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Complex number arithmetic

# Complex number arithmetic - Consistency of Arithmetic...

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Consistency of Arithmetic Operations on Complex Numbers in Both Rectangular and Polar (Exponential) Forms Consider two complex numbers, N 1 and N 2 , written in both rectangular and polar (exponential) forms: N 1 = a + j b = a 2 + b 2 e j ! 1 where ! 1 = tan –1 ( b / a ) cos ! 1 ( ) = a a 2 + b 2 sin ! 1 ( ) = b a 2 + b 2 N 2 = c + j d = c 2 + d 2 e j ! 2 where ! 2 = tan –1 ( d / c ) cos ! 2 ( ) = c c 2 + d 2 sin ! 2 ( ) = d c 2 + d 2 a Real Imag b N 1 a 2 + b 2 ! 1 c Real Imag d N 2 c 2 + d 2 ! 2 (0, 0) (0, 0)

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The rectangular and polar (exponential) forms of complex numbers shown above are consistent with the Euler relation: Ae j ! = A cos ! + jA sin ! A. The rectangular and polar forms of complex numbers yield identical results when added or subtracted. Rectangular form : N 1 ± N 2 = ( a + j b ) ± ( c + j d ) = ( a ± c ) + j ( b ± d ) Polar (exponential) form : N 1 ± N 2 = a 2 + b 2 e j ! 1 ± c 2 + d 2 e j ! 2 = a 2 + b 2 cos ! 1 + j a 2 + b 2 sin ! 1 ( ) ± c 2 + d 2 cos ! 2 + j c 2 + d 2 sin ! 2 ( ) = a 2 + b 2 ! a a 2 + b 2 + j a 2 + b 2 ! b a 2 + b 2 " # \$ % & ± c 2 + d 2 ! c c 2 + d 2 + j c 2 + d 2 !
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