Download presentation

Published byDaphne Simpkin Modified over 4 years ago

1
**Copyright © 2013, 2009, 2005 Pearson Education, Inc.**

Equations and Inequalities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

2
**1.3 Complex Numbers Basic Concepts of Complex Numbers**

Operations on Complex Numbers

3
**Basic Concepts of Complex Numbers**

There is no real number solution of the equation since no real number, when squared, gives To extend the real number system to include solutions of equations of this type, the number i is defined to have the following property.

4
**Basic Concepts of Complex Numbers**

If a and b are real numbers, then any number of the form a + bi is a complex number. In the complex number a + bi , a is the real part and b is the imaginary part.

5
**Basic Concepts of Complex Numbers**

Two complex numbers a + bi and c + di are equal provided that their real parts are equal and their imaginary parts are equal; that is if and only if and

6
**Basic Concepts of Complex Numbers**

For complex number a + bi, if b = 0, then a + bi = a. Thus, the set of real numbers is a subset of the set of complex numbers.

7
**Basic Concepts of Complex Numbers**

If a = 0 and b ≠ 0, the complex number is said to be a pure imaginary number. A pure imaginary number, or a number like 7 + 2i with a ≠ 0 and b ≠ 0, is a nonreal complex number. A complex number written in the form a + bi (or a + ib) is in standard form.

8
**Copyright © 2013, 2009, 2005 Pearson Education, Inc.**

8

9
THE EXPRESSION

10
**Write as the product of a real number and i, using the definition of**

Example 1 WRITING AS Write as the product of a real number and i, using the definition of (a) Solution

11
**Write as the product of a real number and i, using the definition of**

Example 1 WRITING AS Write as the product of a real number and i, using the definition of (b) Solution

12
**Write as the product of a real number and i, using the definition of**

Example 1 WRITING AS Write as the product of a real number and i, using the definition of (c) Solution Product rule for radicals

13
**Operations on Complex Numbers**

Products or quotients with negative radicands are simplified by first rewriting for a positive number a. Then the properties of real numbers and the fact that are applied

14
**Operations on Complex Numbers**

Caution When working with negative radicands, use the definition before using any of the other rules for radicals.

15
**Operations on Complex Numbers**

Caution In particular, the rule is valid only when c and d are not both negative. is correct, while is incorrect.

16
**First write all square roots in terms of i.**

FINDING PRODUCTS AND QUOTIENTS INVOLVING Example 2 Multiply or divide, as indicated. Simplify each answer. (a) Solution First write all square roots in terms of i. i 2 = −1

17
**Multiply or divide, as indicated. Simplify each answer.**

FINDING PRODUCTS AND QUOTIENTS INVOLVING Example 2 Multiply or divide, as indicated. Simplify each answer. (b) Solution

18
**Multiply or divide, as indicated. Simplify each answer.**

FINDING PRODUCTS AND QUOTIENTS INVOLVING Example 2 Multiply or divide, as indicated. Simplify each answer. (c) Solution Quotient rule for radicals

19
**Multiply or divide, as indicated. Simplify each answer.**

FINDING PRODUCTS AND QUOTIENTS INVOLVING Example 2 Multiply or divide, as indicated. Simplify each answer. (d) Solution

20
**Write in standard form a + bi. Solution**

SIMPLIFYING A QUOTIENT INVOLVING Example 3 Write in standard form a + bi. Solution

21
**Be sure to factor before simplifying**

SIMPLIFYING A QUOTIENT INVOLVING Example 3 Write in standard form a + bi. Solution Be sure to factor before simplifying Factor. Lowest terms

22
**Addition and Subtraction of Complex Numbers**

For complex numbers a + bi and c + di, and

23
**Find each sum or difference.**

ADDING AND SUBTRACTING COMPLEX NUMBERS Example 4 Find each sum or difference. (a) Add imaginary parts. Solution Add real parts. Commutative, associative, distributive properties

24
**Find each sum or difference.**

ADDING AND SUBTRACTING COMPLEX NUMBERS Example 4 Find each sum or difference. (b) Solution

25
**Multiplication of Complex Numbers**

The product of two complex numbers is found by multiplying as though the numbers were binomials and using the fact that i2 = – 1, as follows. FOIL Distributive property; i 2 = – 1

26
**Multiplication of Complex Numbers**

For complex numbers a + bi and c + di,

27
**Find each product. (a) Solution MULTIPLYING COMPLEX NUMBERS Example 5**

FOIL Multiply. Combine like terms; i2 = −1 Standard form

28
**Remember to add twice the product of the two terms.**

MULTIPLYING COMPLEX NUMBERS Example 5 Find each product. (b) Remember to add twice the product of the two terms. Solution Square of a binomial Multiply. i 2 = −1 Standard form

29
**Find each product. (c) Solution MULTIPLYING COMPLEX NUMBERS Example 5**

Product of the sum and difference of two terms i 2 = −1 Multiply. Standard form

30
**Example 5(c) showed that**

The numbers 6 + 5i and 6 – 5i differ only in the sign of their imaginary parts and are called complex conjugates. The product of a complex number and its conjugate is always a real number. This product is the sum of the squares of real and imaginary parts.

31
**Property of Complex Conjugates**

For real numbers a and b,

32
**Write each quotient in standard form a + bi.**

DIVIDING COMPLEX NUMBERS Example 6 Write each quotient in standard form a + bi. (a) Solution Multiply by the complex conjugate of the denominator in both the numerator and the denominator. Multiply.

33
**Write each quotient in standard form a + bi.**

DIVIDING COMPLEX NUMBERS Example 6 Write each quotient in standard form a + bi. (a) Solution Combine like terms; i 2 = −1

34
**Write each quotient in standard form a + bi.**

DIVIDING COMPLEX NUMBERS Example 6 Write each quotient in standard form a + bi. (a) Solution Write in lowest terms and standard form. Check

35
**Write each quotient in standard form a + bi.**

DIVIDING COMPLEX NUMBERS Example 6 Write each quotient in standard form a + bi. (b) Solution – i is the conjugate of i. Multiply.

36
**Write each quotient in standard form a + bi.**

DIVIDING COMPLEX NUMBERS Example 6 Write each quotient in standard form a + bi. (b) Solution i 2 = −1(−1) = 1 Standard form

37
**Simplifying Powers of i**

Powers of i can be simplified using the facts

38
Powers of i and so on.

39
Powers of i cycle through the same four outcomes (i, –1, –i, and 1) since i4 has the same multiplicative property as 1. Also, any power of i with an exponent that is a multiple of 4 has value 1. As with real numbers, i0 = 1.

40
**Simplify each power of i.**

SIMPLIFYING POWERS OF i Example 7 Simplify each power of i. (a) Solution Since i 4 = 1, write the given power as a product involving i 4.

41
**Simplify each power of i.**

SIMPLIFYING POWERS OF i Example 7 Simplify each power of i. (b) Solution Multiply by 1 in the form of i 4 to create the least positive exponent for i.

Similar presentations

Presentation is loading. Please wait....

OK

Introduction to Complex Numbers

Introduction to Complex Numbers

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google