 2 + 3i 2 + 3i
= = = = .
z 2 − 3i 2 − 3i 2 + 3i 4+9 13

Answer should be in the form a + ib. ☞ That is,

1
= z −1 =
2 3
+ i.
z 13 13

You should take a few seconds to verify the multiplication

2
zz −1 = (2 − 3i) 13 3
+ 13 i = 1.

Remarks Comparison with Real Analysis

(i ) Many of the properties of the real number system R hold in the

complex number system C, but there are some truly remarkable
differences as well. For example, the concept of order in the
real number system does not carry over to the complex number
system. In other words, we cannot compare two complex numbers
z1 = a1 + ib1 , b1 = 0, and z2 = a2 + ib2 , b2 = 0, by means of
1.1 Complex Numbers and Their Properties 7

inequalities. Statements such as z1 < z2 or z2 ≥ z1 have no

meaning in C except in the special case when the two num-
bers z1 and z2 are real. See Problem 55 in Exercises 1.1.
Therefore, if you see a statement such as z1 = αz2 , α > 0,
it is implicit from the use of the inequality α > 0 that the sym-
bol α represents a real number.

(ii ) Some things that we take for granted as impossible in real analysis,
such as ex = −2 and sin x = 5 when x is a real variable, are per-
fectly correct and ordinary in complex analysis when the symbol x
is interpreted as a complex variable. See Example 3 in Section 4.1
and Example 2 in Section 4.3.

We will continue to point out other differences between real analysis and
complex analysis throughout the remainder of the text.

EXERCISES 1.1 Answers to selected odd-numbered problems begin on page ANS-2.

1. Evaluate the following powers of i.

(a) i8 (b) i11
(c) i42 (d) i105

2. Write the given number in the form a + ib.

(a) 2i3 − 3i2 + 5i (b) 3i5 − i4 + 7i3 − 10i2 − 9
 3
5 2 20 2
(c) + 3 − 18 (d) 2i6 + + 5i−5 − 12i
i i i −i

In Problems 3–20, write the given number in the form a + ib.

3. (5 − 9i) + (2 − 4i) 4. 3(4 − i) − 3(5 + 2i)
5. i(5 + 7i) 6. i(4 − i) + 4i(1 + 2i)
 
7. (2 − 3i)(4 + i) 8. 12 − 14 i 23 + 53 i
1 i
9. 3i + 10.
2−i 1+i
2 − 4i 10 − 5i
11. 12.
3 + 5i 6 + 2i
(3 − i)(2 + 3i) (1 + i)(1 − 2i)
13. 14.
1+i (2 + i)(4 − 3i)

(5 − 4i) − (3 + 7i) (4 + 5i) + 2i3

15. 16.
(4 + 2i) + (2 − 3i) (2 + i)2

17. i(1 − i)(2 − i)(2 + 6i) 18. (1 + i)2 (1 − i)3

 2
1 2−i
19. (3 + 6i) + (4 − i)(3 + 5i) + 20. (2 + 3i)
2−i 1 + 2i
8 Chapter 1 Complex Numbers and the Complex Plane

In Problems 21–24, use the binomial theorem∗

n n−1 n(n − 1) n−2 2
(A + B)n = An + A B+ A B + ···
1! 2!
n(n − 1)(n − 2) · · · (n − k + 1) n−k k
+ A B + · · · + Bn,
k!
where n = 1, 2, 3, . . . , to write the given number in the form a + ib.
 3
21. (2 + 3i)2 22. 1 − 12 i
23. (−2 + 2i)5 24. (1 + i)8

In Problems 25 and 26, find Re(z) and Im(z).

  
i 1 1
25. z = 26. z =
3−i 2 + 3i (1 + i)(1 − 2i)(1 + 3i)
In Problems 27–30, let z = x + iy. Express the given quantity in terms of x and y.
27. Re(1/z) 28. Re(z 2 )
29. Im(2z + 4z̄ − 4i) 30. Im(z̄ 2 + z 2 )

In Problems 31–34, let z = x + iy. Express the given quantity in terms of the
symbols Re(z) and Im(z).
31. Re(iz) 32. Im(iz)
33. Im((1 + i)z) 34. Re(z 2 )

In Problems 35 and 36, show that the indicated numbers satisfy the given equation.
In each case explain why additional solutions can be found.
√ √
2 2
35. z 2 + i = 0, z1 = − + i. Find an additional solution, z2 .
2 2
36. z 4 = −4; z1 = 1 + i, z2 = −1 + i. Find two additional solutions, z3 and z4 .

In Problems 37–42, use Definition 1.2 to solve each equation for z = a + ib.
37. 2z = i(2 + 9i) 38. z − 2z̄ + 7 − 6i = 0
2
39. z = i 40. z̄ 2 = 4z
2−i z
41. z + 2z̄ = 42. = 3 + 4i
1 + 3i 1 + z̄
In Problems 43 and 44, solve the given system of equations for z1 and z2 .
43. iz1 − iz2 = 2 + 10i 44. iz1 + (1 + i)z2 = 1 + 2i
−z1 + (1 − i)z2 = 3 − 5i (2 − i)z1 + 2iz2 = 4i

Focus on Concepts

45. What can be said about the complex number z if z = z̄? If (z)2 = (z̄)2 ?
46. Think of an alternative solution to Problem 24. Then without doing any sig-
nificant work, evaluate (1 + i)5404 .

∗ Recall that the coefficients in the expansions of (A + B)2 , (A + B)3 , and so on, can

also be obtained using Pascal’s triangle.

1.1 Complex Numbers and Their Properties 9

47. For n a nonnegative integer, in can be one of four values: 1, i, −1, and −i. In
each of the following four cases, express the integer exponent n in terms of the
symbol k, where k = 0, 1, 2, . . . .
(a) in = 1 (b) in = i
(c) in = −1 (d) in = −i

48. There is an alternative to the procedure given in (7). For example, the quotient
(5 + 6i)/(1 + i) must be expressible in the form a + ib:
5 + 6i
= a + ib.
1+i
Therefore, 5 + 6i = (1 + i)(a + ib). Use this last result to find the given quotient.
Use this method to find the reciprocal of 3 − 4i.
√
49. Assume for the moment that 1 + i makes sense in the complex number system.
How would you then demonstrate the validity of the equality
√ √ √
1+i= 1
2
+ 1
2
2+i − 12 + 1
2
2?

50. Suppose z1 and z2 are complex numbers. What can be said about z1 or z2 if
z1 z2 = 0?
51. Suppose the product z1 z2 of two complex numbers is a nonzero real constant.
Show that z2 = kz̄1 , where k is a real number.
52. Without doing any significant work, explain why it follows immediately from
(2) and (3) that z1 z̄2 + z̄1 z2 = 2Re(z1 z̄2 ).
53. Mathematicians like to prove that certain “things” within a mathematical sys-
tem are unique. For example, a proof of a proposition such as “The unity in
the complex number system is unique” usually starts out with the assumption
that there exist two different unities, say, 11 and 12 , and then proceeds to show
that this assumption leads to some contradiction. Give one contradiction if it
is assumed that two different unities exist.
54. Follow the procedure outlined in Problem 53 to prove the proposition “The zero
in the complex number system is unique.”
55. A number system is said to be an ordered system provided it contains a
subset P with the following two properties:
First, for any nonzero number x in the system, either x or −x is (but not both)
in P.
Second, if x and y are numbers in P, then both xy and x + y are in P.
In the real number system the set P is the set of positive numbers. In the real
number system we say x is greater than y, written x > y, if and only if x − y
is in P . Discuss why the complex number system has no such subset P . [Hint:
Consider i and −i.]
10 Chapter 1 Complex Numbers and the Complex Plane

1.2 Complex Plane

1.2 z = x + iy is uniquely determined by an ordered pair
A complex number of real numbers
(x, y). The first and second entries of the ordered pairs correspond, in turn, with the
real and imaginary parts of the complex number. For example, the ordered pair (2, −3)
corresponds to the complex number z = 2 − 3i. Conversely, z = 2 − 3i determines the
ordered pair (2, −3). The numbers 7, i, and −5i are equivalent to (7, 0), (0, 1), (0, −5),
respectively. In this manner we are able to associate a complex number z = x + iy with a
point (x, y) in a coordinate plane.

y-axis
or Complex Plane Because of the correspondence between a complex
imaginary axis
number z = x + iy and one and only one point (x, y) in a coordinate plane,
z = x + iy or we shall use the terms complex number and point interchangeably. The coor-
y (x, y) dinate plane illustrated in Figure 1.1 is called the complex plane or simply
the z -plane. The horizontal or x-axis is called the real axis because each
point on that axis represents a real number. The vertical or y-axis is called
x
x-axis the imaginary axis because a point on that axis represents a pure imaginary
or
real axis number.

Figure 1.1 z-plane

Vectors In other courses you have undoubtedly seen that the numbers
in an ordered pair of real numbers can be interpreted as the components of
y
a vector. Thus, a complex number z = x + iy can also be viewed as a two-
z = x + iy dimensional position vector, that is, a vector whose initial point is the origin
and whose terminal point is the point (x, y). See Figure 1.2. This vector
interpretation
 prompts us to define the length of the vector z as the distance
x2 + y 2 from the origin to the point (x, y). This length is given a special
name.
x

Figure 1.2 z as a position vector

Definition 1.3 Modulus

The modulus of a complex number z = x + iy, is the real number


|z| = x2 + y 2 . (1)

The modulus |z| of a complex number z is also called the absolute value
of z. We shall use both words modulus and absolute value throughout this
text.

EXAMPLE 1 Modulus of a Complex Number

 2 − 3i, then √from (1) we find the modulus of thenumber to be
If z =
|z| = 22 + (−3)2 = 13. If z = −9i, then (1) gives |−9i| = (−9)2 = 9.