1. in a math class far, far away..
A long long time ago,
in a math class far, far away..
There was no way
to take the square root
of a negative number

2. Every time we squared a negative number
We got a positive.

3. (-1) = 1
(-2) = 4
(-3) = 9

4. that when multiplied by itself
Is there a number,
that when multiplied by itself
gives you a negative???

5. Can we in fact, take the square root
of a negative number?
WE CAN!!!!

6. Ladies and Gentlemen of Analytic Geometry
I present to you
a NEW number...
A number so complex...

7. It stretches the imagination..
I present to you:

9. You can't take the square root of a negative number, right?
When we were young and still in Coordinate Algebra, no numbers that, when multiplied by themselves, gave us a negative answer. 
Squaring a negative number always gives you a positive.   (-1)² = 1. (-2)² = 4 (-3)² = 9

10. So here’s what the math people did: They used the letter “i” to represent the square root of (-1). “i” stands for “imaginary.”
So, does
really exist?

11. Examples of how we use

12. Examples of how we use

14. The first four powers of i establish an important pattern and should be memorized.

15. Powers of i Divide the exponent by 4 No remainder: answer is 1.
Remainder of 1: answer is i.
Remainder of 2: answer is –1.
Remainder of 3: answer is –i.

16. Powers of i
Find i23
Find i2006
Find i37
Find i828

17. Complex Number System Reals Rationals (fractions, decimals) Integers
Imaginary
i, 2i, -3-7i, etc.
Rationals
(fractions, decimals)
Integers
(…, -1, -2, 0, 1, 2, …)
Irrationals
(no fractions)
pi, e
Whole
(0, 1, 2, …)
Natural
(1, 2, …)

18. Express these numbers in terms of i.

19. You try… 4. 5. 6.

20. Multiplying 7. 8. 9.

21. To mult. imaginary numbers or an imaginary number by a real number, it’s important to 1st express the imaginary numbers in terms of i.

22. a + bi imaginary real Complex Numbers
The complex numbers consist of all sums a + bi, where a and b are real numbers and i is the imaginary unit. The real part is a, and the imaginary part is bi.

23. Add or Subtract
10.
11.
12.

24. Examples

25. Multiplying Treat the i’s like variables, then change any that are not to the first power
Ex:
Ex: