append - APPENDIX CN ! 1. Complex Numbers Complex Plane The...

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APPENDIX CN Complex Numbers ! Complex Plane 1. The complex numbers 33 + i , 4 i , 2, and 1 i are plotted as the respective points 3 3 , () , 0 4 , 2, 0 , and 1 1 in the complex plane (see figure). –4 –4 4 4 Re( ) z Im( ) z (1, –1) (2, 0) (3, 3) (0, 4) ! Complex Operations 2. (a) 2 3 4 8 2 12 3 11 10 2 + =− + − = + ii i i i i (b) 2 3 1 2 2 3 3 1 5 2 + + =+ + + = −+ i i i i (c) Rationalizing the denominator, multiply the numerator and denominator by 1 i yielding 1 1 1 1 1 2 1 22 + = =− i i i . (d) Rationalizing the denominator, we multiply the numerator and denominator by 3 i yielding 2 3 3 3 7 10 7 10 10 + + = + =+ i i i i . ! Complex Exponential Numbers 3. (a) Using Euler’s formula, we write ei i i 2 1 0 1 π ππ = + () = cos sin . 793
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794 APPENDIX (b) Using Euler’s formula, we write ei i i i π ππ 2 22 01 =+ = + () = cos sin . (c) Using Euler’s formula, we write i i =− () +− = cos sin cos sin 1. (d) Using the property ee e ab + = and using Euler’s formula, we write e e ie i e i e ii 24 242 2 2 2 44 2 2 2 2 2 2 2 2 + == + F H I K F H G I K J af cos sin . ! Magnitudes and Angles 4. (a) Absolute value: 12 1 2 5 += += i . Polar angle: θ =≈ ° tan 1 2 1 63 or roughly 63 180 radians. (b) Absolute value: −= = i 1 2 2 . Polar angle: The complex number – i is located at the point 0 1 , in the complex plane so the angle is 3 2 radians (or 270 ° ). (c) Absolute value: −− = − ()+− = 11 1 2 i . Polar angle: θπ tan 1 1 bg and because the number −− 1 i is in the third quadrant in the complex plane, we have = 5 4 radians (or 225 ° ). (d) Absolute value: −+ = − += 23 2 3 1 3 2 3 i . Polar angle: F H G I K J ≈° tan 1 3 2 124 or 124 180 radians. (e) e i 2 . We write the exponential as i 2 cos sin .
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SECTION CN Complex Numbers 795 Magnitude is ei i 22 2 1 =+ = + = cos sin cos sin . Polar angle is θ = F H I K = () = −− tan sin cos tan tan 11 2 2 . (f) 2 1 + + i i . We rationalize the denominator to get 2 1 1 1 3 + + =− i i i i i . Magnitude is 2 1 3 2 1 2 1 2 10 + + = F H I K +− F H I K = i i . Polar angle is F H I K ≈− ° tan . 1 1 3 184 or 341.6 ° . ! Complex Verification I 5. We check the first root zi =− + 1 by direct substitution: −+ +−+ +=− −−+ += 12 1 2 1 2 1 2 2 2 0 2 iii i . The second root 1 i is left to the reader. ! Complex Verification II 6. By direct substitution we have 1 2 1 4 1 1 4 1 4 12 112 1 1 4 41 4 42 2 2 + F H I K + == i ii i i bg . ! Real and Complex Parts 7. Calling the complex number zai b , we write z z a ib a ib a b iab a ib a b a i ab b 2 2 2 2 2 +=+ ++ + + . (a) Re zz a ba 2 += + (b) Im zzb a 2 1 +
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796 APPENDIX ! Absolute Value Revisited 8. Using the formula zz z = , yields 42 4242 1 64 25 += + () =+ = ii i . ! Roots of Unity 9. The m roots of z m = 1 (called the roots of unity) are the m values z k m i k m k m = F H G I K J + F H G I K J F H G I K J 1 22 1 cos sin ππ , km =− 01 1 , ! . Note that for z = 1 yields polar angle θ = 0 for the previous formula. (a) z 2 1 = has two roots z k i k ki k k = F H I K + F H I K cos sin cos sin 2 2 2 2 a f a f , k = 0, 1 or z 1.
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This note was uploaded on 04/04/2008 for the course APPM 2360 taught by Professor Williamheuett during the Fall '07 term at Colorado.

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append - APPENDIX CN ! 1. Complex Numbers Complex Plane The...

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