11.3 Powers of Complex Numbers, DeMoivre's Theorem Objective To use De Moivre’s theorem to find powers of complex numbers.

Slides:

Advertisements

Similar presentations

Equations in Quadratic Form

Advertisements

< < < > > >         . There are two kinds of notation for graphs of inequalities: open circle or filled in circle notation and interval notation.

Operations on Functions

Solving Quadratic Equations.

Parallel and Perpendicular Lines. Gradient-Intercept Form Useful for graphing since m is the gradient and b is the y- intercept Point-Gradient Form Use.

LINES. gradient The gradient or gradient of a line is a number that tells us how “steep” the line is and which direction it goes. If you move along the.

TRIGONOMETRY FUNCTIONS

PAR TIAL FRAC TION + DECOMPOSITION. Let’s add the two fractions below. We need a common denominator: In this section we are going to learn how to take.

Let's find the distance between two points. So the distance from (-6,4) to (1,4) is 7. If the.

REAL NUMBERS. {1, 2, 3, 4,... } If you were asked to count, the numbers you’d say are called counting numbers. These numbers can be expressed using set.

SETS A = {1, 3, 2, 5} n(A) = | A | = 4 Sets use “curly” brackets The number of elements in Set A is 4 Sets are denoted by Capital letters 3 is an element.

DOUBLE-ANGLE AND HALF-ANGLE FORMULAS

(r,  ). You are familiar with plotting with a rectangular coordinate system. We are going to look at a new coordinate system called the polar coordinate.

SPECIAL USING TRIANGLES Computing the Values of Trig Functions of Acute Angles.

SOLVING LINEAR EQUATIONS. If we have a linear equation we can “manipulate” it to get it in this form. We just need to make sure that whatever we do preserves.

TRIGONOMETRIC IDENTITIES

You walk directly east from your house one block. How far from your house are you? 1 block You walk directly west from your house one block. How far from.

Logarithmic Functions. y = log a x if and only if x = a y The logarithmic function to the base a, where a > 0 and a  1 is defined: exponential form logarithmic.

INVERSE FUNCTIONS.

The definition of the product of two vectors is: 1 This is called the dot product. Notice the answer is just a number NOT a vector.

Dividing Polynomials.

exponential functions

GEOMETRIC SEQUENCES These are sequences where the ratio of successive terms of a sequence is always the same number. This number is called the common ratio.

VECTORS. A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude.

The standard form of the equation of a circle with its center at the origin is Notice that both the x and y terms are squared. Linear equations don’t.

A binomial is a polynomial with two terms such as x + a. Often we need to raise a binomial to a power. In this section we'll explore a way to do just.

ARITHMETIC SEQUENCES These are sequences where the difference between successive terms of a sequence is always the same number. This number is called the.

When trying to figure out the graphs of polar equations we can convert them to rectangular equations particularly if we recognize the graph in rectangular.

LINEAR Linear programming techniques are used to solve a wide variety of problems, such as optimising airline scheduling and establishing telephone lines.

Properties of Logarithms

Logarithmic and Exponential Equations. Steps for Solving a Logarithmic Equation If the log is in more than one term, use log properties to condense Re-write.

A polynomial function is a function of the form: All of these coefficients are real numbers n must be a positive integer Remember integers are … –2, -1,

Powers and Roots of Complex Numbers. Remember the following to multiply two complex numbers:

The Complex Plane; DeMoivre's Theorem- converting to trigonometric form.

The Complex Plane; DeMoivre's Theorem. Real Axis Imaginary Axis Remember a complex number has a real part and an imaginary part. These are used to plot.

Library of Functions You should be familiar with the shapes of these basic functions. We'll learn them in this section.

SEQUENCES A sequence is a function whose domain in the set of positive integers. So if I gave you a function but limited the domain to the set of positive.

COMPLEX Z R O S. Complex zeros or roots of a polynomial could result from one of two types of factors: Type 1 Type 2 Notice that with either type, the.

Sum and Difference Formulas. Often you will have the cosine of the sum or difference of two angles. We are going to use formulas for this to express in.

Solving Quadratics and Exact Values. Solving Quadratic Equations by Factoring Let's solve the equation First you need to get it in what we call "quadratic.

Surd or Radical Equations. To solve an equation with a surd First isolate the surd This means to get any terms not under the square root on the other.

COMPOSITION OF FUNCTIONS “SUBSTITUTING ONE FUNCTION INTO ANOTHER”

VECTORS. A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude.

Remainder and Factor Theorems. REMAINDER THEOREM Let f be a polynomial function. If f (x) is divided by x – c, then the remainder is f (c). Let’s look.

Dividing Polynomials Using Synthetic Division. List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put.

INTRODUCING PROBABILITY. This is denoted with an S and is a set whose elements are all the possibilities that can occur A probability model has two components:

The sum f + g This just says that to find the sum of two functions, add them together. You should simplify by finding like terms. Combine like terms &

Let's just run through the basics. x axis y axis origin Quadrant I where both x and y are positive Quadrant II where x is negative and y is positive Quadrant.

Solving Trigonometric Equations Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 x y π π 6 -7 π 6 π 6.

We’ve already graphed equations. We can graph functions in the same way. The thing to remember is that on the graph the f(x) or function value is the.

The sum f + g This just says that to find the sum of two functions, add them together. You should simplify by finding like terms. Combine like terms &

(r,  ). You are familiar with plotting with a rectangular coordinate system. We are going to look at a new coordinate system called the polar coordinate.

TRIGONOMETRIC IDENTITIES

Start Up Day 54 PLOT the complex number, z = -4 +4i

Systems of Inequalities.

RATIONAL FUNCTIONS II GRAPHING RATIONAL FUNCTIONS.

THE DOT PRODUCT.

INVERSE FUNCTIONS.

SIMPLE AND COMPOUND INTEREST

INVERSE FUNCTIONS Chapter 1.5 page 120.

Solving Quadratic Equations.

INVERSE FUNCTIONS.

Symmetric about the y axis

exponential functions

Symmetric about the y axis

The Complex Plane; DeMoivre's Theorem

Rana karan dev sing.

Presentation transcript:

11.3 Powers of Complex Numbers, DeMoivre's Theorem Objective To use De Moivre’s theorem to find powers of complex numbers.

(r cisθ)³ = (r cisθ)(r cisθ)(r cisθ) Remember: multiply absolute values and add polar angles. = (r² cis 2θ) (r cisθ) = r³ cis 3θ (r cis θ)ⁿ = rⁿ cis nθ

Abraham de Moivre ( ) DeMoivre’s Theorem You can repeat this process raising complex numbers to powers. Abraham DeMoivre did this and proved the following theorem: This says to raise a complex number to a power, raise the modulus to that power and multiply the argument by that power.

This theorem is used to raise complex numbers to powers. It would be a lot of work to find you would need to FOIL and multiply all of these together and simplify powers of i --- UGH! Instead let's convert to polar form and use DeMoivre's Theorem. but in Quad II

Note: In order to use De Moivre’s theorem, the equation must be in polar form.

Solve the following over the set of complex numbers : We know that if we cube root both sides we could get 1 but we know that there are 3 roots. So we want the complex cube roots of 1. Using DeMoivre's Theorem with the power being a rational exponent (and therefore meaning a root), we can develop a method for finding complex roots. This leads to the following formula:

Let's try this on our problem. We want the cube roots of 1. We want cube root so our n = 3. Can you convert 1 to polar form? (hint: 1 = 1 + 0i) We want cube root so use 3 numbers here Once we build the formula, we use it first with k = 0 and get one root, then with k = 1 to get the second root and finally with k = 2 for last root.

Here's the root we already knew. If you cube any of these numbers you get 1. (Try it and see!)

We found the cube roots of 1 were: Let's plot these on the complex plane about 0.9 Notice each of the complex roots has the same magnitude (1). Also the three points are evenly spaced on a circle. This will always be true of complex roots. each line is 1/2 unit

Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. Shawna has kindly given permission for this resource to be downloaded from and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar

Assignment P. 410 #1, 5, 7, 17