Download presentation

Presentation is loading. Please wait.

Published byNoah McMillan Modified over 4 years ago

1
Topic 6 Topic 6 Real and Complex Number Systems II 9.1 – 9.5, 12.1 – 12.2 Algebraic representation of complex numbers including: Cartesian, trigonometric (mod-arg) and polar form Cartesian, trigonometric (mod-arg) and polar form definition of complex numbers including standard and trigonometric form definition of complex numbers including standard and trigonometric form geometric representation of complex numbers including Argand diagrams geometric representation of complex numbers including Argand diagrams powers of complex numbers powers of complex numbers operations with complex numbers including addition, subtraction, scalar multiplication, multiplication and conjugation operations with complex numbers including addition, subtraction, scalar multiplication, multiplication and conjugation

2
Topic 6 Real and Complex Number Systems II

3
Definition i 2 = -1 i = -1 A complex number has the form z = a + bi (standard form) where a and b are real numbers We say that Re(z) = a [the real part of z] and that Im(z) = b [the imaginary part of z]

4
i = i i 2 = -1 i 3 = -i i 4 = 1 i 5 = i i 6 = -1 i 7 = -i i 8 = 1 Question : What is the value of i 2003 ?

6
Equality If a + b i = c + d i then a = c and b = d then a = c and b = d Addition a+b i + c+d i = (a+c) + (b+d) i e.g. 3+4 i + 2+6 i = 5+10 i e.g. 3+4 i + 2+6 i = 5+10 i e.g. 2+6 i – (4-5 i ) = 2+6 i -4+5 i e.g. 2+6 i – (4-5 i ) = 2+6 i -4+5 i = -2+11 i = -2+11 i Scalar Multiplication 3(4+2 i ) = 12+6 i

8
Multiplication (3+4i)(2+5i) = 6+8i+15i+20i 2 = 6+8i+15i+20i 2 = 6 + 23i + -20 = 6 + 23i + -20 = -14 + 23i = -14 + 23i (2+3i)(4-5i) = 8-10i+12i-15i 2 = 8-10i+12i-15i 2 = 8 + 2i -15 i 2 = 8 + 2i -15 i 2 = 23 + 2i = 23 + 2i In general (a+bi)(c+di) = (ac-bd) + (ad+bc)i

9
Exercise FM P 168 Exercise 12.1 Exercise 12.1 ExerciseExerciseExerciseExercise NewQ P 227, 234 Exercise 9.1, 9.3 Exercise 9.1, 9.3

10
Determine the nature of the roots of each of the following quadratics: (a) x 2 – 6x + 9 = 0 (b) x 2 + 7x + 6 = 0 (c) x 2 + 4x + 2 = 0 (d) x 2 + 4x + 8 = 0

11
Determine the nature of the roots of each of the following quadratics: (a) x 2 – 6x + 9 = 0 (b) x 2 + 7x + 6 = 0 (c) x 2 + 4x + 2 = 0 (d) x 2 + 4x + 8 = 0 (a) x 2 – 6x + 9 = 0 = 36 – 4x1x9 = 0 The roots are real and equal [ x = 3 ]

12
Determine the nature of the roots of each of the following quadratics: (a) x 2 – 6x + 9 = 0 (b) x 2 + 7x + 6 = 0 (c) x 2 + 4x + 2 = 0 (d) x 2 + 4x + 8 = 0 (b) x 2 + 7x + 6 = 0 = 49 – 4x1x6 = 25 The roots are real and unequal [ x = -1 or -6 ]

13
Determine the nature of the roots of each of the following quadratics: (a) x 2 – 6x + 9 = 0 (b) x 2 + 7x + 6 = 0 (c) x 2 + 4x + 2 = 0 (d) x 2 + 4x + 8 = 0 (c) x 2 + 4x + 2 = 0 = 16 – 4x1x2 = 8 The roots are real, unequal and irrational [ x = -2 2 ]

14
Determine the nature of the roots of each of the following quadratics: (a) x 2 – 6x + 9 = 0 (b) x 2 + 7x + 6 = 0 (c) x 2 + 4x + 2 = 0 (d) x 2 + 4x + 8 = 0 = 16 – 4x1x8 = -16 The roots are complex and unequal [ x = -2 4i ]

15
Exercise FM P 232 Exercise 9.2 Exercise 9.2

16
Division of complex numbers Try this on your GC

17
ExerciseExerciseExerciseExercise NewQ P 239 Exercise 9.4 Exercise 9.4

18
Exercise Prove that the set of complex numbers under addition forms a group Prove that the set of complex numbers under addition forms a group Prove that the set of complex numbers under multiplication forms a group Prove that the set of complex numbers under multiplication forms a group

19
Model : Show that the set {1,-1,i,-i} forms a group under multiplication Since every row and column contains every element, it must be a group Since every row and column contains every element, it must be a group x1i-i 11i-i1-ii ii-i1 -i-ii1

20
ExerciseExerciseExerciseExercise NewQ P 245 Exercise 9.5 Exercise 9.5

21
Argand Diagrams Model : Represent the complex number 3+2 i on an Argand diagram or

22
Model : Show the addition of 4+i and 1+2i on an Argand diagram

23
Draw the 2 lines representing these numbers

24
Complete the parallelogram and draw in the diagonal. This is the line representing the sum of the two numbers

26
Exercise New Q P300 Ex 12.1

27
Model : Express z=8+2 i in mod-arg form (8,2)

28
Model : Express z=8+2 i in mod-arg form (8,2)

29
Model : Express z=8+2 i in mod-arg form (8,2)

30
r θ

31
Model: Express 3 cis /3 in standard form

32
Exercise New Q P306 Ex 12.2

Similar presentations

Presentation is loading. Please wait....

OK

0 x0 0. 0 x2 0 0 x1 0 0 x3 0 1 x7 7 2 x0 0 9 x0 0.

0 x0 0. 0 x2 0 0 x1 0 0 x3 0 1 x7 7 2 x0 0 9 x0 0.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on power sharing in india download music Ppt on statistics and probability formulas Ppt on schottky diode operation Ppt on cartesian product of sets Ppt on special education in india Ppt on gunn diode oscillator Ppt on representation of rational numbers on number line Human digestive system anatomy and physiology ppt on cells Ppt on biodiversity in india Ppt on taj mahal tea