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5.6 – Complex Numbers

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What is a Complex Number??? A complex number is made up of two parts – a real number and an imaginary number. Imaginary numbers are defined to be the square root of -1 a + bi Real PartImaginary Part

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COMPLEX NUMBERS Main Rules Where i is imaginary Examples 1) 2)

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The Complex Number Plane 12-2 i 2i -i -2i Because a complex number is made up of a real and an imaginary value, the complex number plane is different than an xy coordinate plane. Say we want to know where 2 – 2i would be We would go left or right for the real part and up or down for the imaginary part.

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Finding Absolute Value The Absolute Value of a complex number is the distance away from the origin on the complex number plane. Ex: |3 - 4i| You can find the absolute value by using the Pythagorean Theorem. In general, |a + bi|= 5

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Additive Inverses of Complex Numbers Remember that to get the additive inverse of something, you simply multiply everything by a negative Ex: The additive Inverse of -5 is 5 Therefore, what is the additive inverse of 5 – 2i? i

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Complex Number Operations Combining like terms (adding or subtracting) (5 + 7i) + (-2 + 6i) (Hint: treat the imaginary i like a variable) i Multiplying Complex Numbers (12i)(7i) 84 i 2 =84 (-1) = -84

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You can even FOIL Complex Numbers! (6 – 5i)(4 – 3i) = 24 – 20i -18i + 15i 2 24 – 38i + 15(-1) 24 – 15 – 38i 9 – 38i Now, try a couple on your own: A)(2 + 3i)(-3 + 5i)B)(4 – 9i)(4 + 3i) i43 – 24i

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SOLVING EQUATIONS WITH COMPLEX SOLUTIONS

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