1. M.1 U.1 Complex Numbers

2. What are imaginary numbers?
Viewed the same way negative numbers once were
How can you have less than zero? 
Numbers which square to give negative real numbers.
“I dislike the term “imaginary number” — it was considered an insult, a slur, designed to hurt i‘s feelings. The number i is just as normal as other numbers, but the name “imaginary” stuck so we’ll use it.”
Imaginary numbers deal with rotations, complex numbers deal with scaling and rotations simultaneously
(we’ll discuss this further later in the week)

3. Imaginary Numbers What is the square root of 9?

4. Imaginary Numbers
The constant, i, is defined as the square root of negative 1:

5. Imaginary Numbers
The square root of -9 is an imaginary number...

6. Imaginary Numbers
Simplify these radicals:
=6xi
=2y√5yi

7. Multiples of i
Consider multiplying two imaginary numbers:
So...

8. Multiples of i
Powers of i:

9. Powers of i - Practice
i28
i75
i113
i86
i1089 1 -i i -1

10. Solutions Involving i
Solve:

11. Complex Numbers Have a real and imaginary part .
Write complex numbers as a + bi
Examples: 3 - 7i, i, -4i, 5 + 2i
Real = a
Imaginary = bi

12. Add & Subtract
Like Terms
Example: (3 + 4i) + (-5 - 2i) =
-2 + 2i

13. Practice (4 + 7i) - (2 - 3i) (3 - i) + (7i) (-3 + 2i) - (-3 + i)
Add these Complex Numbers:
(4 + 7i) - (2 - 3i)
(3 - i) + (7i)
(-3 + 2i) - (-3 + i)
= 2 +10i
= 3 + 6i
= i

14. Multiplying
FOIL and replace i2 with -1:

15. Practice
Multiply:
5i(3 - 4i)
(7 - 4i)(7 + 4i)
= i
= 65

16. Division/Standard Form
A complex number is in standard form when there is no i in the denominator.
Rationalize any fraction with i in the denominator.
Monomial Denominator:
Binomial Denominator:

17. Rationalizing
Monomial: multiply the top & bottom by i.

18. Complex #: Rationalize
Binomial: multiply the numerator and denominator by the conjugate of the denominator ...
conjugate is formed by negating the imaginary term of a binomial

20. Practice
Simplify:

21. Absolute Value of Complex Numbers
Absolute Value is a numbers distance from zero on the coordinate plane.
a = x-axis
b = y axis
Distance from the origin (0,0) =
|z| = √x2+y2
Modulus

22. Graphing Complex Numbers

23. Exit Ticket Simplify Write the following in standard form
(-2+4i) –(3+9i)
Write the following in standard form
8+7i
3+4i
Find the absolute value
4-5i

24. Check Your Answers

25. Homework Complex Numbers worksheet
For #7, remember the quadratic formula!