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Ch 4 Complex Numbers 4.1 Definitions Study Book 4.1, and Appendix B, Sec 8.1

Ch 4 Complex Numbers 4.1 Definitions Study Book 4.1, and Appendix B, Sec 8.1. Objectives: know standard form, a + ib , of a complex number real & imaginary parts, a & b the geometrical representation

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Ch 4 Complex Numbers 4.1 Definitions Study Book 4.1, and Appendix B, Sec 8.1

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  1. Ch 4 Complex Numbers4.1 DefinitionsStudy Book 4.1, and Appendix B, Sec 8.1 Objectives: know • standard form, a + ib , of a complex number • real & imaginary parts, a & b • the geometrical representation • definitions for equality, addition , subtraction, scalar multiplication & multiplication • how to use the quadratic formula to find complex-valued solutions / zeros.

  2. To solve polynomial eqns like x2 = -1 or x2 = -2, we need numbers that square to give negatives, ie roots of negative numbers. Real numbers give positives when squared so we need new non-real numbers. We define a number i to have the property i2 = -1 and write i =  - 1. And we allow multiples of i too, so that • - 4 = 2 i ,  - 9 = 3 i , etc. Geometrically the pure real numbers, a, fill the real x-axis. We plot pure imaginary numbers, i b, on a perpendicular axis, & fill the quadrants between with sums a + ib (see slide 3).

  3. So we can plot 4i, 2 + 5 i , -3 + 1 i , 6 (= 6 + 0i) etc Imaginary axis -3 + 1i 1i -3 Real axis Note that b i = 0 + b i,i = 0 + 1 i , 1 = 1 + 0 i, etc. The set of all numbers of form a + i b, (a, b real) is called the set of Complex numbers , C. Complex numbers are similar to 2-D vectors geometrically, both represented by ordered pairs in the plane. a is the real part of z = a + i b, written a = Re (z). b is the imaginary part, written b = Im (z).

  4. To do arithmetic in C, we need definitions: • Equality: a + bi = c + di if & only if a=c & b=d. Note how equating real & imag parts gives 2 real equations.) • Addition: (a + i b) + (c + d i) = (a + c) + i ( b + d) • Subtraction is similar. • Scalar multiplication: c (a + i b) = a c + i (b c) • Multiplication: ( a + i b ) ( c + d i ) = ( a c - b d ) + i ( b c + a d ) Hence the usual rules for arithmetic hold, with-1 replacing i2 wherever it occurs. i 2 = -1, hence i 3 = - i, i 4 = 1, i5 = i , etc. Examples: ( -2 + 3 i) 2 - 3 ( 5 - 2 i ) = - 20 - 6 i ( -2 + 3 i) 2 - 3 i ( 5 - 2 i ) = - 11 - 27 i

  5. Note that addition, subtraction & scalar multiplication in C has the same geometry as vectors: See Fig 8.3, p 434, App B. But complex numbers multiply differently. Complex numbers are often used in matrices and other algebras. The quadratic formula is easily proved, so can be used. Examples: Find all complex-valued solutions: 1) x2 + 4 = 0 2) x2 + 2 = 0 (Ans: x = +2i or -2i) (Ans: x =  2 i or - 2 i ) 3) x2 - 6x + 13 = 0 4) x2 - 6x - 13 = 0 5) x3 = 6x2 - 13x (start with x3 - 6x2 + 13x = 0 )

  6. Homework Study Book Appendix B, Section 8.1, p 438: • Master 1 - 54, 56. Try Q 58. • Write full solutions to Q 2, 3, 5, 13, 17, 27, 29, 35, 37, 43, 51, 56.

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