SECTION A.7 Complex Numbers; Quadratic Equations in the Complex Number System A61 Proof If = + z a bi, then ( )( ) ( ) ( ) = + − = − + − = − = − − = + zz a bi a bi a abi abi bi a b i a b a b 1 2 2 2 2 2 2 2 2 2 Now Work problem 89 To express the reciprocal of a nonzero complex number z in standard form, multiply the numerator and denominator of z 1 by z. That is, if = + z a bi is a nonzero complex number, then + = = ⋅ = = − + = + − + a bi z z z z z zz a bi a b a a b b a b i 1 1 1 2 2 2 2 2 2 ↑ Use (5). Writing the Reciprocal of a Complex Number in Standard Form Write + i 1 3 4 in standard form; that is, find the reciprocal of + i 3 4 . Solution EXAMPLE 4 The idea is to multiply the numerator and denominator by the conjugate of + i 3 4 , that is, by the complex number − i 3 4 . The result is + = + ⋅ − − = − + = − i i i i i i 1 3 4 1 3 4 3 4 3 4 3 4 9 16 3 25 4 25 Writing the Quotient of Two Complex Numbers in Standard Form Write each quotient in standard form. (a) + − i i 1 4 5 12 (b) − − i i 2 3 4 3 Solution EXAMPLE 5 (a) + − = + − ⋅ + + = + + + + = − + = − + i i i i i i i i i i i 1 4 5 12 1 4 5 12 5 12 5 12 5 12 20 48 25 144 43 32 169 43 169 32 169 2 (b) − − = − − ⋅ + + = + − − + = − = − i i i i i i i i i i i 2 3 4 3 2 3 4 3 4 3 4 3 8 6 12 9 16 9 17 6 25 17 25 6 25 2 Writing Other Expressions in Standard Form If = − z i 2 3 and = + w i 5 2 , write each expression in standard form. (a) z w (b) +z w (c) +z z Solution EXAMPLE 6 (a) z w z w w w i i i i i i i i i 2 3 5 2 5 2 5 2 10 4 15 6 25 4 4 19 29 4 29 19 29 2 ( )( ) ( )( ) = ⋅ ⋅ = − − + − = − − + + = − = − (b) ( ) ( ) + = − + + = − = + z w i i i i 2 3 5 2 7 7 (c) ( ) ( ) + = − + + = z z i i 2 3 2 3 4 Figure 25 shows the result of Example 4 using a TI-84 Plus CE graphing calculator. To express the quotient of two complex numbers in standard form, multiply the numerator and denominator of the quotient by the conjugate of the denominator. Now Work problem 29 Figure 25
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