17.1 Complex Numbers

INTRODUCTION

You have undoubtedly encountered complex numbers in your earlier courses in mathematics. When you first learned to solve a quadratic equation ax2 + bx + c = 0 by the quadratic formula, you saw that the roots of the equation are not real, that is, complex, whenever the discriminant b2 − 4ac is negative. So, for example, simple equations such as x2 + 5 = 0 and x2 + x + 1 = 0 have no real solutions. For example, the roots of the last equation are and . If it is assumed that , then the roots are written and .

A Definition

Two hundred years ago, around the time that complex numbers were gaining some respectability in the mathematical community, the symbol i was originally used as a disguise for the embarrassing symbol . We now simply 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 17.1.1 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.

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

Terminology

The number i in Definition 17.1.1 is called the imaginary unit. The real number x in z = x + iy is called the real part of z; the real number y 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 z = 4 − 9i, then Re(z) = 4 and Im(z) = −9. A real constant multiple 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 real and imaginary parts are equal. Since this simple concept is sometimes useful, we formalize the last statement in the next definition.

Note: The imaginary part of z = 4 − 9i is −9, not −9i.

DEFINITION 17.1.2 Equality

Complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal, z1 = z2, if

Re(z1) = Re(z2)     and     Im(z1) = Im(z2).

A complex number x + iy = 0 if x = 0 and y = 0.

Arithmetic Operations

Complex numbers can be added, subtracted, multiplied, and divided. If z1 = x1 + iy1 and z2 = x2 + iy2, these operations are defined as follows.

The familiar commutative, associative, and distributive laws hold for complex numbers.

In view of these laws, there is no need to memorize the definitions of addition, subtraction, and multiplication. To add (subtract) two complex numbers, we simply add (subtract) the corresponding real and imaginary parts. To multiply two complex numbers, we use the distributive law and the fact that i2 = −1.

EXAMPLE 1 Addition and Multiplication

If z1 = 2 + 4i and z2 = −3 + 8i, find (a) z1 + z2 and (b) z1z2.

SOLUTION

(a) By adding the real and imaginary parts of the two numbers, we get

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

(b) Using the distributive law, we have

≡

There is also no need to memorize the definition of division, but before discussing that we need to introduce another concept.

Conjugate

If z is a complex number, then the number obtained by changing the sign of its imaginary part is called the complex conjugate or, simply, the conjugate of z. If z = x + iy, then its conjugate is

For example, if z = 6 + 3i, then = 6 − 3i; if z = −5 − i, then = −5 + i. If z is a real number, say z = 7, then = 7. From the definition of addition it can be readily shown that the conjugate of a sum of two complex numbers is the sum of the conjugates:

.

Moreover, we have the additional three properties

The definitions of addition and multiplication show that the sum and product of a complex number z and its conjugate are also real numbers:

(1)

(2)

The difference between a complex number z and its conjugate is a pure imaginary number:

(3)

Since x = Re(z) and y = Im(z), (1) and (3) yield two useful formulas:

However, (2) is the important relationship that enables us to approach division in a more practical manner: To divide z1 by z2, we multiply both numerator and denominator of z1/z2 by the conjugate of z2. This procedure is illustrated in the next example.

EXAMPLE 2 Division

If z1 = 2 − 3i and z2 = 4 + 6i, find (a) and (b) .

SOLUTION

In both parts of this example we shall multiply both numerator and denominator by the conjugate of the denominator and then use (2).

(a)

(b) ≡

Geometric Interpretation

A complex number z = x + iy is uniquely determined by an ordered pair 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). In this manner we are able to associate a complex number z = x + iy with a point (x, y) in a coordinate plane. But, as we saw in Section 7.1, an ordered pair of real numbers can be interpreted as the components of a vector. Thus, a complex number z = x + iy can also be viewed as a vector whose initial point is the origin and whose terminal point is (x, y). The coordinate plane illustrated in FIGURE 17.1.1 is called the complex plane or simply the z-plane. The horizontal or x-axis is called the real axis and the vertical or y-axis is called the imaginary axis. The length of a vector z, or the distance from the origin to the point (x, y), is clearly . This real number is given a special name.

A graph. The horizontal axis is labeled x and the vertical axis is labeled y. The graph shows a point labeled z = x plus i y at the center followed by an arrow from the origin.

FIGURE 17.1.1 z as a position vector

DEFINITION 17.1.3 Modulus or Absolute Value

The modulus or absolute value of z = x + iy, denoted by |z|, is the real number

(4)

EXAMPLE 3 Modulus of a Complex Number

If z = 2 − 3i, then |z| = ≡

As FIGURE 17.1.2 shows, the sum of the vectors z1 and z2 is the vector z1 + z2. For the triangle given in the figure, we know that the length of the side of the triangle corresponding to the vector z1 + z2 cannot be longer than the sum of the remaining two sides. In symbols this is

(5)

A graph. The horizontal axis is labeled x and the vertical axis is labeled y. The graph shows two increasing arrows, namely, z subscript 1 and z subscript 2, which are closer to the horizontal and vertical axis, respectively. Another rising arrow between them is labeled z subscript 1 plus z subscript 2. A dashed rising arrow, also labeled z subscript 1, from the head of z subscript 2 touches the head of z subscript 1 plus z subscript 2 forming a triangle.

FIGURE 17.1.2 Sum of vectors

The result in (5) is known as the triangle inequality and extends to any finite sum:

(6)

Using (5) on z1 + z2 + (−z2), we obtain another important inequality:

(7)

REMARKS

Many of the properties of the real system hold in the complex number system, but there are some remarkable differences as well. For example, we cannot compare two complex numbers z1 = x1 + iy1, y1 ≠ 0, and z2 = x2 + iy2, y2 ≠ 0, by means of inequalities. In other words, statements such as z1 < z2 and z2 ≥ z1 have no meaning except in the case when the two numbers z1 and z2 are real. We can, however, compare the absolute values of two complex numbers. Thus, if z1 = 3 + 4i and z2 = 5 − i, then |z1| = 5 and |z2| = , and consequently |z1| < |z2|. This last inequality means that the point (3, 4) is closer to the origin than is the point (5, −1).

17.1 Exercises Answers to selected odd-numbered problems begin on page ANS-44.

In Problems 1–4, evaluate the given power of

In Problems 5–32, write the given number in the form a + ib.

  1. 2i3 − 3i2 + 5i
  2. 3i5 − i4 + 7i3 − 10i2 − 9
  3. 4i11 − 5i3 + 10i−2
  4. 6i−4 + 3i15 − i6 + 7i2
  5. (5 − 9i) + (2 − 4i)
  6. 3(4 − i) − 3(5 + 2i)
  7. i(5 + 7i)
  8. i(4 − i) + 4i(1 + 2i)
  9. (2 − 3i)(4 + i)
  10. ( − i)( + i)
  11. (2 + 3i)2
  12. (1 − i)3












  21. i(1 − i)(2 − i)(2 + 6i)
  22. (1 + i)2(1 − i)3
  23. (3 + 6i) + (4 − i)(3 + 5i) +








In Problems 33–38, let z = x + iy. Find the indicated expression.

  1. Re(1/z)
  2. Re(z2)
  3. Im(2z + 4 − 4i)
  4. Im( + z2)
  5. |z − 1 − 3i|
  6. |z + 5|

In Problems 39–44, use Definition 17.1.2 to find a complex number z satisfying the given equation.

  1. 2z = i(2 + 9i)
  2. z − 2 + 7 − 6i = 0
  3. z2 = i
  4. = 4z

In Problems 45 and 46, determine which complex number is closer to the origin.

  1. 10 + 8i, 11 − 6i
  2. − i, + i
  3. Prove that |z1 − z2| is the distance between the points z1 and z2 in the complex plane.
  4. Show for all complex numbers z on the circle x2 + y2 = 4 that |z + 6 + 8i| ≤ 12.

Discussion Problems

  1. For n a nonnegative integer, in can be one of four values: i, −1, −i, and 1. In each of the following four cases express the integer exponent n in terms of the symbol k, where k = 0, 1, 2, . . . .
    1. in = i
    2. in = −1
    3. in = −i
    4. in = 1
    1. Without doing any significant work, such as multiplying out or using the binomial theorem, think of an easy way of evaluating (1 + i)8.
    2. Use your method in part (a) to evaluate (1 + i)64.