Complex Numbers Notes PDF

CHAPTER 3 : COMPLEX NUMBERS 1. INTRODUCTION TO COMPLEX NUMBER a. CATRESIAN/ALGEBRIC COMPLEX NUMBER b. THE ARGAND DIAGRAM c. OPERATION OF COMPLEX NUMBER 2. TRIGOMETRIC, POLAR & EXPONENTIAL FORM a. MODULUS/MAGNITUDE b. ARGUMENTED/AMPLITUDE c. MULTIPLICATION & DIVISI...

CHAPTER 3 : COMPLEX NUMBERS 1. INTRODUCTION TO COMPLEX NUMBER a. CATRESIAN/ALGEBRIC COMPLEX NUMBER b. THE ARGAND DIAGRAM c. OPERATION OF COMPLEX NUMBER 2. TRIGOMETRIC, POLAR & EXPONENTIAL FORM a. MODULUS/MAGNITUDE b. ARGUMENTED/AMPLITUDE c. MULTIPLICATION & DIVISION d. DE MOIVRE’S THEOREM  TECHNICAL MATHEMATICS 1 (WQD10103) Lecture 3 1 3 COMPLEX NUMBERS 3.1 INTRODUCTION TO COMPLEX NUMBER 3 3 COMPLEX NUMBERS 3.1 INTRODUCTION TO COMPLEX NUMBER What is the usage of complex numbers in real life? 1. As an electrical engineer, I can confidently say that we did a LOT of complex number Maths during my degree. It’s used, for example, when working with AC electricity because the voltage and current aren’t always in phase, and using complex numbers helps with that. We did complex algebra, calculus, trigonometry and all sorts of stuff with complex numbers. It was great fun. 4 https://www.quora.com/What-is-the-usage-of-complex-numbers-in-real-life 3 COMPLEX NUMBERS 3.1 INTRODUCTION TO COMPLEX NUMBER What is the usage of complex numbers in real life? 2. Complex numbers are intimately connected to trigonometric functions and, as practically everything that forms part of any engineering solution is modelled with trig one way or another (especially things that 'vibrate' like mechanical parts or electronic signals), complex numbers are at the root of engineering. 5 https://www.quora.com/What-is-the-usage-of-complex-numbers-in-real-life 3 COMPLEX NUMBERS 3.1.1 CARTESIAN/ALGEBRIC COMPLEX NUMBER i.e. i2 = -1 6 3 COMPLEX NUMBERS 3.1.1 CARTESIAN/ALGEBRIC COMPLEX NUMBER 7 3 COMPLEX NUMBERS 3.1.1 CARTESIAN/ALGEBRIC COMPLEX NUMBER EXAMPLE 1 State the real and imaginary parts of each of the following complex numbers. a. 3 + 4i b. -2 + 5i The real part = 3 The real part = -2 The imaginary part = 4. The imaginary part = 5. c. -4 – 3i d. 6i e. 4 – 2i f. 5 + 7i 8 3 COMPLEX NUMBERS 3.1.1 CARTESIAN/ALGEBRIC COMPLEX NUMBER EXAMPLE 2 Express the following square roots in terms of i. a. − 9 b. - 0.25 Solution : Solution : − 9 = 9 × -1 = 9 −1 - 0.25 = 0.25 -1 = 3i = 0.5i c. − 5 =2.23i d. - 36 Solution : Solution : − 5 = 5 −1 = 5i - 36 = 36 -1 = i is not under the radical = 6i e. − 72 f. - 49 =7i =8.48i 9 3 COMPLEX NUMBERS 3.1.2 ARGAND DIAGRAM GRAPH OF A COMPLEX NUMBER z = a + bi  For example: z = 4 + 2i (4,2) 2 4 10 3 COMPLEX NUMBERS 3.1.2 ARGAND DIAGRAM EXAMPLE 3 a. 11 3 COMPLEX NUMBERS Solution: 3.1.2 ARGAND DIAGRAM b. Plot the complex number listed in Example 4.1 on an Argand diagram a. 3 + 4i b. -2 + 5i c. 6i d. -4 – 3i e. 4 – 2i f. 5 + 7i  Note that purely real numbers lie on the real axis and purely imaginary numbers lie on the imaginary axis.  Complex conjugate such as a and d are symmetrical about the x axis 12 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER POWERS OF i  Study the following powers of i. i = ( -1) 2  ( -1) =  (−1) 1 2 2   = (−1) i2 = 13   3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER POWERS OF i EXAMPLE 4 Simplify each of the following expressions. = (i2)5 = (-1)5 = -1 14 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER POWERS OF i Simplify each of the following expressions. e. i2-i6 f. i9+i15 15 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER Addition: (a + bi) + (c + di) = a + c + bi + di Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i a + bi - c – di = a - c + bi – di Multiplication: (a + bi)(c + di) = ac + adi + bic + bdi2 = ac + adi + bci + bd(-1) = ac + adi + bci – bd = ac – bd + adi + bci = (ac - bd) + (ad + bc)i 16 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM numerator Division: (a + bi) = (a + bi)(c - di) (c + di) (c + di)(c - di) denominator = ac – adi + bci - bdi2 c2 – cdi + dic – (di)2 = ac – bd (-1) – adi + bci c2 – cdi + cdi – (d)2(-1) = ac + bd – adi + bci c2 + d2 = (ac + bd) + (bc – ad)i c2 + d2 c2 + d2 17 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM COMPLEX CONJUGATE  What is complex conjugate? 18 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM EXAMPLE 5 Given z1 = 12 + 5i, z2 = 9-4i and z3 = -2 + i, determine : a. z1 + z2 b. z2 - z1 Solution : Solution : z1 + z2 = (12 + 5i) + (9 − 4i) z2 - z1 = (9 − 4i) - (12 + 5i) 19 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM Given z1 = 12 + 5i, z2 = 9-4i and z3 = -2 + i, determine : c. z 1 + z 2 d. z 2 − z 1 Solution : Solution : 20 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM Given z1 = 12 + 5i, z2 = 9-4i and z3 = -2 + i, determine : e. z 1 - z 2 f. z 2 − z 3 21 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM Given z1 = 12 + 5i, z2 = 9-4i and z3 = -2 + i, determine : g. z 1 + z 3 h. z 2 + z 3 - z1 22 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM EXAMPLE 6 23 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM e. (9 - 5i ) - (-10 + 8i ) + 3i f. 5i + (-8 + i ) - (3 + 7i ) 24 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM EXAMPLE 7 Given M = 4 – 5i, N = 2 + 6i, O=-6+2i and P = 3i, determine the following: a. MN = (4 – 5i)(2 +6i) = 8 + 24i -10i -30i2 = 8 + 14i + 30 = 38 + 14i b. MP = (4 – 5i)(3i) 25 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM c. NO d. OM 26 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM e. M N M = (4 – 5i) = (4 – 5i) (2 -6i) N (2 +6i) (2 + 6i) (2 -6i) 27 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM f. N O N = (2 + 6i) = O (-6 +2i) 28 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM g. O M 29 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM EQUALITY OF TWO COMPLEX NUMBERS Two complex numbers, a + bi and c + di are EQUAL, if and only if a = c and b = d. Example: If a + bi = 5 – 3i, find the values of a and b a + bi = 5 – 3i a=5 bi = -3i b = -3 30 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM EQUALITY OF TWO COMPLEX NUMBERS EXAMPLE 8 Determine each of the following unknown: a. 2a + 7b i = 4 − 21i b. (2x - 3y) + (x + 5y ) i = 11- 14 i Solution : 2a = 4 a=2 7b = -21 b = -3 31 3 COMPLEX NUMBERS 3.1.3 OPERATION OF COMPLEX NUMBER – ALGEBRIC FORM EQUALITY OF TWO COMPLEX NUMBERS Determine each of the following unknown: c. (5x + 1) - 3yi = x + (2y + 4)i d. t 2 - 3t + 5ui = −2 - 10i 32 3 COMPLEX NUMBERS 2. TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS 1. MODULUS/MAGNITUDE 33 3 COMPLEX NUMBERS 1. MODULUS/MAGNITUDE 2. ARGUMENT/AMPLITUDE 34 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS 35 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS With Euler’s formula, and can rewrite the polar form of a complex number into its exponential form as follows, 36 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS EXAMPLE 9 Determine the modulus and the argument for each the following complex Number. Then express to trigonometric, polar and exponential form. z = r (cos θ + i sin θ) = 6.71 Z = 6.71 z = r < θ Z = 6.71 Exponential form: 37 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS z = r (cos θ + i sin θ) X= (cos 338.2o+isin338.2o) z = r < θ X= ∠338.2 o Exponential form: θ’ θ’ (2/5) θ’ 4th quardrant θ = 360o - θ’ 38 θ = 360 o – 21.8 o = 338.2 o 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS c. M = - 4 - 6i d. N = - 7 + 5i 39 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS e. L = - 4i f. Y = 5 - 12i 40 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS EXAMPLE 10 1. Determine the real and imiginary parts of M = 5e2i. Solution : r = 5, θ = 2 rad change 2 rad to degree : 180ο 2x = 114.58 ο π Algebraic form = a + bi a = Re b = Im ο a = 5 cos114.58ο = −2.08 b = 5 sin 114.58ο = 4.55 41 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS 2. Express S = 4.6e5.7i in polar form and algebraic form. Solution : r = 4.6, θ = 5.7 rad in polar form : 180ο 5.7 x = 326.54 ο (in degrees) π S = 4.6e5.7i = 4.6∠326.54ο in algebraic form : a + bi a = 4.6 cos 326.54ο = 3.84 b = 4.6 sin 326.54ο = −2.54 S = 3.84 - 2.54i 42 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS 3. Express the following complex numbers in the standard form a +bi. π 7π − i i πi a. 2e b. e 3 c. 3e 6 43 3 COMPLEX NUMBERS 3.2.3 MULTIPLICATION AND DIVISION 44 3 COMPLEX NUMBERS 3.2.3 MULTIPLICATION AND DIVISION 45 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS 3.4.1 MULTIPLICATION AND DIVISION 46 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS 3.2.3 MULTIPLICATION AND DIVISION EXAMPLE 11 Given: A = 4(cos50ο + i sin 50ο) B = 8(cos15ο + i sin 15ο) C = 16(cos(-45)ο + i sin (-45ο) Determine: a. AB AB = 4(8) (cos [50ο + 15ο] + i sin [50ο + 15ο]) = 32 (cos 65ο + i sin 65ο) 47 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS 3.2.3 MULTIPLICATION AND DIVISION b. BC c. CA 48 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS 3.2.3 MULTIPLICATION AND DIVISION d. B/A e. A/C f. C/B 49 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS EXAMPLE 12 Evaluate the following, giving your answers in polar form. a. b. 50 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS EXAMPLE 13 c. Z2Z3 d. Z1 Z2 51 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS c. Z2Z3 d. Z1 Z2 52 3 COMPLEX NUMBERS 3.2 TRIGONOMETRIC, POLAR AND EXPONENTIAL FORMS e. Z2 f. Z3 Z1 Z1 53 3 COMPLEX NUMBERS  SUMMARY Polar Form : z=r