1. Imaginary & Complex Numbers

2. Once upon a time…

3. -In the set of real numbers, negative numbers do not have square roots.
-Imaginary numbers were invented so that negative numbers would have square roots and certain equations would have solutions.
-These numbers were devised using an imaginary unit named i.

4. -The imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i² = -1.
-The first four powers of i establish an important pattern and should be memorized.
Powers of i

5. 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.

6. Powers of i
1.) Find i23
2.) Find i2006
3.) Find i37
4.) Find i828

7. 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, …)

8. Simplify.
-Express these numbers in terms of i.
3.)
4.)
5.)

9. You try… 6. 7. 8.

10. To multiply imaginary numbers or an imaginary number by a real number, it is important first to express the imaginary numbers in terms of i.

11. Multiplying 9.
10.
11.

12. 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.

13. Add or Subtract
12.
13.
14.

14. Multiplying & Dividing Complex Numbers

15. REMEMBER: i² = -1
Multiply 1) 2)

16. You try… 3) 4)

17. Multiply 5)

18. You try… 6)

19. You try… 7)

20. Conjugate -The conjugate of a + bi is a – bi

21. Find the conjugate of each number…8) 9)
10)
11)

22. Divide…
12)

23. You try…
13)