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11.2 Geometric Representations of Complex Numbers
Graphing Complex Numbers In Algebra 2, you learned how to graph a complex number… Real Part Imaginary Part
Graphing Complex Numbers Graph the point 5 - 7i
A point representing a complex number can be written 2 different ways: 1.) Rectangular Form: z = a + bi 2.) Polar Form z = r cos( ) + r sin( )i Polar form is often simplified by factoring out the radius. z = r cis( )
Covert each complex number to polar form:
Covert each complex number to rectangular form:
Products of Complex Numbers Back in Algebra 2, you learned how to multiply complex numbers… Foil This
Products of Complex Numbers in Polar Form Let’s multiply 2 complex numbers: Reorder the terms…
Products of Complex Numbers in Polar Form Reorder the terms…
Products of Complex Numbers in Polar Form Translation: When you multiply complex numbers in polar form, you simply multiply the radii & add the angles.
Multiply using the polar form of the complex number:
11.2 GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS Can be confusing, polar form has a degree with it, rectangular form does not, this all takes place in.
GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS A Complex Number is in the form: z = a+bi We can graph complex numbers on the axis shown below: Real axis.
8.2 Trigonometric (Polar) Form of Complex Numbers.
DeMoivre’s Theorem The Complex Plane. Complex Number A complex number z = x + yi can be interpreted geometrically as the point (x, y) in the complex plane.
Algebra Operations with Complex Numbers. Vocabulary Imaginary Number i -
Complex Numbers. Solve the Following 1. 2x 2 = 8 2. x = 0.
1 © 2011 Pearson Education, Inc. All rights reserved 1 © 2010 Pearson Education, Inc. All rights reserved © 2011 Pearson Education, Inc. All rights reserved.
11.2a Geometric Representation of a Complex Number Write complex numbers in polar form.
IMAGINARY NUMBERS AND DEMOIVRE’S THEOREM Dual 8.3.
Precalculus Section 1.5 Perform basic operations with complex numbers The basic imaginary unit is i. i = i 2 = -1 A complex number is any number that can.
Introduction to Complex Numbers AC Circuits I. Outline Complex numbers basics Rectangular form – Addition – Subtraction – Multiplication – Division Rectangular.
Lesson 78 – Polar Form of Complex Numbers HL2 Math - Santowski 11/16/15.
5.6 Complex Numbers. Solve the following quadratic: x = 0 Is this quadratic factorable? What does its graph look like? But I thought that you could.
Section 6.5 Complex Numbers in Polar Form. Overview Recall that a complex number is written in the form a + bi, where a and b are real numbers and While.
Complex Numbers. Numbers that are not real are called Imaginary. They use the letter i. i = √-1 or i 2 = -1 Simplify each: √-81 √-10 √-32 √-810.
Algebra II Honors Problem of the Day Homework: p odds Solve the following: No real solution.
Complex Numbers We haven’t been allowed to take the square root of a negative number, but there is a way: Define the imaginary number For example,
Complex Numbers Write imaginary numbers using i. 2.Perform arithmetic operations with complex numbers. 3.Raise i to powers.
Section 5.4 Imaginary and Complex Numbers
5-7: COMPLEX NUMBERS Goal: Understand and use complex numbers.
9.7 PRODUCTS AND QUOTIENTS OF COMPLEX NUMBERS IN POLAR FORM By the end of this section students will be able to multiply and divide complex numbers in.
Trigonometric Form of Complex Numbers. Real Axis Imaginary Axis Remember a complex number has a real part and an imaginary part. These are used to plot.
Complex Numbers in Polar Form; DeMoivre’s Theorem 6.5
Complex Numbers in Polar Form; DeMoivre’s Theorem
Copyright © 2007 Pearson Education, Inc. Slide Trigonometric (Polar) Form of Complex Numbers The Complex Plane and Vector Representations Call.
Complex Numbers Any number in form a+bi, where a and b are real numbers and i is imaginary. What is an imaginary number?
Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
The Complex Plane; DeMoivre's Theorem- converting to trigonometric form.
Copyright © 2009 Pearson Addison-Wesley Complex Numbers, Polar Equations, and Parametric Equations.
The Geometry of Complex Numbers Section 9.1. Remember this?
Copyright © 2009 Pearson Education, Inc. CHAPTER 8: Applications of Trigonometry 8.1The Law of Sines 8.2The Law of Cosines 8.3Complex Numbers: Trigonometric.
Simplify. Complex Numbers Complex Numbers Intro Definition of Pure Imaginary Numbers: For any positive real number, “b” Where i is the imaginary unit.
10.4A Polar Equations Rectangular: P (x, y) Polar: P (r, ) r = radius (distance from origin) = angle (radians)
Pre-Calculus Chapter 6 Additional Topics in Trigonometry.
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 8 Complex Numbers, Polar Equations, and Parametric Equations.
10.4 Trigonometric (Polar) Form of Complex Numbers
Complex Numbers. Complex number is a number in the form z = a+bi, where a and b are real numbers and i is imaginary. Here a is the real part and b is.
Trigonometric (Polar) Form of Complex Numbers
9.7 Products and Quotients of Complex Numbers in Polar Form
Trig form of Complex Numbers Objective: Be able to operate with complex numbers, and be able to convert complex numbers into Trig Form and vise versa.
Pre Clac Chapter 2 Section 4. Imaginary #’s Let’s pretend that x = 0 had a solution That would mean x 2 = -1 That can’t be… if you square a number.
8.3 THE COMPLEX PLANE & DEMOIVRE’S THEOREM Precalculus.
Complex Numbers in Polar Form
Objectives Define and use imaginary and complex numbers.
Jeopardy! for the Classroom. Real Numbers Complex Numbers Polar Equations Polar Graphs Operations w/ Complex Numbers C & V
5.9 Complex Numbers Alg 2. Express the number in terms of i. Factor out –1. Product Property. Simplify. Multiply. Express in terms of i.
Solve the equation. 1.) 3x = 23 2.) 2(x + 7) 2 = 16 Warm Up.
Complex Numbers - Definition The number i is given by: We can also write the following: Numbers in the form of bi, where b is a real number, are called.
5.9 C OMPLEX N UMBERS Algebra II w/ trig. I. Imaginary numbers:(it is used to write the square root of a negative number) A. B. If r is a positive real.
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