Copyright © Cengage Learning. All rights reserved. 2.4 Complex Numbers.

Copyright © Cengage Learning. All rights reserved.

2.4 Complex Numbers

2

What You Should Learn

• Use the imaginary unit i to write complex numbers.

• Add, subtract, and multiply complex numbers.

• Use complex conjugates to write the quotient of two complex numbers in standard form.

• Find complex solutions of quadratic equations.

3

The Imaginary Unit i

4


Some quadratic equations have no real solutions. For instance, the quadratic equation x2 + 1 = 0 has no real solution because there is no real number that can be squared to produce –1.

To overcome this deficiency, mathematicians created an expanded system of numbers using the imaginary unit i, defined as

where i2 = –1. By adding real numbers to real multiples of this imaginary unit, you obtain the set of complex numbers.

Imaginary unit

5


Each complex number can be written in the standard form a + bi. For instance, the standard form of the complex number is –5 + 3i because

In the standard form a + bi, the real number a is called the real part of the complex number a + bi and the number bi (where b is a real number) is called the imaginary part of the complex number.

6


7


The set of real numbers is a subset of the set of complex numbers, as shown in Figure 2.32. This is true because every real number a can be written as a complex number using b = 0. That is, for every real number a, you can writea = a + 0i.

Figure 2.32

8

Operations with Complex Numbers

9


To add (or subtract) two complex numbers, you add (or subtract) the real and imaginary parts of the numbers separately.

10

Example 1 – Adding and Subtracting Complex Numbers

Perform the operation(s) and write the result in standard form.

a. (3 – i ) + (2 + 3i )

b. (1 + 2i ) – (4 + 2i )

c. 3 – (–2 + 3i ) + (–5 + i )

d. (3 + 2i ) + (4 – i ) – (7 + i )

11

Example 1 – Solution

a. (3 – i ) + (2 + 3i ) = 3 – i + 2 + 3i

= (3 + 2) + (–i + 3i )

= 5 + 2i

b. (1 + 2i ) – (4 + 2i ) = 1 + 2i – 4 – 2i

= (1 – 4) + (2i – 2i )

= –3 + 0i

= –3

Remove parentheses.

Group like terms.

Write in standard form.

Remove parentheses.

Group like terms.

Simplify.


12


c. 3 – (–2 + 3i ) + (–5 + i ) = 3 + 2 – 3i – 5 + i

= (3 + 2 – 5) + (– 3i + i )

= 5 + 2i

d. (3 + 2i ) + (4 – i ) – (7 + i ) = 3 + 2i + 4 – i – 7 – i

= (3 + 4 – 7) + (2i – i – i )

= 0 + 0i

= 0

cont’d

13


Many of the properties of real numbers are valid for complex numbers as well. Here are some examples.

Associative Properties of Addition and Multiplication

Commutative Properties of Addition and Multiplication

Distributive Property of Multiplication over Addition

Notice how these properties are used when two complex numbers are multiplied.

14


(a + bi ) + (c + di ) = a(c + di ) + bi(c + di )

= ac + (ad)i + (bc)i + (bd)i2

= ac + (ad)i + (bc)i + (bd)(–1)

= ac – bd + (ad)i + (bc)i

= (ac – bd) + (ad + bc)i

The procedure above is similar to multiplying two polynomials and combining like terms, as in the FOIL Method.

Distributive Property


i

2 = –1

Commutative Property

Associative Property

15

Example 2 – Multiplying Complex Numbers

Perform the operation(s) and write the result in standard form.

a. 5(–2 + 3i )

b. (2 – i )(4 + 3i )

c. (3 + 2i )(3 – 2i )

d. 4i(–1 + 5i )

e. (3 + 2i )2

16


a. 5(– 2 + 3i ) = 5(–2) + 5(3i )

= –10 + 15i

b. (2 – i )(4 + 3i ) = 2(4 + 3i ) – i(4 + 3i )

= 8 + 6i – 4i – 3i2

= 8 + 6i – 4i – 3(–1)

= 8 + 3 + 6i – 4i

= 11 + 2i


Simplify.


Product of binomials

Group like terms.


i 2 = –1

17


c. (3 + 2i ) (3 – 2i ) = 3(3 – 2i ) + 2i(3 – 2i )

= 9 – 6i + 6i – 4i2

= 9 – 4(–1)

= 9 + 4

= 13

d. 4i(–1 + 5i ) = 4i(–1) – 4i(5i )

= –4i + 20i2

= –4i + 20(–1)

= –20 – 4i


Simplify.




i 2 = –1

Simplify.

i 2 = –1

cont’d

18


e. (3 + 2i)2 = 9 + 6i + 6i + 4i2

= 9 + 12i + 4(–1)

= 9 – 4 + 12i

= 5 + 12i



i 2 = – 1

Group like terms

cont’d

19

Complex Conjugates

20

Complex Conjugates

Notice in Example 2(c) that the product of two complex numbers can be a real number.

This occurs with pairs of complex numbers of the forms a + bi and a – bi called complex conjugates.

(a + bi )(a + bi ) = a2 – abi + abi – b2i2

= a2 – b2(–1)

= a2 + b2

21

Example 3 – Multiplying Conjugates

Multiply each complex number by its complex conjugate.

a. (1 + i )

b. 3 – 5i

22


a. The complex conjugate of 1 + i is 1 – i.

(1 + i )(1 – i ) = 12 – i2

= 1 – (–1)

= 2

b. The complex conjugate of 3 – 5i is 3 + 5i.

(3 – 5i )(3 + 5i ) = 32 – (5i)2

= 9 – 25i2

= 9 – 25(–1)

= 34

23

Complex Conjugates

To write the quotient of a + bi and c + di in standard form, where c and d are not both zero, multiply the numerator and denominator by the complex conjugate of the denominator to obtain

Multiply numerator and denominator by complex conjugate of denominator.

Standard form

24

Example 4 – Writing a Quotient of Complex Numbers in Standard Form

Write the quotient in standard form.

Solution:

Multiply numerator and denominator by complex conjugate of denominator.

Expand.

i

2 = –1

Simplify.


25

Complex Solutions of Quadratic Equations

26


When using the Quadratic Formula to solve a quadratic equation, you often obtain a result such as , which you know is not a real number. By factoring out i = , you can write this number in standard form.

The number is called the principal square root of –3.

27


28

Example 5 – Writing Complex Numbers in Standard Form

a.

b.

c.

29

Example 6 – Complex Solutions of a Quadratic Equation

Solve (a) x2 + 4 = 0 and (b) 3x2 – 2x + 5 = 0

Solution:a. x2 + 4 = 0

x2 = –4

x = 2i

Write original equation.

Subtract 4 from each side.

Extract square roots.

30


b. 3x2 – 2x + 5 = 0 Write original equation.

Quadratic Formula.

Simplify.


cont’d

Copyright © Cengage Learning. All rights reserved. 2.4 Complex Numbers.

Documents

Transcript of Copyright © Cengage Learning. All rights reserved. 2.4 Complex Numbers.