1. 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 standard form, a + ib, of a complex number real & imaginary parts, a & b real & imaginary parts, a & b the geometrical representation the geometrical representation definitions for equality, addition, subtraction, scalar multiplication & multiplication definitions for equality, addition, subtraction, scalar multiplication & multiplication how to use the quadratic formula to find complex-valued solutions / zeros. how to use the quadratic formula to find complex-valued solutions / zeros.

2. 2 To solve polynomial eqns like x 2 = -1 or x 2 = -2, we need numbers that square to give negatives, ie roots of negative numbers. Real numbers give positives when squared Real numbers give positives when squared so we need new non-real numbers. We define a number i to have the property i 2 = -1 We define a number i to have the property i 2 = -1 and write i =  - 1. And we allow multiples of i too, so that 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. 3 So we can plot 4i, 2 + 5 i, -3 + 1 i, 6 (= 6 + 0i) etc Imaginary axis Imaginary axis -3 + 1i 1i -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). a is the real part of z = a + i b, written a = Re (z). b is the imaginary part, written b = Im (z).

4. 4 To do arithmetic in C, we need definitions: Equality: a + bi = c + di if & only if a=c & b=d. Equality: a + bi = c + di if & only if a=c & b=d. Note how equating real & imag parts gives 2 real equations.) Note how equating real & imag parts gives 2 real equations.) Addition: (a + i b) + (c + d i) = (a + c) + i ( b + d) Addition: (a + i b) + (c + d i) = (a + c) + i ( b + d) Subtraction is similar. Subtraction is similar. Scalar multiplication: c (a + i b) = a c + i (b c) Scalar multiplication: c (a + i b) = a c + i (b c) Multiplication: 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 i 2 wherever it occurs. i 2 = -1, hence i 3 = - i, i 4 = 1, i 5 = i, etc. i 2 = -1, hence i 3 = - i, i 4 = 1, i 5 = 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 ( -2 + 3 i) 2 - 3 i ( 5 - 2 i ) = - 11 - 27 i

5. 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) x 2 + 4 = 02) x 2 + 2 = 0 (Ans: x = +2i or -2i) (Ans: x =  2 i or -  2 i ) (Ans: x = +2i or -2i) (Ans: x =  2 i or -  2 i ) 3) x 2 - 6x + 13 = 04) x 2 - 6x - 13 = 0 5)x 3 = 6x 2 - 13x (start with x 3 - 6x 2 + 13x = 0 )

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