3.b - Appendix B: Complex Numbers Definition: A complex...

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Unformatted text preview: Appendix B: Complex Numbers Definition: A complex number is a number z = a + bi where a, b R and i is a symbol satisfying the relation i 2 =- 1. a is the real part of z , denoted Re z and b is the imaginary part of z , denoted Im z EX: Complex number z = 3- 4 i Re z = 3 , Im z =- 4 Complex number system : C = set of all complex numbers For C , have the standard arithmetic operations + , : ( a + bi ) + ( c + di ) = ( a + c ) + ( b + d ) i , ( a + bi ) ( c + di ) = ( ac- bd ) + ( ad + bc ) i EX: (2 + 4 i )(- 6 + 3 i ) = (2 + 4 i )(- 6) + (2 + 4 i )(3 i ) = (- 12- 24 i ) + (6 i + 12 i 2 ) =- 12- 18 i + 12(- 1) =- 24- 18 i 1 Definition: Given z = a + bi C , conjugate of z is z = a- bi EX: For z =- 2 + 3 i , the conjugate z =- 2- 3 i Definition: absolute value or modulus of z = a + bi is | z | = zz = radicalbig ( a- bi )( a + bi ) = radicalbig a 2 + b 2 Note: for z = a + 0 i R , | z | = zz = a 2 = | a | Properties: 1. z = z iff z R 2. w +...
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3.b - Appendix B: Complex Numbers Definition: A complex...

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