College Algebra Twelfth Edition Chapter 1 Equations and
College Algebra Twelfth Edition Chapter 1 Equations and Inequalities Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 1
Section 1. 3 Complex Numbers • Basic Concepts of Complex Numbers • Operations on Complex Numbers Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 2
Basic Concepts of Complex Numbers (1 of 5) There is no real number solution of the equation since no real number, when squared, gives − 1. To extend the real number system to include solutions of equations of this type, the number i is defined to have the following property. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 3
Basic Concepts of Complex Numbers (2 of 5) If a and b are real numbers, then any is a complex number of the form number. In the complex number a is the real part and b is the imaginary part. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 4
Basic Concepts of Complex Numbers (3 of 5) Two complex numbers are equal provided that their real parts are equal and their imaginary parts are equal; that is if and only if a = c and b = d. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 5
Basic Concepts of Complex Numbers (4 of 5) For complex number if b = 0, then Thus, the set of real numbers is a subset of the set of complex numbers. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 6
Basic Concepts of Complex Numbers (5 of 5) the complex number is said If a = 0 and to be a pure imaginary number. A pure imaginary number, or a number like is a nonreal complex number. A complex number written in the form is in standard form. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 7
Complex Numbers a + bi, for a and b Real Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 8
The Expression the Square Root of −a Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 9
Example 1: Writing the Square Root of −a as i Times the Square Root of a (1 of 3) Write as the product of a real number and i, using the definition of (a) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 10
Example 1: Writing the Square Root of −a as i Times the Square Root of a (2 of 3) Write as the product of a real number and i, using the definition of (b) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 11
Example 1: Writing the Square Root of −a as i Times the Square Root of a (3 of 3) Write as the product of a real number and i, using the definition of (c) Solution Product rule for radicals Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 12
Operations on Complex Numbers (1 of 3) 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 Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 13
Operations on Complex Numbers (2 of 3) Caution When working with negative radicands, use the definition before using any of the other rules for radicals. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 14
Operations on Complex Numbers (3 of 3) Caution In particular, the rule is valid only when c and d are not both negative. is correct, while is incorrect. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 15
Example 2: Finding Products and Quotients Involving the Square Root of −a (1 of 4) Multiply or divide, as indicated. Simplify each answer. (a) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 16
Example 2: Finding Products and Quotients Involving the Square Root of −a (2 of 4) Multiply or divide, as indicated. Simplify each answer. (b) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 17
Example 2: Finding Products and Quotients Involving the Square Root of −a (3 of 4) Multiply or divide, as indicated. Simplify each answer. (c) Solution Quotient rule for radicals Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 18
Example 2: Finding Products and Quotients Involving the Square Root of −a (4 of 4) Multiply or divide, as indicated. Simplify each answer. (d) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 19
Example 3: Simplifying a Quotient Involving the Square Root of −a (1 of 2) Write in standard form Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 20
Example 3: Simplifying a Quotient Involving the Square Root of −a (2 of 2) Write in standard form Solution Factor. Lowest terms Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 21
Addition and Subtraction of Complex Numbers For complex numbers and Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 22
Example 4: Adding and Subtracting Complex Numbers (1 of 2) Find each sum or difference. Write answers in standard form. (a) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 23
Example 4: Adding and Subtracting Complex Numbers (2 of 2) Find each sum or difference. (b) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 24
Multiplication of Complex Numbers (1 of 2) The product of two complex numbers is found by multiplying as though the numbers were binomials and using the fact that as follows. FOIL Distributive property; Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 25
Multiplication of Complex Numbers (2 of 2) For complex numbers Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 26
Example 5: Multiplying Complex Numbers (1 of 4) Find each product. (a) Solution FOIL Multiply. Combine like terms; Standard form Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 27
Example 5: Multiplying Complex Numbers (2 of 4) Find each product. (b) Solution Square of a binomial Multiply. Standard form Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 28
Example 5: Multiplying Complex Numbers (3 of 4) Find each product. (c) Solution Product of the sum and difference of two terms Multiply. Standard form Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 29
Example 5: Multiplying Complex Numbers (4 of 4) Example 5(c) showed that The numbers 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. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 30
Simplifying Powers of i can be simplified using the facts Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 31
Powers of i (1 of 2) and so on. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 32
Powers of i (2 of 2) Powers of i cycle through the same four has the outcomes (i, − 1, −i, and 1) since 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, Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 33
Example 7: Simplifying Powers of i (1 of 2) Simplify each power of i. (a) Solution Since involving write the given power as a product Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 34
Example 7: Simplifying Powers of i (2 of 2) Simplify each power of i. (b) Solution by 1 in the form of Multiply least positive exponent for i. to create the Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 35
- Slides: 35