Review of Complex Numbers - Review of Complex Numbers...

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Review of Complex Numbers Introduction This is a short review of the main concepts of complex numbers . Complex numbers are used throughout mathematics and its applications. In particular, when we try to solve differential equations it is often convenient and natural to use complex numbers to express the solutions. Here we review those ideas and results from the theory of complex numbers that will be used in Math 216. A complex number z may be expressed as an ordered pair of real numbers: z = ( x, y ) = x + iy where i := - 1 (so i 2 = - 1) and x and y are real numbers. The following notations are often used: x = Re( z ) or x = < ( z ) denotes the real part of the complex number z y = Im( z ) or y = = ( z ) denotes the imaginary part of the complex number z Recall that two complex numbers are equal if and only if both the real and the imaginary parts are equal. In other words, z 1 := ( x 1 , y 1 ) equals z 2 := ( x 2 , y 2 ) if and only if x 1 = x 2 and y 1 = y 2 . A convenient way of thinking about complex numbers is to imagine them as points in the ( x, y ) plane (in this case it is called the complex plane ), as illustrated in the following figure. In the complex plane, the line y = 0 is frequently called the real axis , and the line y=Im(z) x=Re(z) z=(x,y)=x+iy Figure 1: The complex number z = x + iy plotted in the complex plane. x = 0 is frequently called the imaginary axis . Doing arithmetic with complex numbers Addition and multiplication of two complex numbers z 1 = ( x 1 , y 1 ) and z 2 = ( x 2 , y 2 ) are defined by the following rules: 1
Addition: z 1 + z 2 := ( x 1 + x 2 , y 1 + y 2 ) = ( x 1 + x 2 ) + i ( y 1 + y 2 ). Multiplication: z 1 z 2 := ( x 1 x 2 - y 1 y 2 , x 1 y 2 + x 2 y 1 ) = ( x 1 x 2 - y 1 y 2 ) + i ( x 1 y 2 + x 2 y 1 ). Note that if we interpret z 1 and z 2 as points in the complex plane as in Figure 1, then addition of complex numbers is the same as vector addition in the plane; we are just adding the real and imaginary parts componentwise. On the other hand, the multiplication of two complex numbers may perhaps seem different than what you might have expected it to be; this is only an illusion, however, and when we introduce exponential forms for complex numbers later, the multiplication will make perfect sense. Although complex numbers obey different rules of arithmetic than do ordinary real num- bers, it is very important to keep in mind that the complex numbers simply generalize the notion of the real numbers. Indeed, we can think of the real number x as the complex number ( x, 0) = x + i 0. Such a complex number whose imaginary part is zero is said to be purely real . If we add or multiply two purely real complex numbers, then according to the rules for complex arithmetic, we have ( x 1 , 0) + ( x 2 , 0) = ( x 1 + x 2 , 0) and ( x 1 , 0)( x 2 , 0) = ( x 1 x 2 , 0) so in each case the result is also a purely real complex number, and the real part in each case is exactly what we would have found by applying the usual rules of addition and multiplication for real numbers to the real parts. This shows that all the new operations defined for complex numbers when applied to purely real numbers give the usual familiar corresponding operations. One way to think of (0 , 1) is as the new

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