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Unformatted text preview: MAT A26 Lecture 37 1 Complex Numbers One of the things we learn in high school is that the symbol √ 1 cannot be a real number, since the square of a real number is always ≥ 0. In this lecture we are going to describe a number system C (the “complex numbers”), generalizing the real numbers R , and large enough to give a meaning to √ 1. In fact we have already mentioned them in our discussion of the method of partial fractions. definition 1.1. The set of complex numbers C is the set of all ordered pairs ( x, y ) of real numbers x, y ∈ R equipped with the following operations 1. Addition: If z 1 = ( x 1 , y 1 ) ∈ C and z 2 = ( x 2 , y 2 ) ∈ C then we define z 1 + z 2 def = ( x 1 + x 2 , y 1 + y 2 ) ∈ C 2. Multiplication: If z 1 = ( x 1 , y 1 ) ∈ C and z 2 = ( x 2 , y 2 ) ∈ C then we define z 1 z 2 def = ( x 1 x 2 y 1 y 2 , x 1 y 2 + y 1 x 2 ) ∈ C example 1.2. If a = (1 , 2) and b = (3 , 4) are complex numbers, calculate a + b and ab . solution. We have a + b = (1 , 2) + (3 , 4) = (1 + 3 , 2 + 4) = (4 , 6) and ab = (1 , 2)(3 , 4) = (1 · 3 2 · 4 , 1 · 4 + 2 · 3) = ( 5 , 10) Of course complex numbers may be represented as points on the x, y plane in the usual way....
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This note was uploaded on 03/04/2012 for the course MATH 10250 taught by Professor Himonas during the Fall '08 term at Notre Dame.
 Fall '08
 HIMONAS
 Calculus, Complex Numbers

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