week10 - Complex Numbers Original Notes adopted from...

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Complex Numbers Original Notes adopted from November 13, 2001 (Week 10) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong Polynomial Equations with Integer Coefficients Eg. 3x + 2 = 0. No Solution in Z, solution in Q Every linear equation with coeffients in Z, has a root in Q. Eg. x 2 – 2 = 0. No root in Q. Has root in R. Eg. x 2 + 1 = 0 , no root in R. To define set of complex numbers, C C = { a + bi, a,b R } i 2 = -1 Define ( a + bi ) + (c + di) = (a + c) + ( b + d) i Define ( a + bi ) ( c + di ) = ( ac – bd ) + ( ad + bc) i Given a + bi, we say " a is the real part" of a + bi, & write R ( a + bi ) , & we say "b is the imaginary part" of a + bi, & write Im( a + bi) = b. a + 0i, write as a, R is embedded in C: 0 + bi, write as bi to a R , corresponds a + 0i. Numbers of form bi for b 0 are "pure imaginary" numbers 0 = 0 + 0i a + bi + 0 = a + bi 2200 a + bi. Given a + bi, there is an additive inverse . (a + bi ) + (-a + (-b)i) = a + (-a) + ( b + (-b)) i = 0. 1 = 1 + 0i
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week10 - Complex Numbers Original Notes adopted from...

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