10.8 The Complex Numbers.

Slides:



Advertisements
Similar presentations
Complex Numbers Objectives Students will learn:
Advertisements

Section 1.4 Complex Numbers
Complex Numbers.
Section 2.4 Complex Numbers
7.5 – Rationalizing the Denominator of Radicals Expressions
Complex Numbers Section 0.7. What if it isnt Real?? We have found the square root of a positive number like = 4, Previously when asked to find the square.
§ 7.7 Complex Numbers.
Section 7.8 Complex Numbers  The imaginary number i  Simplifying square roots of negative numbers  Complex Numbers, and their Form  The Arithmetic.
6.2 – Simplified Form for Radicals
Review and Examples: 7.4 – Adding, Subtracting, Multiplying Radical Expressions.
Complex Numbers OBJECTIVES Use the imaginary unit i to write complex numbers Add, subtract, and multiply complex numbers Use quadratic formula to find.
Section 5.4 Imaginary and Complex Numbers
Lesson 1-5 The Complex Numbers. Objective: Objective: To add, subtract, multiply, and divide complex numbers.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
1.3 Complex Number System.
Copyright © Cengage Learning. All rights reserved. Quadratic Equations, Quadratic Functions, and Complex Numbers 9.
Sullivan Algebra and Trigonometry: Section 1.3 Quadratic Equations in the Complex Number System Objectives Add, Subtract, Multiply, and Divide Complex.
Section 2.2 The Complex Numbers.
MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.
§ 7.7 Complex Numbers. Blitzer, Intermediate Algebra, 4e – Slide #94 Complex Numbers The Imaginary Unit i The imaginary unit i is defined as The Square.
§ 7.7 Complex Numbers. Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.7 Complex Numbers The Imaginary Unit i The imaginary unit i is defined.
Imaginary Number: POWERS of i: Is there a pattern?
Imaginary and Complex Numbers 18 October Question: If I can take the, can I take the ? Not quite…. 
1 Complex Numbers Digital Lesson. 2 Definition: Complex Number The letter i represents the numbers whose square is –1. i = Imaginary unit If a is a positive.
Section 7.7 Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”.
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.
Lesson 2.1, page 266 Complex Numbers Objective: To add, subtract, multiply, or divide complex numbers.
Complex Number System Adding, Subtracting, Multiplying and Dividing Complex Numbers Simplify powers of i.
Imaginary and Complex Numbers Negative numbers do not have square roots in the real-number system. However, a larger number system that contains the real-number.
Complex Numbers Write imaginary numbers using i. 2.Perform arithmetic operations with complex numbers. 3.Raise i to powers.
 PERFORM COMPUTATIONS INVOLVING COMPLEX NUMBERS. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 3.1 The Complex Numbers.
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.
7.7 Complex Numbers. Imaginary Numbers Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution”
Imaginary Number: POWERS of i: Is there a pattern? Ex:
Chapter 5.9 Complex Numbers. Objectives To simplify square roots containing negative radicands. To solve quadratic equations that have pure imaginary.
Chapter 1 Equations and Inequalities Copyright © 2014, 2010, 2007 Pearson Education, Inc Complex Numbers.
NOTES 5.7 FLIPVOCABFLIPVOCAB. Notes 5.7 Given the fact i 2 = ________ The imaginary number is _____ which equals _____ Complex numbers are written in.
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.
Complex Numbers n Understand complex numbers n Simplify complex number expressions.
5.9 Complex Numbers Objectives: 1.Add and Subtract complex numbers 2.Multiply and divide complex numbers.
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.
6.6 – Complex Numbers Complex Number System: This system of numbers consists of the set of real numbers and the set of imaginary numbers. Imaginary Unit:
Radicals and Complex Numbers N-CN.1 Know there is a complex number i such that i 2 = –1, and every complex number has the form a + bi with a and b real.
Holt McDougal Algebra 2 Operations with Complex Numbers Perform operations with complex numbers. Objective.
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,
Section 2.4 – The Complex Numbers. The Complex Number i Express the number in terms of i.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
CHAPTER 3: Quadratic Functions and Equations; Inequalities
Complex Numbers Objectives Students will learn:
PreCalculus 1st Semester
Copyright © 2006 Pearson Education, Inc
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
6.7 Imaginary Numbers & 6.8 Complex Numbers
Imaginary Numbers.
Section 9.7 Complex Numbers.
Ch 6 Complex Numbers.
Roots, Radicals, and Complex Numbers
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Complex Numbers Objectives Students will learn:
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
College Algebra Chapter 1 Equations and Inequalities
Lesson 2.4 Complex Numbers
Section 10.7 Complex Numbers.
Imaginary Numbers though they have real world applications!
CHAPTER 3: Quadratic Functions and Equations; Inequalities
Licensed Electrical & Mechanical Engineer
Section 3.1 The Complex Numbers
Express each number in terms of i.
Introduction to Complex Numbers
Presentation transcript:

10.8 The Complex Numbers

The Number i Imaginary Numbers i is the unique number for which and so we have i 2 = –1. Imaginary Numbers An imaginary number is a number that can be written in the form a + bi, where a and b are real numbers and We can now express the square root of a negative number in terms of i.

Example Express in terms of i: Solution

Complex Numbers A complex number is any number that can be written in the form a + bi, where a and b are real numbers. (Note that a and b both can be 0.) The following are examples of imaginary numbers: Here a = 7, b =2.

Addition and Subtraction The complex numbers obey the commutative, associative, and distributive laws. Thus we can add and subtract them as we do binomials.

Example Add or subtract and simplify.

Multiplication Caution! To multiply square roots of negative real numbers, we first express them in terms of i. For example, Caution! With complex numbers, simply multiplying radicands is incorrect when both radicands are negative:

Example Multiply and simplify. When possible, write answers in the form a + bi. Solution

Solution continued

Conjugate of a Complex Number The conjugate of a complex number a + bi is a – bi, and the conjugate of a – bi is a + bi.

Example Find the conjugate of each number. Solution The conjugate is 4 – 3i. The conjugate is –6 + 9i. The conjugate is –i.

Conjugates and Division Conjugates are used when dividing complex numbers. The procedure is much like that used to rationalize denominators.

Example Divide and simplify to the form a + bi. Solution

Solution continued

Powers of i Simplifying powers of i can be done by using the fact that i 2 = –1 and expressing the given power of i in terms of i 2. Consider the following: i 23 = (i 2)11i1 = (–1)11i = –i

Example Simplify: Solution