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Unformatted text preview: MATH 135: COMPLEX NUMBERS (WINTER, 2008) The introduction of complex numbers in the 16th century made it possible to solve the equation x 2 + 1 = 0. These notes 1 present one way of defining complex numbers. 1. The Complex Plane A complex number z is given by a pair of real numbers x and y and is written in the form z = x + iy , where i satisfies i 2 = 1. The complex numbers may be represented as points in the plane (sometimes called the Argand diagram). The real number 1 is represented by the point (1 , 0), and the complex number i is represented by the point (0 , 1). The xaxis is called the “real axis”, and the yaxis is called the “imaginary axis”. For example, the complex numbers 3 + 4 i and 3 4 i are illustrated in Fig 1a . Fig 1a r 3 + 4 i 7 r 3 4 i S S S S Sw Fig 1b : 3 r r r 4 + i 2 + 3 i 6 + 4 i Complex numbers are added in a natural way: If z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 , then z 1 + z 2 = ( x 1 + x 2 ) + i ( y 1 + y 2 ) (1) Fig 1b illustrates the addition (4 + i ) + (2 + 3 i ) = (6 + 4 i ). Multiplication is given by z 1 z 2 = ( x 1 x 2 y 1 y 2 ) + i ( x 1 y 2 + x 2 y 1 ) Note that the product behaves exactly like the product of any two algebraic expressions, keeping in mind that i 2 = 1. Thus, (2 + i )( 2 + 4 i ) = 2( 2) + 8 i 2 i + 4 i 2 = 8 + 6 i We call x the real part of z and y the imaginary part , and we write x = Re z , y = Im z . ( Remem ber : Im z is a real number.) The term “imaginary” is an historical holdover; it took mathematicians Date : November 26, 2007. 1 These notes are based on notes written by Bob Phelps, with modifications by Tom Duchamp. 1 2 (WINTER, 2008) some time to accept the fact that i (for “imaginary”, naturally) was a perfectly good mathematical object. Electrical engineers (who make heavy use of complex numbers) reserve the letter i to denote electric current and they use j for √ 1. There is only one way we can have z 1 = z 2 , namely, if x 1 = x 2 and y 1 = y 2 . An equivalent statement (one that is important to keep in mind) is that z = 0 if and only if Re z = 0 and Im z = 0. If a is a real number and z = x + iy is complex, then az = ax + iay (which is exactly what we would get from the multiplication rule above if z 2 were of the form z 2 = a + i 0). Division is more complicated (although we will show later that the polar representation of complex numbers makes it easy). To find z 1 /z 2 it suffices to find 1 /z 2 and then multiply by z 1 . The rule for finding the reciprocal of z = x + iy is given by: 1 x + iy = 1 x + iy · x iy x iy = x iy ( x + iy )( x iy ) = x iy x 2 + y 2 (2) The expression x iy appears so often and is so useful that it is given a name. It is called the complex conjugate of z = x + iy and a shorthand notation for it is z ; that is, if z = x + iy , then z = x iy . For example, 3 + 4 i = 3 4 i , as illustrated in the Fig 1a . Note that z = z and z 1 + z 2 = z 1 + z 2 . Exercise (3b) is to show that z 1 z 2 =...
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 Complex Numbers, Complex number, de Moivre, Tom Duchamp, Bob Phelps, $ 2

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