Chapter 2
Complex Numbers
2.1 The Square Root of —1
The study of complex numbers begins when we are bold enough to ask a very childish question:
what is the square root of —1? Forbidden by sixth grade teachers the world over, the expression
is nevertheless the key to a whole branch of math.
Historically, people were led to write by the quest to solve equations. Clearly, an equation like
x2 + 1 = 0 (2.1)
C
has only the solutions x = ± M I. In order to be able to solve all equations, it was decided to
accept as a legitimate number.
The square root of —1 is usually written as i. This weird number shows its weird properties
almost from the beginning, as we shall see. It is not a real number in the mathematical sense. This
is not to say it is not real, at least any less than negatives are. If we multiply i by a real number
like 2 or 71 , we get a number like 2i or 71i•, there is no way to simplify this product. Numbers
like this, formed by multiplying i by a real, are called pure imaginary numbers, though you
should not let this prejudice of name keep you from accepting them as regular numbers. Treat the
word imaginary as a purely mathematical definition.
If we multiply i by itself, we get = —1, as we would expect. But
notice that if we try to combine = the radicals and write = 1, we will get
the wrong answer. Manipulations like this are forbidden.
If we keep taking powers of i we get z
3
i(—i) 5 _ ii4
etc. The powers of i go in cycles of 4: i, —1, —i, 1, i, —1, —i, 1, etc.
EXERCISE 2-1 What is i17 ? How about i69 ? i1971 ?
14CHAPTER 2. COMPLEX NUMBERS
is just 0. Those complex numbers which are not real are called imaginary numbers. (This is not
exactly the same as pure imaginary numbers; can you write a number which is imaginary but not
pure imaginary?)
1 .2 Complex Number Operations
The so-called complex numbers are just the numbers you get when you add a real to an
imaginary, like + 3i or —17 + Vi. Every real number is also a complex number; the imaginary
component
< 13
EXAMPLE 2-1 Let's clear up these confusing definitions by looking at some examples. 3 is both
real and complex, but not imaginary. 3i is not real, but is complex, imaginary, and pure
imaginary. 3 + 3i is neither real nor pure imaginary, but is imaginary and complex. (We realize
this is unnecessarily complicated, but they are called complex numbers... )
Complex variables are usually designated by z or w, for no other reason than that letters near
the end of the alphabet are best for variables, and x and y are already typically used for reals.
To add two complex numbers together, all we have to do is add their real and imaginary parts
separately, as in the following examples.
EXAMPLE 2-2 Let's add 3 + 4i to -3 + 8i. The sum is just 3 -3+4i + 8i = 12i.
EXERCISE 2-3 Find the general formula for the sum (Zl + Z2i) + (WI + W2i).
Subtraction follows easily from addition. Furthermore, we can multiply two complex numbers
with the distributive law.
EXAMPLE 2-3 Let's multiply 3 + 4i by —3 + 8i. The product is
3(-3 + 8i) + 4i(-3 + 8i)
= 12i = + 12i. -9 +
-
(Note the negative sign of the 32; it comes from i times i.)
EXERCISE 2-5 Find the general formula for the product (Zl + + W2i).
EXERCISE 2-6 Simplify (Zl + - Z2i).
When we divide two complex numbers, we clear all instances of i from the denominator in
exactly the same way as rationalizing a denominator which contains square roots. We use the fact
that the complex number a + bi multiplied by a — bi is real, just as a + multiplied by a — gets rid
of the square root. (You showed this in Exercise 2-6 above, right?)
the ART of PROBLEM SOLVING 15
EXAMPLE 2-4 Let's divide 3 + 4i by -3 + 8i. The quotient is
23 - 36i 23 36
73 73 73
1
EXERCISE 2-7 What is•
2- 3 1
EXERCISE 2-8 Find the general formula for the quotient (Zl + + W2i).
We can do more complicated operations, like taking square or cube roots of complex
numbers, but we'll let that wait for now. We should define a couple of basic notations, however.
Consider an arbitrary complex number z = a + bi. We denote the number a — bi by Z, and call it
the conjugate of z. We call the number a the real part of z, and denote it by Re(z). Similarly, the
number bi is the imaginary part of z. WARNING: The expression Im(z) refers to the real
coefficient of this imaginary part, not the imaginary part itself. Thus Im(a + bi) = b, NOT bi.
EXERCISE 2-9 Prove that = z for all complex z.
EXERCISE 2-10 What is the conjugate of a real number a? of a pure imaginary number bi?
EXERCISE 2-11 Show that z + w = Z + for all z and w. Does this fact surprise you?
EXERCISE 2-12 Show that = for all z and w. Does this surprise you?
EXERCISE 2-13 How about (z/w)? Surprising?
EXAMPLE 2-5 Consider Im(z) + Im(z). Let z = a + bi, so that Z = a — bi. Then Im(z) = b and
Im(Z) = —b, so that Im(z) + Im(Z) = 0, no matter what z is.
EXERCISE 2-14 What is Re(z) + ilm(z)?
Problems to Solve for Chapter 2
17. Find (MAe 1987)
18. Which are true? (MAO 1987) (Don't look back at the text!)
i) Zl + z2 =
ii) ZIZ2 =
iii) Zl/Z2 =
Zi/Zä
19. Evaluate . 1991)
20. Find i¯18 + + + 19 + i18 . (MAO 1991)
21. Find Re [(a + bi)(c + di)l in terms of a, b, c, and d. (MAO 1991)
22. Evaluate (2 + i)3 . (MA€ 1991)
23. Find (1 + - 2i)3 . (MG 1987)
24. Simplify . (MAe 1990)
25. If F(x) = 3x3 - 2x2 + x - 3, find + i). (MG 1990)
26. Which of the following are true? (MAO 1987)
i) Z +3i =z—
3i ii) iz = —E iii) (2
+ i)
