With a different method

Slides:

Advertisements

Similar presentations

Complex Numbers Any number in form a+bi, where a and b are real numbers and i is imaginary. What is an imaginary number?

Advertisements

Complex Numbers.

Section 2.4 Complex Numbers

Complex Numbers.

Complex Numbers The imaginary number i is defined as so that Complex numbers are in the form a + bi where a is called the real part and bi is the imaginary.

Complex Numbers OBJECTIVES Use the imaginary unit i to write complex numbers Add, subtract, and multiply complex numbers Use quadratic formula to find.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

1.3 Complex Number System.

Section 2.2 The Complex Numbers.

Warm-Up: December 13, 2011  Solve for x:. Complex Numbers Section 2.1.

Unit 2 – Quadratic, Polynomial, and Radical Equations and Inequalities

Imaginary Number: POWERS of i: Is there a pattern?

Math is about to get imaginary!

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.

M3U3D4 Warm Up Divide using Synthetic division: (2x ³ - 5x² + 3x + 7) /(x - 2) 2x² - x /(x-2)

Complex Numbers Add and Subtract complex numbers Multiply and divide complex numbers.

Entry task- Solve two different ways 4.8 Complex Numbers Target: I can identify and perform operations with complex numbers.

4-8 Complex Numbers Today’s Objective: I can compute with complex numbers.

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.

Complex Numbers warm up 4 Solve the following Complex Numbers Any number in form a+bi, where a and b are real numbers and i is imaginary. What is an.

5-7: COMPLEX NUMBERS Goal: Understand and use complex numbers.

Imaginary & Complex Numbers. Once upon a time… -In the set of real numbers, negative numbers do not have square roots. -Imaginary numbers were invented.

Complex Number System Reals Rationals (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …) Whole (0, 1, 2, …) Natural (1, 2, …) Irrationals.

Introduction to Complex Numbers Adding, Subtracting, Multiplying Complex Numbers.

Unit 2 – Quadratic, Polynomial, and Radical Equations and Inequalities Chapter 5 – Quadratic Functions and Inequalities 5.4 – Complex Numbers.

Quick Crisp Review Simplifying Square Roots √24√-72.

Chapter 4 Section 8 Complex Numbers Objective: I will be able to identify, graph, and perform operations with complex numbers I will be able to find complex.

5.9 Complex Numbers Objectives: 1.Add and Subtract complex numbers 2.Multiply and divide complex numbers.

Chapter 4.6 Complex Numbers. Imaginary Numbers The expression does not have a real solution because squaring a number cannot result in a negative answer.

January 17, 2012 At the end of the today, you will be able to work with complex numbers. Warm-up: Correct HW 2.3: Pg. 160 # (2x – 1)(x + 2)(x.

Simplify. Complex Numbers Complex Numbers Intro Definition of Pure Imaginary Numbers: For any positive real number, “b” Where i is the imaginary unit.

Complex Numbers Any number in form a+bi, where a and b are real numbers and i is imaginary. What is an imaginary number?

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.

SOL Warm Up 1) C 2) B 3) (4x + y) (2x – 5y) 4) x = 7 ½ and x = -1/2 Answers.

Any questions about the practice? Page , 11, 13, 21, 25, 27, 39, 41, 53.

Complex Numbers Simplifying Addition & Subtraction 33 Multiplication.

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,

Imaginary & Complex Numbers

Roots, Radicals, and Complex Numbers

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Imaginary & Complex Numbers

Perform Operations with Complex Numbers

Imaginary & Complex Numbers

Imaginary & Complex Numbers Mini Unit

Imaginary & Complex Numbers

Ex. Factor a) x2 + 5x + 6 b) x2 + 3x – 40 c) 5x2 – 17x + 6 d) 9x2 – 25.

6.7 Imaginary Numbers & 6.8 Complex Numbers

Complex Numbers.

Math is about to get imaginary!

Imaginary & Complex Numbers

Complex Numbers Any number in form a+bi, where a and b are real numbers and i is imaginary. What is an imaginary number?

Complex Numbers Using Complex Conjugates in dividing complex numbers and factoring quadratics -- Week 15 11/19.

9-5 Complex Numbers.

Sec Math II Performing Operations with Complex Numbers

Imaginary & Complex Numbers

Roots, Radicals, and Complex Numbers

Warm Up Take out your notes from last class and underline or highlight important information that you need to remember when solving and graphing quadratic.

3.2 Complex Numbers.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Imaginary & Complex Numbers

Section 4.6 Complex Numbers

Imaginary & Complex Numbers

Chapter 9 Section 4.

Section 2.4 Complex Numbers

Complex Numbers.

Lesson 2.4 Complex Numbers

Chapter 9 Section 4.

5.4 Complex Numbers.

Introduction to Complex Numbers

Presentation transcript:

With a different method Entry Task With a different method

Target: I can identify and perform operations with complex numbers

-In the set of real numbers, negative numbers do not have square roots. -Imaginary numbers were invented so that negative numbers would have square roots and certain equations would have solutions. -These numbers were devised using an imaginary unit named i. Watch Me

it is a symbol for a specific number Imaginary numbers: i is not a variable it is a symbol for a specific number

With your a/b partner determine the values for the cycle of i 1 1 1 i i i -1 -1 -1 -i -i 1

Definition of Imaginary Numbers Any number in form a+bi, where a and b are real numbers and i is imaginary unit.

Definition of Pure imaginary numbers: Any positive real number b, where i is the imaginary unit and bi is called the pure imaginary number.

Simplify the expression.

Simplify each expression. Remember Remember

When adding or subtracting complex numbers, combine like terms.

Simplify.

Simplify.

Multiplying complex numbers. To multiply complex numbers, you use the same procedure as multiplying polynomials.

Simplify. F O I L

Simplify. F O I L

-Express these numbers in terms of i. Try These

Conjugates In order to simplify a fractional complex number, use a conjugate. What is a conjugate?

are said to be conjugates of each other.

Lets do an example: Rationalize using the conjugate Next

Reduce the fraction

Lets do another example Next

Try these problems.

Homework Pg. 253 – #9,11,19,21,23,27,29,39,41,43,51,61

MULTIPLYING COMPLEX NUMBERS

ANSWERS (-1)

(-1) =

Use the quadratic formula to solve the following: a=3, b= -2, c=5 4 14

Let’s Review You need to be able to: 1) Recognize what i, i2, i3 ect. is equal to (slide 5) 2) Simplify Complex numbers 3) Combine like terms (add or subtract) 4) Multiply (FOIL) complex numbers 5) Divide (multiply by complex conjugates)