1 Complex Numbers
2 Copy feedback Complete homework everyday.Mark questions when we go through them. Take down solution if you get it wrong. Complete all of your homework in the same copy. Title and date each page. Hand up copies every Friday at the end of class.
3 Use of Complex Numbers in Real LifeDigital communications Computer Science Image Processing (Photo shopping) Sound Waves in music – Piano. Electrical engineering Electromagnetic waves and oscillations.
4 What are Complex Numbers?Quadratic Equations with Complex Roots Quadratic Equations with Real Roots
5 You cant get the square root of a negative number!Where do they come from? Solve the following equation using the quadratic formula: x2 + 6x + 10 = 0 a = 1 b = 6 c = 10 − 6 ± √ (6)2 − 4 (1) (10) 2(1) = − 6 ± √ 36 − 40 2 Syntax Error − 6 ± √ −4 2 You cant get the square root of a negative number! =
6 Square root of a negative numberThe square root of any negative number is known as an Imaginary Number E.g. √−25 or √−4 We use the symbol i to represent the imaginary number √− Hence i = √−1 and i2 = −1. Example: √−25 = 5i √−4 = 2i
7 Examples of Complex NumbersWrite the following in the form a + bi: 3 + √ −4 − 4 + √ −9 6 − √ −16 = 3 + 2i Where 3 is the Real Part and 2i is the Imaginary part of the complex number. = − i = 6 − 4i
8 Adding & Subtracting Complex NumbersIf z1 = 1 + 2i and z2 = 3 − 4i, evaluate the following: z1 + z2 z1 − z2 3z z1 − 2z2
9 Multiplying two Complex Numbers by each other
10 Multiplying two Complex NumbersIf z1 = 1 + 2i, z2 = 3 − 4i and z3 = − 4 + 3i evaluate the following: z1z2 z1z3 Exercise: Evaluate z2z3 **N.B. i2 = −1
11 Dividing Complex Numbers (Complex Conjugate)
12 Complex Conjugate If z = a +bi is a complex number then a − bi is known as the complex conjugate of z. The conjugate is written as z. Find the complex conjugate of the following: z1 = 2 + 3i z2 = 3 − 4i z1 = 2 − 3i z2 = 3 + 4i
13 Division of two Complex NumbersIf z1 = 1 + 2i, z2 = 3 − 4i evaluate the following: z z2 Exercise: Evaluate z2 z1 **N.B. i2 = −1