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8. complex

# 8. complex - Complex Numbers and Exponentials Definition...

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Complex Numbers and Exponentials Definition and Basic Operations A complex number is nothing more than a point in the xy –plane. The sum and product of two complex numbers ( x 1 , y 1 ) and ( x 2 , y 2 ) is defined by ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 + x 2 , y 1 + y 2 ) ( x 1 , y 1 ) ( x 2 , y 2 ) = ( x 1 x 2 - y 1 y 2 , x 1 y 2 + x 2 y 1 ) respectively. It is conventional to use the notation x + iy (or in electrical engineering country x + jy ) to stand for the complex number ( x, y ). In other words, it is conventional to write x in place of ( x, 0) and i in place of (0 , 1). In this notation, the sum and product of two complex numbers z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 is given by z 1 + z 2 = ( x 1 + x 2 ) + i ( y 1 + y 2 ) z 1 z 2 = x 1 x 2 - y 1 y 2 + i ( x 1 y 2 + x 2 y 1 ) The complex number i has the special property i 2 = (0 + 1 i )(0 + 1 i ) = (0 × 0 - 1 × 1) + i (0 × 1 + 1 × 0) = - 1 For example, if z = 1 + 2 i and w = 3 + 4 i , then z + w = (1 + 2 i ) + (3 + 4 i ) = 4 + 6 i zw = (1 + 2 i )(3 + 4 i ) = 3 + 4 i + 6 i + 8 i 2 = 3 + 4 i + 6 i - 8 = - 5 + 10 i Addition and multiplication of complex numbers obey the familiar algebraic rules z 1 + z 2 = z 2 + z 1 z 1 z 2 = z 2 z 1 z 1 + ( z 2 + z 3 ) = ( z 1 + z 2 ) + z 3 z 1 ( z 2 z 3 ) = ( z 1 z 2 ) z 3 0 + z 1 = z 1 1 z 1 = z 1 z 1 ( z 2 + z 3 ) = z 1 z 2 + z 1 z 3 ( z 1 + z 2 ) z 3 = z 1 z 3 + z 2 z 3 The negative of any complex number z = x + iy is defined by - z = - x + ( - y ) i , and obeys z + ( - z ) = 0. Other Operations The complex conjugate of z is denoted ¯ z and is defined to be ¯ z = x - iy . That is, to take the complex conjugate, one replaces every i by - i . Note that z ¯ z = ( x + iy )( x - iy ) = x 2 - ixy + ixy + y 2 = x 2 + y 2 is always a positive real number. In fact, it is the square of the distance from x + iy (recall that this is the point ( x, y ) in the xy –plane) to 0 (which is the point (0 , 0)). The distance from z = x + iy to 0 is denoted | z | and is called the absolute value, or modulus, of z . It is given by | z | = p x 2 + y 2 = z ¯ z December 17, 2007 Complex Numbers and Exponentials 1

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Since z 1 z 2 = ( x 1 + iy 1 )( x 2 + iy 2 ) = ( x 1 x 2 - y 1 y 2 ) + i ( x 1 y 2 + x 2 y 1 ), | z 1 z 2 | = p ( x 1 x 2 - y 1 y 2 ) 2 + ( x 1 y 2 + x 2 y 1 ) 2 = q x 2 1 x 2 2 - 2 x 1 x 2 y 1 y 2 + y 2 1 y 2 2 + x 2 1 y 2 2 + 2 x 1 y 2 x 2 y 1 + x 2 2 y 2 1 = q x 2 1 x 2 2 + y 2 1 y 2 2 + x 2 1 y 2 2 + x 2 2 y 2 1 = q ( x 2 1 + y 2 1 )( x 2 2 + y 2 2 ) = | z 1 || z 2 | for all complex numbers z 1 , z 2 . Since | z | 2 = z ¯ z , we have z ( ¯ z | z | 2 ) = 1 for all complex numbers z 6 = 0 . This says that the multiplicative inverse, denoted z - 1 or 1 z , of any nonzero complex number z = x + iy is z - 1 = ¯ z | z | 2 = x - iy x 2 + y 2 = x x 2 + y 2 - y x 2 + y 2 i It is easy to divide a complex number by a real number. For example 11+2 i 25 = 11 25 + 2 25 i In general, there is a trick for rewriting any ratio of complex numbers as a ratio with a real denominator.
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8. complex - Complex Numbers and Exponentials Definition...

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