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

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Unformatted text preview: 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 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 + 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 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 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 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 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 Theorem...
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mmlec12b - Objectives (1) Know what a complex number is....

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