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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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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. 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 i 23 2.) Find i ) Find i 37 4.) Find i 828

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

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

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You try…

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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

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a + bi Complex Numbers real imaginary 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

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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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5) Multiply

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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 -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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13) You try…

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