complex_basics

complex_basics - Some Basics in Operations with Complex...

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Some Basics in Operations with Complex Numbers Consider two complex numbers a and b : a = α + i β b = δ + i γ where i = 1 . Addition a + b = + i ( )+ + i ( ) = + i + ( ) Subtraction a b = + i ( ) + i ( ) = i ( ) Multiplication ab = + i ( ) + i ( ) = αδ + i αγ + i βδ + i 2 βγ = + i + ( ) since i 2 = 1 . Division a b = + i + i = + i + i i i = + + i + ( ) 2 + 2
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The complex exponential If we make a plot of the complex number a in the complex plane, we have: From this plot, we see that: a = α 2 + β 2 tan φ = or, alternately, = a cos = a sin and a = + i = a cos + i a sin = a cos + i sin ( ) = a e i where Euler’s identity ( e i = cos + i sin ) has been used in the last equality. The complex exponential form of complex numbers often times will simplify the
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complex_basics - Some Basics in Operations with Complex...

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