Imaginary & Complex Numbers Honors Algebra II Keeper 1

What is an imaginary number? A number "𝑖" that, when squared, results in a negative. Defined as 𝑖= −1

If 𝑖= −1 , then …. 𝑖 2 = ___________________________ 𝑖 3 = ___________________________ 𝑖 4 = ___________________________ 𝑖 5 = ___________________________ 𝑖 6 = ___________________________

How could you simplify imaginary numbers with any exponent? *Take the exponent and divide by 4. *If your remainder is... 1 use ________ 2 use ________ 3 use ________ 0 use ________

Example Simplify 𝑖 38

Example Simplify 𝑖 45

Example Simplify 𝑖 400

Example Simplify 𝑖 23

We can also use properties of exponents with imaginary numbers! Example: Simplify completely. 𝒊 𝟓 ∙𝟐 𝒊 𝟒 ∙𝟖 𝒊 𝟏𝟎

Example: Simplify completely. 3 𝑖 3 2

Example: Simplify completely. 𝑖 25 𝑖 7

Example Simplify 𝑖 23 𝑖 12 + 𝑖 5

We see the most benefit from imaginary numbers when simplifying radicals. Imaginary numbers allow us to simplify radicals with negatives under an EVEN index!

Simplifying Radicals with Negatives: *Take out the negative from under the radical and write as an "𝒊" on the outside. *Simplify the radical like normal.

Example: Simplify completely. −25

Example: Simplify completely. −28

Example: Simplify completely. −63

Example: Simplify completely. −200

Example: Simplify completely. 3 −48

What is a complex number? A combination of a Real Number and an Imaginary Number.

Graphing A Complex Number To graph the complex number 𝒂+𝒃𝒊 plot the point (𝒂,𝒃)

Example Graph each of the following complex numbers. a) 3 + 2i b) –4 – 5i c) –3i d) –1 + 3i e) 2

Absolute Value The absolute value of a complex number a + bi is

Example Find the absolute value of each of the following. 3+4𝑖 −2−𝑖 4 5 𝑖

Distance Formula For Complex Numbers

Example Find the distance between the two points: 2+3𝑖 𝑎𝑛𝑑 5−2𝑖 4−3𝑖 𝑎𝑛𝑑 2+2𝑖 1+2𝑖 𝑎𝑛𝑑 −1+4𝑖

Operations with Imaginary & Complex Numbers: When adding, subtracting, and multiplying...treat these exactly how you would if the problem contained an "𝑥" instead of an "𝑖." Just remember to simplify "𝒊" completely!

Adding/Subtracting: *Combine like-terms* Example: Simplify completely. (5 – 2𝑖) – (–4 −𝑖) 2 −3 − −27 +3 −12

Example: Simplify completely. 5+ 𝑖 3 −(3− 𝑖 3 )

Example: Simplify completely. 2+𝑖 −( 𝑖 4 + 𝑖 3 )

Example: Simplify completely. 3+ 𝑖 2 + 𝑖 4

Example: Simplify completely. 3+2𝑖 −(7+6𝑖)

Example: Simplify completely. 2−3 −48 − 5+ −75