Complex Number.

Slides:



Advertisements
Similar presentations
5.4 Complex Numbers (p. 272).
Advertisements

Section 1.4 Complex Numbers
Complex Numbers.
1.4. i= -1 i 2 = -1 a+b i Real Imaginary part part.
Objective Perform operations with complex numbers.
Complex Numbers OBJECTIVES Use the imaginary unit i to write complex numbers Add, subtract, and multiply complex numbers Use quadratic formula to find.
Complex Numbers i.
Section 5.4 Imaginary and Complex Numbers
2-9 Operations with complex numbers
Section 2-5 Complex Numbers.
4.6 – Perform Operations with Complex Numbers Not all quadratic equations have real-number solutions. For example, x 2 = -1 has no real number solutions.
Sullivan Algebra and Trigonometry: Section 1.3 Quadratic Equations in the Complex Number System Objectives Add, Subtract, Multiply, and Divide Complex.
Imaginary Number: POWERS of i: Is there a pattern?
10.8 The Complex Numbers.
Complex Numbers 2-4.
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.
1 What you will learn  Lots of vocabulary!  A new type of number!  How to add, subtract and multiply this new type of number  How to graph this new.
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.
Express each number in terms of i.
Complex Numbers Definitions Graphing 33 Absolute Values.
Imaginary Number: POWERS of i: Is there a pattern? Ex:
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.
Lesson 1.8 Complex Numbers Objective: To simplify equations that do not have real number solutions.
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.
Complex Numbers C.A-1.5. Imaginary numbers i represents the square root of – 1.
Multiply Simplify Write the expression as a complex number.
Imaginary Numbers Review Imaginary Numbers Quadratic Forms Converting Standard Form.
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.
ALGEBRA TWO CHAPTER FIVE QUADRATIC FUNCTIONS 5.4 Complex Numbers.
Complex Numbers TENNESSEE STATE STANDARDS:
CHAPTER 1 COMPLEX NUMBER.
Complex & Imaginary Numbers
4.4: Complex Numbers -Students will be able to identify the real and imaginary parts of complex numbers and perform basic operations.
Perform Operations with Complex Numbers
Complex Numbers Real part Imaginary part
Complex Numbers Imaginary Numbers Vectors Complex Numbers
Operations with Complex Numbers
4.4 Complex Numbers.
Warm-up 7-7.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Introduction to Complex Numbers
(Sections 4-5 pt. 1 & 2, 4-6 pt. 1, 4-7, 4-8 pt. 1 & 2)
What are imaginary and complex numbers?
Complex Numbers.
The imaginary unit i is defined as
Section 2.1 Complex Numbers
Imaginary Numbers.
4.6 Complex Numbers (p. 275).
Section 1.4 Complex Numbers
Complex Numbers Dave Saha 15 Jan 99.
Roots, Radicals, and Complex Numbers
Objectives Student will learn how to define and use imaginary and complex numbers.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Solving Quadratics Using Square Roots
Complex Number and Roots
1 Preliminaries Precalculus Review I Precalculus Review II
Sec. 1.5 Complex Numbers.
Lesson 2.4 Complex Numbers
Warmup.
Complex Numbers What you’ll learn
4.6 Complex Numbers Algebra II.
Number Lines.
Complex Numbers and i is the imaginary unit
Introduction to Complex Numbers
4.6 – Perform Operations with Complex Numbers
Complex Numbers.
7.7 Complex Numbers.
Presentation transcript:

Complex Number

Complex Number Form: 𝑥+𝑦𝑖

Complex Number Form: 𝑥+𝑦𝑖 x and y are real number i is the imaginary unit: square root of -1 (𝑖= −1 )

Complex Number Elementary operations Conjugate: negative the imaginary part Say 𝑧=𝑥+𝑦𝑖, then the conjugate of z will be: 𝑧 =𝑥−𝑦𝑖

Complex Number Elementary operations Conjugate: 𝑧 =𝑥−𝑦𝑖 Addition and Subtraction: complex number a and b 𝑎±𝑏= 𝑅𝑒 𝑎 ±𝑅𝑒 𝑏 + 𝐼𝑚 𝑎 ±𝐼𝑚 𝑏

Complex Number Elementary operations 𝑎±𝑏= 𝑅𝑒 𝑎 ±𝑅𝑒 𝑏 + 𝐼𝑚 𝑎 ±𝐼𝑚 𝑏 Conjugate: 𝑧 =𝑥−𝑦𝑖 𝑎±𝑏= 𝑅𝑒 𝑎 ±𝑅𝑒 𝑏 + 𝐼𝑚 𝑎 ±𝐼𝑚 𝑏 Multiplication and Division: follow the distributive property 𝑥+𝑦𝑖 × 𝑢+𝑣𝑖 =𝑥𝑢+𝑥𝑣𝑖+𝑦𝑢𝑖+𝑦𝑖𝑣𝑖 =𝑥𝑢+ 𝑥𝑣+𝑦𝑢 𝑖+𝑦𝑣 𝑖 2 = 𝑥𝑢−𝑦𝑣 + 𝑥𝑣+𝑦𝑢 𝑖

Complex Number Elementary operations 𝑎±𝑏= 𝑅𝑒 𝑎 ±𝑅𝑒 𝑏 + 𝐼𝑚 𝑎 ±𝐼𝑚 𝑏 Conjugate: 𝑧 =𝑥−𝑦𝑖 𝑎±𝑏= 𝑅𝑒 𝑎 ±𝑅𝑒 𝑏 + 𝐼𝑚 𝑎 ±𝐼𝑚 𝑏 Multiplication and Division: follow the distributive property 𝑥+𝑦𝑖 × 𝑢+𝑣𝑖 = 𝑥𝑢−𝑦𝑣 + 𝑥𝑣+𝑦𝑢 𝑖 𝑥+𝑦𝑖 𝑢+𝑣𝑖 = 𝑥𝑢+𝑦𝑣 𝑢 2 + 𝑣 2 + 𝑦𝑢−𝑥𝑣 𝑢 2 + 𝑣 2 𝑖

Complex Number Complex Number on Cartesian Coordinate System x is the real axis, y is the imaginary axis

Complex Number Complex Number on Cartesian Coordinate System x is the real axis, y is the imaginary axis If z = x + yi, then the angle of z is the 𝜃= tan −1 ( 𝑦 𝑥 )

Complex Number Complex Number on Cartesian Coordinate System x is the real axis, y is the imaginary axis If z = x + yi, then the angle of z is the 𝜃= tan −1 ( 𝑦 𝑥 ) Absolute value of z: 𝑧 = 𝑥 2 + 𝑦 2

Complex Number Complex Number on Cartesian Coordinate System x is the real axis, y is the imaginary axis If z = x + yi, then the angle of z is the 𝜃= tan −1 ( 𝑦 𝑥 ) Absolute value of z: 𝑧 = 𝑥+𝑦𝑖 = 𝑥 2 + 𝑦 2 Square root of z: 𝑧 = 𝑥+𝑦𝑖 = 𝑥+𝑦𝑖 +𝑥 2 ±𝑖 𝑥+𝑦𝑖 −𝑥 2