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Algebra - Complex Numbers Leaving Cert Helpdesk 27 th September 2012
Why do we need Complex Numbers?
Where are Complex Numbers used? Electronic Engineering Aircraft Design Medicine Movie/Computer game graphics
Adding and Subtracting Complex Numbers Group the real parts together and group the imaginary parts together. Examples:
Multiplication of Complex Numbers
Examples of Multiplication
The Complex Conjugate
Example of Division
Dividing one Complex Number by Another
Examples of the use of Complex Numbers
Modulus of a Complex Number
Polar Form of a Complex Number
Converting a Complex Number to Polar Form
De Moivres Theorem
De Moivres Theorem – Ex. 1
De Moivres Theorem - Example 2
Roots of a Complex Number
Proof by Induction
Proving De Moivres Theorem by Induction
Complex numbers Definitions Conversions Arithmetic Hyperbolic Functions.
Manipulate real and complex numbers and solve equations AS
Complex Numbers 2 The Argand Diagram. Representing Complex Numbers Real numbers are usually represented as positions on a horizontal number line
11.4 Roots of Complex Numbers To find roots of complex numbers.
1Complex numbersV-01 Reference: Croft & Davision, Chapter 14, p Introduction Extended the set of real numbers to.
Sec. 6.6b. One reason for writing complex numbers in trigonometric form is the convenience for multiplying and dividing: T The product i i i involves.
Complex Numbers If we wish to work with, we need to extend the set of real numbers Definitions i is a number such that i 2 = -1 C is the set of numbers.
Complex Numbers in Polar Form. Imaginary and Complex Numbers Imaginary Number – Basic definition: i = √-1 = Sqrt(-1) i 2 = -1 i 3 = i*i 2 = i*(-1) = -i.
Section 1.4 Complex Numbers. The Imaginary Unit i.
Operations with Complex Numbers. Aim: Add, subtract, multiply and/or divide complex numbers in order to simplify. Simplify: 1. 2x + 3y – x + y = x + 4y.
4.6 Perform Operations With Complex Numbers. Vocabulary: Imaginary unit “i”: defined as i = √-1 : i 2 = -1 Imaginary unit is used to solve problems that.
Complex Numbers. Simplify: Answer: Simplify: Answer: Simplify: Answer:
Complex numbers. Extending the number system Operations with complex numbers Do Q1-Q5, pp.226.
CHAPTER 1 COMPLEX NUMBER. CHAPTER OUTLINE 1.1 INTRODUCTION AND DEFINITIONS 1.2 OPERATIONS OF COMPLEX NUMBERS 1.3 THE COMPLEX PLANE 1.4 THE MODULUS AND.
Complex Numbers. Allow us to solve equations with a negative root NB: these are complex conjugates usually notated as z and z.
Lesson 1.8 Complex Numbers Objective: To simplify equations that do not have real number solutions.
IMAGINARY NUMBERS AND DEMOIVRE’S THEOREM Dual 8.3.
Complex Roots Solve for z given that z 4 =81cis60°
Complex Numbers C.A-1.5. Imaginary numbers i represents the square root of – 1.
2.2 Complex Numbers Fri Feb 20 Do Now 1) 2x^2 -8 = 0 2) (sinx)^2 – 2sinx + 1 = 0.
Complex Numbers Ideas from Further Pure 1 What complex numbers are The idea of real and imaginary parts –if z = 4 + 5j, Re(z) = 4, Im(z) = 5 How to add,
Complex Zeros; Fundamental Theorem of Algebra Objective: SWBAT identify complex zeros of a polynomials by using Conjugate Root Theorem SWBAT find all real.
Copyright © 2009 Pearson Addison-Wesley Complex Numbers, Polar Equations, and Parametric Equations.
Section 8.7 Complex Numbers. Overview In previous sections, it was not possible to find the square root of a negative number using real numbers: is not.
2.4 Complex Numbers What is an imaginary number What is a complex number How to add complex numbers How to subtract complex numbers How to multiply complex.
Algebra 2. Solve for x Algebra 2 (KEEP IN MIND THAT A COMPLEX NUMBER CAN BE REAL IF THE IMAGINARY PART OF THE COMPLEX ROOT IS ZERO!) Lesson 6-6 The Fundamental.
Complex Analysis Prepared by Dr. Taha MAhdy. Complex analysis importance Complex analysis has not only transformed the world of mathematics, but surprisingly,
The Complex Plane; De Moivre’s Theorem. Polar Form.
6.6 The Fundamental Theorem of Algebra. Fundamental Theorem of Algebra A polynomial of degree > 1 has at least one complex root Including complex roots.
Chapter 1 Equations and Inequalities Copyright © 2014, 2010, 2007 Pearson Education, Inc Complex Numbers.
Let’s do some Further Maths What sorts of numbers do we already know about? Natural numbers: 1,2,3,4... Integers: 0, -1, -2, -3 Rational numbers: ½, ¼,...
Sullivan Algebra and Trigonometry: Section 1.3 Quadratic Equations in the Complex Number System Objectives Add, Subtract, Multiply, and Divide Complex.
9.9 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra Every polynomial equation with complex coefficients and positive degree n has.
Lesson 2.5 The Fundamental Theorem of Algebra. For f(x) where n > 0, there is at least one zero in the complex number system Complex → real and imaginary.
Imaginary Number: POWERS of i: Is there a pattern? Pattern repeats every 4 th power: Divide power by 4 and use remainder Ex:
Advanced Precalculus Notes 8.3 The Complex Plane: De Moivres Theorem Absolute value of a number: distance from zero (origin) Complex number: z = a + bi.
Know what is meant by proof by Induction Learning Outcomes: PROOF BY INDUCTION Be able to use proof by induction to prove statements.
I Complex Numbers. Given that x² = 2 Then x = ±√2 Or that if ax² + bx + c = 0 How do we solve x² = -2 or a quadratic with b² - 4ac < 0 ? We can introduce.
Warm up: QuestionsAnswers 1) 2). Complex Numbers Lesson 4.8.
Examples: Product Rule for Square Roots 6.2 – Simplified Form for Radicals.
Copyright © 2011 Pearson Education, Inc. Complex Numbers Section P.7 Prerequisites.
Trigonometric Form of a Complex Number. Complex Numbers Recall that a complex number has a real component and an imaginary component. z = a + bi Argand.
Polynomials Review and Complex numbers Newport Math Club Warm Up Problem: The game of assassin is played with 8 people and goes as follows: 1.Each player.
Complex Numbers Modulus-Argument Form. Re Im Modulus-Argument Form y x The complex number z is marked on the Argand diagram. Here is the real and imaginary.
The Complex Plane; DeMoivre's Theorem- converting to trigonometric form.
2.1 Complex Numbers. The Imaginary Unit Complex Numbers the set of all numbers in the form with real numbers a and b; and i, (the imaginary unit), is.
Complex Numbers S. Awad, Ph.D. M. Corless, M.S.E.E. E.C.E. Department University of Michigan-Dearborn Math Review with Matlab: Complex Math.
STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 COMPLEX NUMBERS 2 PROGRAMME 2.
Imaginary Numbers Review Imaginary Numbers Quadratic Forms Converting Standard Form.
1.4. i= -1 i 2 = -1 a+b i Real Imaginary part part.
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