Appendix

Appendix - APPENDIX CN Complex Numbers Complex Plane 1 The...

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916 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 23 4 82 1 2 3 1 11 0 ii i i i i +− = + = + (b) () ( ) 2 231 22 3 3 15 i i iii i ++ = + + + = + (c) Rationalizing the denominator, multiply the numerator and denominator by 1 i yielding 1 1 2 2 2 i −− = =− . (d) Rationalizing the denominator, we multiply the numerator and denominator by 3 i yielding 23 7 7 3 3 10 10 10 i i + −+ = =+ + . ± Complex Exponential Numbers 3. (a) Using Euler’s formula, we write ( ) 2 cos2 sin 2 1 0 1 i ei i π ππ = += + = .
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SECTION CN Complex Numbers 917 (b) Using Euler’s formula, we write () 2 cos sin 0 1 22 i ei i i π ππ = += + = . (c) Using Euler’s formula, we write ( ) ( ) cos sin cos sin 1 i i = −+ −= = . (d) Using the property ab a b ee e + = and using Euler’s formula, we write 24 24 2 2 2 2 2 2 cos sin 44 2 2 2 2 i i e e iei e i e + ⎛⎞ == + = + = + ⎜⎟ ⎝⎠ . ± Magnitudes and Angles 4. (a) Absolute value: 12 1 2 5 i . Polar angle: 1 2 tan 63 1 θ =≈ ° or roughly 63 180 radians. (b) Absolute value: 2 2 01 1 i =+ = . 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: ( ) 1 tan 1 θπ 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: 2 3 23 2 3 1 3 i −+ = − + = . Polar angle: 1 3 tan 124 2 =− − ≈ ° or 124 180 radians. (e) 2 i e . We write the exponential as 2 cos2 sin 2 i .
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918 APPENDIX Magnitude is 22 2 cos2 sin 2 cos 2 sin 2 1 i ei = += . Polar angle is () 11 sin 2 tan tan tan 2 2 θ −− ⎛⎞ == = ⎜⎟ ⎝⎠ . (f) 2 1 i i + + . We rationalize the denominator to get 21 3 2 2 ii i + = + . Magnitude is 23 1 1 10 12 2 2 i i + ⎛⎞ ⎛ ⎞ =+ = ⎜⎟ ⎜ ⎟ + ⎝⎠ ⎝ ⎠ . Polar angle is 1 1 tan 18.4 3 = −≈ ° or 341.6 ° . ± Complex Verification I 5. We check the first root 1 zi =− + by direct substitution: 2 1 2 1 212 122 20 iii i −+ + −+ + =− −− + + = . The second root 1 i is left to the reader. ± Complex Verification II 6. By direct substitution we have 4 42 2 2 1 1 1 2 1 1 2 1 4 1 44 4 4 2 i i i + + = = . ± Real and Complex Parts 7. Calling the complex number zai b , we write ( ) ( ) ( ) ( ) 2 2 2 2 2 z z a ib a ib a b iab a ib ab ai a bb +=+ + + = + + =− ++ + . (a) ( ) 2 Re 2 2 zz a ba + (b) ( ) ( ) 2 Im 2 2 1 zzb a +
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SECTION CN Complex Numbers 919 ± Absolute Value Revisited 8. Using the formula zz z = , yields () 42 42 42 1 6425 ii i += + = + = . ± Roots of Unity 9. The m roots of 1 m z = (called the roots of unity) are the m values 1 22 1c o s s i n m k kk zi mm ππ ⎛⎞ =+ ⎜⎟ ⎝⎠ , 0,1 1 km = " . Note that for 1 z = yields polar angle 0 θ = for the previous formula. (a) 2 1 z = has two roots cos sin cos sin k k i k π = + , 0 k = , 1 or 1 z .
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This note was uploaded on 02/27/2010 for the course CHEM 1132 taught by Professor Cole/clark during the Spring '08 term at UCM.

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Appendix - APPENDIX CN Complex Numbers Complex Plane 1 The...

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