9. • You can't take the square root of a negative
number, right?
• When we were young and still in Coordinate
Algebra, 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 here’s what the math people
did: They used the letter “i” to
represent the square root of (-1).
“i” stands for “imaginary.”
1
i
So, does 1
really exist?
11. Examples of how we use 1
i
16 16 1
4 i
4i
81 81 1
9 i
9i
12. Examples of how we use 1
i
45 45 1
3 5 1
3 5 i
3 5
i
14. The first four powers of i establish an
important pattern and should be
memorized.
Powers of i
1 2
1
i i i
3 4
1
i i i
15. 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.
i4
1
i i
1
i2
1
i i
3
16. Powers of i
Find i23
Find i2006
Find i37
Find i828
i
1
i
1
20. 94i
2
2 5
i
2 5
2
21
i
( 1) 21
21
Multiplying
47 2
i
2 5
i
3 7
2 1 5
i
2 5
i i
i i
3 7
7.
8.
9.
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.
22. a + bi
Complex Numbers
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.
23. 7.) 7 9
i i
16i
8.) ( 5 6 ) (2 11 )
i i
3
5i
9.) (2 3 ) (4 2 )
i i
2 3 4 2
i i
2 i
Add or Subtract
10.
11.
12.
25. Multiplying
Treat the i’s like variables, then change
any that are not to the first power
Ex: )
3
( i
i
2
3 i
i
)
1
(
3
i
i
3
1
Ex: )
2
6
)(
3
2
( i
i
2
6
18
4
12 i
i
i
)
1
(
6
22
12
i
6
22
12
i
i
22
6
26. 3 11
:
1 2
i
Ex
i
)
2
1
)(
2
1
(
)
2
1
)(
11
3
(
i
i
i
i
2
2
4
2
2
1
22
11
6
3
i
i
i
i
i
i
)
1
(
4
1
)
1
(
22
5
3
i
4
1
22
5
3
i
5
5
25 i
5
5
5
25 i
i
5