complexreview - A Quick Review of Complex Numbers Pierre...

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Unformatted text preview: A Quick Review of Complex Numbers Pierre Thibault February 2003 The set of complex numbers, commonly symbolized by C , has many applications in physics. Its appearance occured naturally in the 16th century, when mathematicians wanted to express all the roots of polynomials. Integer numbers ( Z ) could solve equations such as x +1 = 0, rational numbers ( Q ) could solve equations such as 2 x- 1 = 0, and real numbers ( R ), equations as x 2- 2 = 0. Similarly, complex numbers yield the solutions of equations of the form x 2 + 1 = 0. The set of complex numbers is now sufficient to express all roots of any polynomial. First, we give formal definitions and we show some of the most important properties of complex numbers. Then, we present a short list of the many possible applications related to complex numbers. 1 Definition A complex number is an ordered pair of two real numbers, x and y , written z = x + iy, (1) where i = - 1. x is called the real part and y , the imaginary part of z . This is written as <{ z } = x and ={ z } = y . You know already that a pair of real numbers specifes the position of a point on a 2-dimensional cartesian plane. The set of complex numbers defines the complex plane the same way. It is therefore natural to represent a complex number in polar coordinates: z = r (cos + i sin ) , (2) with the usual relations between ( x,y ) and ( r, ). Exercise: Write down these relations. You shouldnt need to think more than 2 or 3 seconds!...
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This note was uploaded on 09/28/2008 for the course PHYS 316 taught by Professor Hoffstaetter during the Spring '05 term at Cornell University (Engineering School).

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complexreview - A Quick Review of Complex Numbers Pierre...

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