A long long time ago,

1
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
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 Algebra 2 I present to
you a NEW number...
A number so complex...
7
It stretches the imagination..
I present to you
8
(No Transcript)
9

• You can't take the square root of a negative
  number, right?
• When we were young and still in Algebra I, 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 heres 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
13
(No Transcript)
14
The first four powers of i establish an important
pattern and should be memorized.
Powers of i
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
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, its important to 1st express
the imaginary numbers in terms of i.
22
Complex Numbers
a bi
imaginary
real
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
MultiplyingTreat the is like variables, then
change any that are not to the first power

• Ex

Ex
26
(No Transcript)
27
Workp. 277 4 10, 17 28, 37 55
28
(No Transcript)