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Exponents and Radicals 7 Exponents and Radicals 7.1 Radical Expressions and Functions 7.2 Rational Numbers as Exponents 7.3 Multiplying Radical Expressions 7.4 Dividing Radical Expressions 7.5 Expressions Containing Several Radical Terms 7.6 Solving Radical Equations 7.7 Geometric Applications 7.8 The Complex Numbers Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

7.8 The Complex Numbers Imaginary and Complex Numbers Addition and Subtraction Multiplication Conjugates and Division 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 complex-number 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

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

Example Express in terms of i: Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

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

The following are examples of imaginary numbers: Here a = 7, b =2. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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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

Example Add or subtract and simplify. Solution Combining the real and the imaginary parts Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

Caution With complex numbers, simply multiplying radicands is incorrect when both radicands are negative: Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Multiply and simplify. When possible, write answers in the form a + bi. Example Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution continued Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

Example Find the conjugate of each number. Solution The conjugate is 4 – 3i. The conjugate is –6 + 9i. The conjugate is –i. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

Example Divide and simplify to the form a + bi. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution continued Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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)11i1 = (–1)11i = –i Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Simplify: Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley