Imaginary Numbers and Negative Square Roots. The imaginary number i is the square root of -1: Example: Evaluate (3 i) 2 Imaginary Numbers.

Slides:

Advertisements

Similar presentations

Imaginary & Complex Numbers

Advertisements

4.5 Complex Numbers Objectives:

MTH 092 Section 15.1 Introduction to Radicals Section 15.2 Simplifying Radicals.

Squares and Square Roots Objective: Students will be able to successfully multiply and simplify expressions using squares and square roots. Warm-Up Evaluate:

Squares and Square Roots Objective: Students will be able to successfully multiply and simplify expressions using squares and square roots. Warm-Up Evaluate:

Complex Numbers. Once upon a time… Reals Rationals (Can be written as fractions) Integers (…, -1, -2, 0, 1, 2, …) Whole (0, 1, 2, …) Natural (1, 2, …)

Imaginary and Complex Numbers. The imaginary number i is the square root of -1: Example: Evaluate i 2 Imaginary Numbers.

Rewrite With Fractional Exponents. Rewrite with fractional exponent:

1.Be able to divide polynomials 2.Be able to simplify expressions involving powers of monomials by applying the division properties of powers.

Lesson 1-5 The Complex Numbers. Objective: Objective: To add, subtract, multiply, and divide complex numbers.

Introduction Until now, you have been told that you cannot take the square of –1 because there is no number that when squared will result in a negative.

7.1, 7.2 & 7.3 Roots and Radicals and Rational Exponents Square Roots, Cube Roots & Nth Roots Converting Roots/Radicals to Rational Exponents Properties.

1.3 Complex Number System.

5.7 Complex Numbers 12/17/2012.

You can't take the square root of a negative number, right? When we were young and still in Algebra I, no numbers that, when multiplied.

5.4 Complex Numbers Until now, you have always been told that you can’t take the square root of a negative number. If you use imaginary units, you can!

Negative Exponents SWBAT express powers with negative exponents as decimals; express decimals as powers with negative exponents; simplify expressions with.

2.5 Introduction to Complex Numbers 11/7/2012. Quick Review If a number doesn’t show an exponent, it is understood that the number has an exponent of.

Objectives Define and use imaginary and complex numbers.

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

5-9 Operations with Complex Numbers Warm Up Lesson Presentation

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.

Square roots of numbers are called numbers. Unit Imaginary Number i is a number whose square is -1. That is, In other words,. negative imaginary.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 3 Quadratic Functions and Equations.

Imaginary & Complex Numbers Obj: Simplify expressions with imaginary numbers; re-write radicals as complex expressions. Why do imaginary numbers exists.

5.4 Complex Numbers. Let’s see… Can you find the square root of a number? A. E.D. C.B.

Exponents and Powers Power – the result of raising a base to an exponent. Ex. 32 Base – the number being raised to the exponent. Ex is the base.

7.4 – Roots of Real Numbers. Real vs. Imaginary Roots – opposite of exponents!

5.7 Complex Numbers 12/4/2013. Quick Review If a number doesn’t show an exponent, it is understood that the number has an exponent of 1. Ex: 8 = 8 1,

Complex Number System Adding, Subtracting, Multiplying and Dividing Complex Numbers Simplify powers of i.

Warm Up for 1.1 in Math 2 2.3x = 23 ANSWER 5, – 5 1. Simplify 3 4 – 5 Solve the equation (x + 7) 2 = 16 Solve the equation. ANSWER.

7-7 Imaginary and Complex Numbers. Why Imaginary Numbers? n What is the square root of 9? n What is the square root of -9? no real number New type of.

Complex Numbers Day 1. You can see in the graph of f(x) = x below that f has no real zeros. If you solve the corresponding equation 0 = x 2 + 1,

OPERATIONS WITH COMPLEX NUMBERS PRE-CALCULUS. IMAGINARY AND COMPLEX NUMBERS The imaginary unit i is defined as the principle square root of -1. i =

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.

Algebra II 6.1: Evaluate nth Roots and Use Rational Exponents.

Drill #81: Solve each equation or inequality

Algebra and Trigonometry III. REAL NUMBER RATIONAL IRRATIONAL INTEGERSNON INTEGERS NEGATIVE …, – 3, – 2, – 1 WHOLE ZERO 0 + Integers Counting or Natural.

Zero and Negative Exponents

Rewrite With Fractional Exponents. Rewrite with fractional exponent:

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

Aim: How do we operate the imaginary numbers? Do Now: 1. Simplify: 2. Simplify: 3. Simplify: Homework: p.208 #20,22,24,26,27,36,38,40 p.209 # 2,4.

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:

Bell Ringer. Complex Numbers Friday, February 26, 2016.

Holt McDougal Algebra 2 Operations with Complex Numbers Perform operations with complex numbers. Objective.

Imaginary & Complex Numbers

Imaginary & Complex Numbers

Imaginary & Complex Numbers

7.1 nth roots and rational Exponents

Imaginary & Complex Numbers

Imaginary & Complex Numbers

Operations with Complex Numbers

Imaginary Numbers.

5-4 Operations with Complex Numbers SWBAT

Imaginary & Complex Numbers

Notes 9-5: Simplifying Complex Numbers

Imaginary & Complex Numbers

Imaginary & Complex Numbers

Imaginary & Complex Numbers

Complex Number and Roots

Evaluating Expressions

Complex Numbers.

Rewrite With Fractional Exponents

Complex Numbers include Real numbers and Imaginary Numbers

Express each number in terms of i.

8.1 – 8.3 Review Exponents.

Presentation transcript:

Imaginary Numbers and Negative Square Roots

The imaginary number i is the square root of -1: Example: Evaluate (3 i) 2 Imaginary Numbers

Generalizing Powers of i PatternExamples After 4, it starts to repeat itself! Evaluate the examples by first finding a pattern for raising i to a power: Divide the exponent by 4 and find the remainder. The remainder is where it falls in the pattern: Divide by 4 Remainder

Evaluating a Negative Square Root “Factor” out a -1 Rewrite Simplify Calculate the value of the expression below:

The Calculator and Imaginary Numbers Hit “Mode” on your calculator. Locate where the word “REAL” is highlighted. Highlight “ a + bi ” instead. Now your calculator will work with imaginary numbers. The imaginary number “ i ” is found on the calculator by pressing “2ND” and “.” Examples: Use the calculator to evaluate the following.