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Friday, 1 August 2014

CHAPTER 3- COMPLEX NUMBERS - The Polar Form of a Complex Number

The unit circle

The fundamental trigonometric identity (i.e the Pythagorean theorem) is
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From this we can see that the complex numbers
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are points on the circle of radius one centered at the origin.
Think of the point tex2html_wrap_inline11 moving counterclockwise around the circle as the real number tex2html_wrap_inline13 moves from left to right. Similarly, the point moves clockwise if tex2html_wrap_inline13 decreases. And whether tex2html_wrap_inline13 increases or decreases, the point returns to the same position on the circle whenever tex2html_wrap_inline13 changes by tex2html_wrap_inline21 or by tex2html_wrap_inline23 or by tex2html_wrap_inline25 where k is any integer.
Exercise: Verify that
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Exercise: Prove de Moivre's formula
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Now picture a fixed complex number on the unit circle
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Consider multiples of z by a real, positive number r.
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As r grows from 1, our point moves out along the ray whose tail is at the origin and which passes through the point z. As r shrinks from 1 toward zero, our point moves inward along the same ray toward the origin. The modulus of the point is r. We call the angle tex2html_wrap_inline32 which this ray makes with the x-axis, the argument of the number z. All the numbers rz have the same argument. We write
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Just as a point in the plane is completely determined by its polar coordinates tex2html_wrap_inline40 , a complex number is completely determined by its modulus and its argument.
Notice that the argument is not defined when r=0 and in any case is only determined up to an integer multiple of tex2html_wrap_inline44
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