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Complex NumbersSection P.7

Prerequisites

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Definition: Imaginary Number iThe imaginary number i is defined by

i2 = –1.

We may also write

Definition: Complex NumbersThe set of complex numbers is the set of all

numbers of the form

a + bi,

where a and b are real numbers.

.1i

Definitions

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A complex number is formed as a real number plus a real multiple of i.

In a + bi, a is called the real part and b is called the imaginary part.

Two complex numbers a + bi and c + di are equal if and only if their real parts are equal (a = c) and their imaginary parts are equal (b = d).

If b = 0, then a + bi is a real number.

If b ≠ 0, then a + bi is an imaginary number.

The form a + bi is the standard form of a complex number.

Definitions

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Definition: Addition, Subtraction, and MultiplicationIf a + bi and c + di are complex numbers, we define their sum, difference, and product as follows:

(a + b i) + (c + d i) = (a + c) + (b + d)i

(a + b i) – (c + d i) = (a – c) + (b – d)i

(a + b i)(c + d i) = (ac – bd) + (bc + ad)i

Addition, Subtraction, and Multiplication

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We can find whole-number powers of i by using the definition of multiplication. Since i1 = i and i2 = –1, we have

i3 = i1i2 = i(–1) = –i

and

i4 = i1i3 = i(–i) = –i2 = 1.

The first eight powers of i are listed here.

i1 = i i2 = –1 i3 = –i i4 = 1

i5 = i i6 = –1 i7 = –i i8 = 1

Powers of Complex Numbers

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Theorem: Complex ConjugatesIf a and b are real numbers, then the product of a + bi and its conjugate a – bi is the real number a2 + b2. In symbols,

(a + bi)(a – bi) = a2 + b2.

The complex numbers a + bi and a – bi are called complex conjugates of each other.

We use the theorem about complex conjugates to divide imaginary numbers, in a process that is similar to rationalizing a denominator.

Division of Complex Numbers

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Definition: Square Root of a Negative NumberFor any positive number b,

.bib

For any positive real number b, we have

and So there are two square roots of –b.

They are and We call the principal

square root of –b and make the following definition.

All operations with complex numbers must be performed after converting to the a + bi form.

bbi 2

.2

bbi bi .bi bi

Roots of Negative Numbers