A62 APPENDIX Review The conjugate of a complex number has certain general properties that will be useful later. For a real number = + a a i0 , the conjugate is = + = − = a a i a i a 0 0 . = i i 1 = ⋅ = ⋅ = i i i i i 1 5 4 = − i 1 2 = ⋅ = − i i i 1 6 4 2 = ⋅ = − ⋅ = − i i i i i 1 3 2 = ⋅ = − i i i i 7 4 3 ( )( ) = ⋅ = − − = i i i 1 1 1 4 2 2 = ⋅ = i i i 1 8 4 4 Evaluating Powers of i (a) ( ) = ⋅ = ⋅ = ⋅ = − i i i i i i i 1 27 24 3 4 6 3 6 3 (b) ( ) = ⋅ = ⋅ = ⋅ = i i i i i i i 1 101 100 1 4 25 25 EXAMPLE 7 Writing the Power of a Complex Number in Standard Form Write ( ) +i 2 3 in standard form. Solution EXAMPLE 8 Use the special product formula for ( ) +x a . 3 ( ) + = + + + x a x ax a x a 3 3 3 3 2 2 3 THEOREM The conjugate of a real number is the real number itself. Other properties that are direct consequences of the definition of the conjugate are given next. In each statement, z and w represent complex numbers. THEOREMS • The conjugate of the conjugate of a complex number is the complex number itself. = z z (6) • The conjugate of the sum of two complex numbers equals the sum of their conjugates. + = + z w z w (7) • The conjugate of the product of two complex numbers equals the product of their conjugates. ⋅ = ⋅ z w z w (8) The proofs of equations (6), (7), and (8) are left as exercises. Powers of i The powers of i follow a pattern that is useful to know. And so on. The powers of i repeat with every fourth power.
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