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mmlec12b

# mmlec12b - Objectives(1 Know what a complex number is(2...

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Objectives (1) Know what a complex number is. (2) Know the ways to express it, be able to change between them, and know when and how to use them. 12 – 1

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Lecture 12 – Complex Numbers Fundamental Theorem of Algebra polynomial of degree n with coefficients in R factors: R : linear and irreducible quadratic C : n linear factors a 0 + a 1 x + a 2 x 2 + · · · + a n x n , a i R 12 – 1
Example . x 3 - 2 x 2 - 2 x - 3 = ( x - 3)( x 2 + x + 1) Solve x 2 + x + 1 = 0 12 – 2

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Define : number i so that i 2 = - 1 = - 3 = p ( - 1)3 = - 1 3 = i 3 x = - 1 2 ± i 3 2 12 – 3
A complex number : z = a + ib a, b R i 2 = - 1 C = { a + ib | a, b R } 12 – 4

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Note: 1) a = Re( z ) and b = Im( z ) are real numbers 2) b = 0 z is real 3) a = 0 z is pure imaginary 12 – 5
z = a + ib Define : ¯ z , complex conjugate of z = a + ib ¯ z = a - ib Example. Roots of x 2 + x + 1 are of the form w , ¯ w : w = - 1 2 + i 3 2 ¯ w = - 1 2 - i 3 2 12 – 6

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How to add, multiply, divide complex numbers? 12 – 7
Addition and multiplication: as polynomials in i i 2 = - 1 Example . Calculate: (1 + i ) + (4 - 2 i ) = (1 + i )(4 - 2 i ) = z z = ? 12 – 8

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z ¯ z = ( a + ib )( a - ib ) = a 2 + b 2 is real. Division is defined by z w = z ¯ w w ¯ w Example . 1 + i 4 - 2 i = (1 + i )(4 + 2 i ) (4 - 2 i )(4 + 2 i ) = 2 + 6 i 16 + 4 = 1 10 + 3 10 i C 12 – 9
Properties of ¯ z : z + ¯ z = z - ¯ z = ¯ ¯ z = z + w = ¯ z + ¯ w zw = ¯ z ¯ w z w · = ¯ z ¯ w 12 – 10

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Theorem . Complex roots of polynomials with real coefficients appear in conjugate pairs.
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mmlec12b - Objectives(1 Know what a complex number is(2...

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