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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 Part Imaginary Part

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

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**The Complex Number Plane**

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. i -2 -1 1 2 -i Say we want to know where 2 – 2i would be -2i 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**

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

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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? -5 + 2i

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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) 3 + 13i Multiplying Complex Numbers (12i)(7i) 84 i2 = 84 (-1) = -84

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**You can even FOIL Complex Numbers!**

(6 – 5i)(4 – 3i) = 24 – 20i -18i + 15i2 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) -21 + i 43 – 24i

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**SOLVING EQUATIONS WITH COMPLEX SOLUTIONS**

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The Complex Number System

The Complex Number System

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