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2.4 Complex Numbers What is an imaginary number What is a complex number How to add complex numbers How to subtract complex numbers How to multiply complex numbers How to rationalize the denominator How to plot complex numbers

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Imaginary Numbers i is an imaginary number and is the solution to the quadratic equation: x 2 = - 1. Any number in the form b i is an imaginary number. Here is the i multiplication table. i 13 i 27 i 42

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Complex Numbers A complex number is in the form a + b i where a is the real part and b is the imaginary part. Every real number is complex, & every imaginary number is complex. Examples

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Adding Complex Numbers Adding complex numbers is really easy. Add the real to real and the imaginary part to the imaginary part. Examples

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Subtracting Complex Numbers Subtracting complex numbers is easy. Simply subtract the real from the real and the imaginary from the imaginary. Examples

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Multiplying Complex Numbers Multiplying complex numbers is just like multiplying binomials. Examples ( 3 – 2 i ) ( 5 + 2 i ) = 15 – 10 i + 6 i – 4 i 2 but i 2 = -1 so we get 15 – 10 i + 6 i + 4 = ? ( 2 + i ) ( - 2 + i ) = - 4 + 2 i - 2 i + i 2 = ? 2 ( -3 + 5 i ) = ? 3 i ( 4 – 2 i ) = 12 i – 6 i 2 = ? 19– 4 i - 5 - 6 + 10 i 6 + 12 i

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Dividing Complex Numbers (rationalizing the denominator) For some reason we dont like is in the denominator. So we rewrite the fraction by multiplying by the complex conjugate. Example

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Plotting Complex Numbers (on the Argand plane) i R -5 – 2 i 2 – 2 i 2 + 3 i -5 -2 + 3 i -3 i

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