Transcript Document
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i
i 1
• 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
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 1
really exist?
i 1
Examples of how we use
16 16 1
4i
4i
i 1
81 81 1
9i
9i
Examples of how we use
i 1
45 45 1
3 3 5 1
3 5 1
3 5 i
3i 5
200 200 1
2 2 2 5 5 1
2 5 2 1
10 2 i
10i 2
The first four powers of i establish an
important pattern and should be
memorized.
Powers of i
i i
i 1
i i
i 1
1
3
2
4
i 1
4
i i
3
i i
1
i 1
2
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.
Powers of i
Find i23
Find i2006
Find i37
Find i828
i
1
i
1
Complex Number System
Reals
Imaginary
i, 2i, -3-7i, etc.
Rationals
(fractions, decimals)
Integers
(…, -1, -2, 0, 1, 2, …)
Whole
(0, 1, 2, …)
Natural
(1, 2, …)
Irrationals
(no fractions)
pi, e
Express these numbers in terms of i.
1.) 5 1 5 1 5 i 5
2.) 7 1 7 1 7
i 7
3.) 99 1 99 1 99
i 3 3 11 3i 11
You try…
4.
5.
6.
7 i 7
36
6i
160 4i 10
7.
8.
Multiplying
47i 2 94i
2i 5 2i 1 5 2i i 5
2i
2
5
2 5
9.
3 7 i 3 i 7 i
2
21
(1) 21 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.
Complex Numbers
a + bi
real
imaginary
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.
Add or Subtract
7.)
10.
7i 9i 16i
8.)
11.
(5 6i) (2 11i) 3 5i
12.
9.)
(2 3i) (4 2i) 2 3i 4 2i
2 i
Examples
2
1. (i 3)
2
2
i ( 3)
1( 3 3)
1(3)
3
2. Solve 3x 10 26
2
3x 36
2
x 12
2
x 12
x i 12
x 2i 3
2
Multiplying
Treat the i’s like variables, then change
any that are not to the first power
Ex:
i(3 i)
3i i
2
3i (1)
1 3i
Ex:
(2 3i)(6 2i)
12 4i 18i 6i
2
12 22i 6(1)
12 22i 6
6 22i
3 11i
Ex :
1 2i
(3 11i )(1 2i )
(1 2i )(1 2i )
3 6i 11i 22i
2
1 2i 2i 4i
3 5i 22(1)
1 4(1)
3 5i 22
1 4
2
25 5i
5
25 5i
5
5
5 i
Work
p. 277
#4 – 10, 17 – 28, 37 – 55