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Chapter 1 Equations and Inequalities © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved 1 SECTION 1.3 Complex Numbers OBJECTIVES 1 2 3 4 Define complex numbers. Add and subtract complex numbers. Multiply complex numbers. Divide complex numbers. © 2010 Pearson Education, Inc. All rights reserved 2 Definition of i The square root of −1 is called i. i 1 so that i 1. 2 The number i is called the imaginary unit. © 2010 Pearson Education, Inc. All rights reserved 3 Complex Numbers A complex number is a number of the form z a bi, where a and b are real numbers and i2 = –1. The number a is called the real part of z, and we write Re(z) = a. The number b is called the imaginary part of z and we write Im(z) = b. © 2010 Pearson Education, Inc. All rights reserved 4 Definitions A complex number z written in the form a + bi is said to be in standard form. A complex number with a = 0 and b ≠ 0, written as bi, is called a pure imaginary number. If b = 0, then the complex number a + bi is a real number. Real numbers form a subset of complex numbers (with imaginary part 0). © 2010 Pearson Education, Inc. All rights reserved 5 Square Root of a Negative Number For any positive number, b b b i i © 2010 Pearson Education, Inc. All rights reserved b. 6 EXAMPLE 1 Identifying the Real and the Imaginary Parts of a Complex Number Identify the real and the imaginary parts of each complex number. 1 c. 3i b. 7 i a. 2 5i 2 f. 3 25 e. 0 d. 9 Solution a. 2 5i real part 2; imaginary part 5 1 b. 7 i 2 c. 3i 1 real part 7; imaginary part 2 real part 0; imaginary part 3 © 2010 Pearson Education, Inc. All rights reserved 7 EXAMPLE 1 Identifying the Real and the Imaginary Parts of a Complex Number Solution continued d. 9 real part –9; imaginary part 0 e. 0 real part 0; imaginary part 0 f. 3 25 real part 3; imaginary part 5 © 2010 Pearson Education, Inc. All rights reserved 8 Equality of Complex Numbers Two complex numbers z = a + bi and w = c + di are equal if and only if a = c and b = d That is, z = w if and only if Re(z) = Re(w) and Im(z) = Im(w). © 2010 Pearson Education, Inc. All rights reserved 9 EXAMPLE 2 Equality of Complex Numbers Find x and y assuming that (3x – 1) + 5i = 8 + (3 – 2y)i. Solution Let z = (3x – 1) + 5i and w = 8 + (3 – 2y)i. Then Re(z) = Re(w) and Im(z) = Im(w). So, 3x – 1 = 8 and 5 = 3 – 2y. 3x = 8 + 1 x=3 So, x = 3 and y = –1. 5 – 3 = –2y –1 = y © 2010 Pearson Education, Inc. All rights reserved 10 ADDITION AND SUBTRACTION OF COMPLEX NUMBERS For all real numbers a, b, c, and d, let z = a + bi and w = c + di. z w a bi c di a c b d i z w a bi c di a c b d i . © 2010 Pearson Education, Inc. All rights reserved 11 EXAMPLE 3 Adding and Subtracting Complex Numbers Write the sum or difference of two complex numbers in standard form. a. 3 7i 2 4i c. 2 b. 5 9i 6 8i 9 2 4 Solution a. 3 7i 2 4i 3 2 7 4 i 5 3i © 2010 Pearson Education, Inc. All rights reserved 12 EXAMPLE 3 Adding and Subtracting Complex Numbers Solution continued b. 5 9i 6 8i 5 6 9 8 i 1 17i c. 2 9 2 4 2 3i 2 2i 2 3i 2 2i 2 2 3 2 i 4i © 2010 Pearson Education, Inc. All rights reserved 13 MULTIPLYING COMPLEX NUMBERS For all real numbers a, b, c, and d, a bi c di ac bd ad bc i . © 2010 Pearson Education, Inc. All rights reserved 14 EXAMPLE 4 Multiplying Complex Numbers Write the following products in standard form. a. 3 5i 2 7i b. 2i 5 9i Solution F O I L a. 3 5i 2 7i 6 21i 10i 35i 2 6 11i 35 41 11i b. 2i 5 9i 10i 18i 2 10i 18 18 10i © 2010 Pearson Education, Inc. All rights reserved 15 WARNING Recall that if a and b are positive real numbers, a b ab . However, this property is not true for nonreal numbers. For example, 9 9 3i 3i 9i 9 1 9, 2 but 9 9 Thus 9 9 81 9. 9 9 . © 2010 Pearson Education, Inc. All rights reserved 16 EXAMPLE 5 Multiplication Involving Roots of Negative Numbers Perform the indicated operation and write the result in standard form. a. 2 8 c. 2 3 3 2 3 b. 2 d. 3 2 1 32 Solution a. 2 8 i 2 i 8 i i 2 2 2 8 16 1 4 4 © 2010 Pearson Education, Inc. All rights reserved 17 EXAMPLE 5 Multiplication Involving Roots of Negative Numbers Solution continued b. c. © 2010 Pearson Education, Inc. All rights reserved 18 EXAMPLE 5 Multiplication Involving Roots of Negative Numbers Solution continued d. © 2010 Pearson Education, Inc. All rights reserved 19 CONJUGATE OF A COMPLEX NUMBER If z = a + bi, then the conjugate (or complex conjugate) of z is denoted by z and defined by z a bi a bi. © 2010 Pearson Education, Inc. All rights reserved 20 EXAMPLE 6 Multiplying a Complex Number by Its Conjugate Find the product zz for each complex number z . a. z 2 5i b. z 1 3i Solution a. zz 2 5i 2 5i 2 5i 4 25i 2 2 2 4 25 29 b. zz 1 3i 1 3i 1 3i 1 9i 2 2 2 1 9 10 © 2010 Pearson Education, Inc. All rights reserved 21 COMPLEX CONJUGATE PRODUCT THEOREM If z = a + bi, then zz a b . 2 2 © 2010 Pearson Education, Inc. All rights reserved 22 DIVIDING COMPLEX NUMBERS To write the quotient of two complex numbers w and z (z ≠ 0), and write w wz z zz and then write the right side in standard form. © 2010 Pearson Education, Inc. All rights reserved 23 EXAMPLE 7 Dividing Complex Numbers Write the following quotients in standard form. 4 25 b. 2 9 1 a. 2i Solution 1 2 i 2i a. 2 2 2 i 2 i 2 1 2i 5 2 1 i 5 5 © 2010 Pearson Education, Inc. All rights reserved 24 EXAMPLE 7 Dividing Complex Numbers Solution continued 4 25 4 5i 4 5i 2 3i b. 2 3i 2 3i 2 3i 2 9 8 12i 10i 15i 22 32 8 15 22i 7 22i 49 13 7 22 i 13 13 2 © 2010 Pearson Education, Inc. All rights reserved 25