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Unformatted text preview: Using Complex Numbers Complex Numbers As a set of mathematical objects, the complex numbers can be con- sidered to coincide exactly with the points in the standard two dimen- sional real vector space R 2 . What is new in designating the points of R 2 as complex numbers is the arithmetic and algebraic structure imposed on that set of points. Complex numbers thus consist of pairs: z = x y ! , x, y real numbers . So written, x is called the real part of the complex number z and y is called the imaginary part of z ; we write x = Re z, y = Im z. We single out particular pairs for special emphasis. Any pair of the form ( r, 0) is considered to be essentially the same as the real number r and will henceforth be denoted simply as r . In particular the pairs (0 , 0) and (1 , 0) will simply be referred to as 0 and 1, respectively. The pair (0 , 1) is denoted by i and any pair (0 , s ) is written as s i or i s ( s j or j s is common in engineering usage); thus we typically write z = x + i y. An important alternative representation of the complex numbers is obtained by going to polar coordinates . For points in the plane Carte- sian coordinates ( x, y ) are related to polar coordinates r, for those same points by the relationships x = r cos , y = r sin , if r 6 = 0 , x = 0 , y = 0 , if r = 0; r = q x 2 + y 2 , = tan- 1 ( y/x ) , x, y not both 0 . 1 Correspondingly we have the polar representation of the complex num- ber z = x + i y = r (cos + i sin ) . This is sometimes written as z = r cis - but not here; later, after we introduce the complex exponential function, we will use the notation z = r e i . The angle , normally given in radian measure, is called the argument of the complex number z as shown, while r = x 2 + y 2 is the absolute value , or modulus of z . The argument is not unique; it is determined only up to an integer multiple of 2 . Complex Arithmetic Additive Operations on Complex Numbers Addition and sub- traction of complex numbers agrees exactly with addition and subtrac- tion of vectors in R 2 ; it is performed componentwise . Thus ( u + i v ) ( x + i y ) = ( u x ) + ( v y ) i....
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