Copyright 2006 Pearson Education Inc Publishing as Pearson
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 1
7 7. 1 7. 2 7. 3 7. 4 7. 5 7. 6 7. 7 7. 8 Exponents and Radicals Radical Expressions and Functions Rational Numbers as Exponents Multiplying Radical Expressions Dividing Radical Expressions Containing Several Radical Terms Solving Radical Equations Geometric Applications The Complex Numbers Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
7. 8 The Complex Numbers n Imaginary and Complex Numbers n Addition and Subtraction n Multiplication n Conjugates and Division n Powers of i Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Imaginary and Complex Numbers Negative numbers do not have square roots in the real-number system. A larger number system that contains the real-number system is designed so that negative numbers do have square roots. That system is called the complex-number system. The complexnumber system makes use of i, a number that is, by definition, a square root of – 1. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 4
The Number i i is the unique number for which and i 2 = – 1. We can now define the square root of a negative number as follows: for any positive number p. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 5
Example Express in terms of i: Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 6
Imaginary Numbers An imaginary number is a number that can be written in the form a + bi, where a and b are real numbers and Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 7
The union of the set of all imaginary numbers and the set of all real numbers is the set of all complex numbers. Complex Numbers A complex number is any number that can be written in the form a + bi, where a and b are real numbers. (Note that a and b both can be 0. ) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 8
The following are examples of imaginary numbers: Here a = 7, b =2. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 9
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 10
Addition and Subtraction The complex numbers obey the commutative, associative, and distributive laws. Thus we can add and subtract them as we do binomials. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 11
Example Add or subtract and simplify. Solution Combining the real and the imaginary parts Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 12
Multiplication To multiply square roots of negative real numbers, we first express them in terms of i. For example, Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 13
Caution With complex numbers, simply multiplying radicands is incorrect when both radicands are negative: Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 14
Example Multiply and simplify. When possible, write answers in the form a + bi. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 15
Solution continued Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 16
Conjugate of a Complex Number The conjugate of a complex number a + bi is a – bi, and the conjugate of a – bi is a + bi. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 17
Example Find the conjugate of each number. Solution The conjugate is 4 – 3 i. The conjugate is – 6 + 9 i. The conjugate is –i. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 18
Conjugates and Division Conjugates are used when dividing complex numbers. The procedure is much like that used to rationalize denominators. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 19
Example Divide and simplify to the form a + bi. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 20
Solution continued Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 21
Powers of i Simplifying powers of i can be done by using the fact that i 2 = – 1 and expressing the given power of i in terms of i 2. Consider the following: i 23 = (i 2)11 i 1 = (– 1)11 i = –i Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 22
Example Simplify: Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 23
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