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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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Introduction to Complex Numbers Adding, Subtracting, Multiplying And Dividing Complex Numbers SPI 3103.2.1 Describe any number in the complex number system.

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