1. Answer parts 4 and 5 on the first page of today’s packet.
Assignment, red pen, highlighter, textbook, GP notebook
U7D5
Have out:
Bellwork:
Simplify: 1) 2) 3) +1 +1
20i2
–3i +2
12i2
–20 +2
–12 +2 4) 5)
Answer parts 4 and 5 on the first page of today’s packet.
total:

2. Recall from yesterday:
Complex Numbers, Part 2
Recall from yesterday:
When a complex number is simplified, it must be written in
_______ form, such that a is the ____ ____ and b is the ________ ____.
a + bi
real
part
imaginary
part
When we add or subtract complex numbers, we just combine the ____ parts and combine the _________ parts.
When we multiply two complex numbers, we use a generic rectangle or the FOIL method to get _____ products, ____ of which are real and ____ imaginary.
real
imaginary
four
two
two
Example:
Example:
2 – 6 – 4i + 10i 15
– 3i
+ 5i
– i2 –4
+ 6i 15
+ 2i
– (–1)
16 + 2i

3. Dividing Complex Numbers
Dividing complex numbers requires some creativity.
For example:
= ???
Recall simplifying expressions such as
To “_________” the denominator, we multiply the numerator and denominator by _____. Then
rationalize =
To simplify complex quotients, we need to make the denominator
______. To do this, we multiply top and bottom by the
_________ _________ of the denominator.
real
complex
conjugate
Definition: For a + bi, the _______ _________ is _____.
complex
conjugate
a – bi

4. How does the complex conjugate work? Multiply (a + bi)(a – bi).
= __________
real
The product of a complex number and its conjugate is ____ and is _______.
a2 + b2

5. 2 – i Back to the example . The conjugate of is ______. (2 – i) 5 – 5i3
– i 2
+ i 2 2 6
– 2i 4
+ 2i
– i
– 3i
+ i2
– i
– 2i
– i2
6 – 2i – 3i – 1
4 – 2i +2i + 1
5 – 5i 5

6. Practice: Simplify. Write all answers in a + bi form.
This is not a + bi form. We must ALWAYS write the answer as 2 terms (or fractions).
6 – 4i
(3 – 2i) 13 9
– 6i
+ 6i
– 4i2
9 + 4 4i
+ i2
(4 + i)
–1 + 4i
(4 + i) 17 16
– 4i
+ 4i
– i2
16 + 1

7. Practice: 6 + 15i + 2i + 5i2 6 + 17i – 5 (2 + 5i) 1 + 17i (2 + 5i) 294
+ 10i
– 10i
– 25i2
4 + 25

8. So what is i? Moreover, what are the powers of i?
Let’s start at the beginning… anything raised to the zero power is 1.
(yeah, yeah, except zero to the zero power)
Therefore i0 = 1
We also know that i1 = by the definition of i.
What is i2?
What about i3, i4, ?

9. Recall the complex plane from yesterday. Label one unit on each axis. 1
i1 = _____ i
Imaginary
i2 = _____ –1
(i2)(i)
(–1)(i) –i
i3 = _____ = _____ = ____
i1 = i i
(i2)(i2)
(–1)(–1) 1
i4 = _____ = _____ = ____
i0 = 1 1
Real
(i4)(i)
(1)(i) i
i5 = _____ = _____ = ____
i4 = 1
i2 = –1 –1 –i
(i4)(i2)
(1)(–1) –1
i3 = –i
i6 = _____ = _____ = ____
(i4)(i3)
(1)(–i) –i
i7 = _____ = _____ = ____
Notice that the units on the plane match our answers for the powers of i. Label the first 4 or so powers on the axis.
(i4)(i4)
(1)(1) 1
i8 = _____ = _____ = ____
i9 = _____ i
i10 = _____ –1

10. The powers of i are in a ______ with _____
elements: ___, ___, ___, ___.
cycle
four i –1 –i 1 IM
To compute a power of i, start at _____, then go around the circle ________________.
i0 = 1
i1 = i
counterclockwise
i0 = 1
The number of ______ is equal to the _______.
jumps RE
power
i2 = –1
i102 will go around the cycle ___ times and land on ___ = ____. 25
i3 = –i i2 –1
i47 = ___ = ___. i3 –i
Just divide the power by ___ and look at the _________. 4
remainder
i157 = ____ = ____. i1 i 39
R:1

11. Negative power means “go the opposite direction”
Another example: 3
R:3
i–15
i–3 i
Negative power means “go the opposite direction” IM
So:
i1 = i
i0 = 1
remainder = 1 i1 RE
i2 = –1
remainder = 2 i2
i3 = –i
remainder = 3 i3
remainder = 0
(no remainder) i4
Of course, reduce these if you can.

12. 1.
i53 i1 i IM 2.
i176 i0 1
i1 = i 3.
i–6
i–2 –1
i0 = 1 4.
i–11
i–3 i RE
i2 = –1
i3 = –i

13. Finish the worksheets.