Transcript Document

2
2
2
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
 4i
 4i
i  1
81  81  1
9i
 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