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**Imaginary & Complex Numbers**

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Once upon a time…

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**-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.

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-The imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i² = -1. -The first four powers of i establish an important pattern and should be memorized. Powers of i

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**Powers of i Divide the exponent by 4 No remainder: answer is 1.**

remainder of 1: answer is i. remainder of 2: answer is –1. remainder of 3:answer is –i.

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Powers of i 1.) Find i23 2.) Find i2006 3.) Find i37 4.) Find i828

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**Complex Number System Reals Rationals (fractions, decimals) Integers**

Imaginary i, 2i, -3-7i, etc. Rationals (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …) Irrationals (no fractions) pi, e Whole (0, 1, 2, …) Natural (1, 2, …)

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Simplify. -Express these numbers in terms of i. 3.) 4.) 5.)

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You try… 6. 7. 8.

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To multiply imaginary numbers or an imaginary number by a real number, it is important first to express the imaginary numbers in terms of i.

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Multiplying 9. 10. 11.

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**a + bi imaginary real Complex Numbers**

The complex numbers consist of all sums a + bi, where a and b are real numbers and i is the imaginary unit. The real part is a, and the imaginary part is bi.

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Add or Subtract 12. 13. 14.

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**Multiplying & Dividing Complex Numbers**

Part of 7.9 in your book

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REMEMBER: i² = -1 Multiply 1) 2)

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You try… 3) 4)

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Multiply 5)

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You try… 6)

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You try… 7)

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**Conjugate -The conjugate of a + bi is a – bi**

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**Find the conjugate of each number…**

8) 9) 10) 11)

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Divide… 12)

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You try… 13)

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Chapter 5.9 Complex Numbers. Objectives To simplify square roots containing negative radicands. To solve quadratic equations that have pure imaginary.

Chapter 5.9 Complex Numbers. Objectives To simplify square roots containing negative radicands. To solve quadratic equations that have pure imaginary.

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