A60 APPENDIX Review Products of complex numbers are calculated as illustrated in Example 2. Multiplying a Complex Number by Its Conjugate Find the product of the complex number = + z i 3 4 and its conjugate z. Solution EXAMPLE 3 Since = − z i 3 4 , we have ( )( ) =+ −=−+− =+= zz i i i i i 3434 912 12 16 9 16 25 2 Multiplying Complex Numbers ( )( ) ( ) ( ) ( ) + + = + + + = + + + = + + − = − + i i i i i i i i i i 5327 527 327 10 35 6 21 10 41 21 1 11 41 2 EXAMPLE 2 Distributive Property Distributive Property i 1 2 =− Simplify. Based on the procedure of Example 2, the product of two complex numbers is defined as follows: Product of Complex Numbers ( )( ) ( ) ( ) + + = − + + a bi c di ac bd ad bc i (4) Do not bother to memorize formula (4). Instead, whenever it is necessary to multiply two complex numbers, follow the usual rules for multiplying two binomials, as we did in Example 2. Just remember that = − i 1. 2 For example, ( )( ) ( ) ( )( ) = = − = − + − = − + − = − i i i i i i i i i 2 2 4 4 1 4 2 1 2 2 3 2 2 Figure 24 shows the result of Example 2 using a TI-84 Plus CE graphing calculator. Now Work problem 21 Algebraic properties for addition and multiplication, such as the commutative, associative, and distributive properties, hold for complex numbers. The property that every nonzero complex number has a multiplicative inverse, or reciprocal, requires a closer look. Conjugates For example, + = − i i 2 3 2 3 and − − = − + i i 6 2 6 2 . DEFINITION Complex Conjugate If = + z a bi is a complex number, then its conjugate , denoted by z, is defined as = + = − z a bi a bi The result obtained in Example 3 has an important generalization. THEOREM Product of Complex Conjugates The product of a complex number and its conjugate is a nonnegative real number. That is, if = + z a bi, then = + zz a b 2 2 (5) In Words The conjugate of a complex number is found by changing the sign of the imaginary part. Figure 24
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