1. Imaginary & Complex Numbers
Honors Algebra II
Keeper 1

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

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

4. 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 ________

5. Example
Simplify 𝑖 38

6. Example
Simplify 𝑖 45

7. Example
Simplify 𝑖 400

8. Example
Simplify 𝑖 23

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

10. Example: Simplify completely.
3 𝑖 3 2

11. Example: Simplify completely.
𝑖 25 𝑖 7

12. Example
Simplify 𝑖 23 𝑖 12 + 𝑖 5

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

14. 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.

15. Example: Simplify completely.
−25

16. Example: Simplify completely.
−28

17. Example: Simplify completely.
−63

18. Example: Simplify completely.
−200

19. Example: Simplify completely.
3 −48

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

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

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

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

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

25. Distance Formula For Complex Numbers

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

27. 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!

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

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

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

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

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

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