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Complex Numbers Michael R. Gustafson II version 2.4 last reviewed: February 1, 2009 1 Introduction In the field of Mathematics, people had to come up with some satisfactory way to deal with the problems that arose when one tried to take the square root of a negative number. Not happy with simply saying “there isn’t one,” the mathematicians came up with an object known as a complex number . The usefulness of this construct blossomed as engineers and physicists saw applications for what the theorists had developed. The idea is that a number n can be made up of two parts: a real part ( ℜ{ n } ) and an imaginary part ( ℑ{ n } ). These parts can be plotted - much like x and y coordinates - in what is called the complex plane . There are many ways to represent a complex number. Of these, three stand out as being particularly useful. The first is the standard or rectangular representation: n = ℜ{ n } + j ℑ{ n } = n r + jn i where j is 1. This gives all the information you need to figure out the number. Now imagine that this is a point plotted in 2D space where the real part is the x coordinate and the imaginary part is the y coordinate. You could figure out a magnitude and direction from the origin to this point and write the location in polar coordinates . The magnitude and direction would be given by: magnitude( n ) = | n | = n = radicalBig ( ℜ{ n } ) 2 + ( ℑ{ n } ) 2 = radicalBig n 2 r + n 2 i direction( n ) = negationslash n = θ n = arctan( ℑ{ n } , ℜ{ n } ) = arctan( n i , n r ) where arctan with two arguments specifies that the answer will be some angle between π and π instead of the typical limits of π/ 2 to π/ 2. This distinction is important because the two-argument version of arctan will give the proper angle between the positive real axis and the complex number. Calculators usually only have the one-argument version. In this case, you will use the ratio n i /n r as your argument, but must make sure your answer is in the proper quadrant. Using the above, a complex number n can be represented: n = n negationslash θ n At this point, you should notice that n is the hypotenuse of a right triangle with an angle of θ n with respect to the horizontal. This means the real and imaginary parts can be written as: ℜ{ n } = n r = n cos ( θ n ) ℑ{ n } = n i = n sin ( θ n ) and n can be written as: n = n cos ( θ n ) + jn sin ( θ n ) = n (cos ( θ n ) + j sin ( θ n ))
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2 Euler Notation In representing complex numbers, you should note the following Maclaurin Series for cos( θ ) and sin( θ ): cos( θ ) = θ 0 0! θ 2 2! + θ 4 4! ... = summationdisplay n =0 , n even ( 1) n/ 2 θ n n ! sin( θ ) = θ 1 1! θ 3 3!
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