Download Imaginary Numbers

Imaginary & Complex Numbers
Once upon a time…
1  no real solution
-In the set of real numbers, negative numbers do
not have square roots.
-Imaginary numbers were invented so that negative
numbers would have square roots and certain
equations would have solutions.
-These numbers were devised using an imaginary
unit named i.
i  1
-The imaginary numbers consist of all numbers bi,
where b is a real number and i is the imaginary unit,
with the property that i² = -1.
-The first four powers of i establish an important
pattern and should be memorized.
Powers of i
i i
1
i  1
2
i  i
3
i 1
4
i 1
4
i  i
3
i
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
1.) Find i23
2.) Find i2006
3.) Find i37
4.) 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
Simplify.
3.)
1.)
-Express these numbers in terms of i.
5  1*5  1 5  i 5
2.)
4.)  7   1* 7   1 7  i 7
3.)
5.)
99  1*99  1 99
 i 3 311
 3i 11
You try…
6.
7.
8.
7  i 7
 36
 6i
160  4i 10
Now that you know how to
take the square root of a
negative number, try solving
this quadratic with completing
the square.
x2 – 4x +6 = 0
Check with your teacher to
see if you got the right
answer.
Stop Here
To multiply imaginary numbers or
an imaginary number by a real
number, it is important first to
express the imaginary numbers in
terms of i.
Multiplying
9.
47i  2  94i
10.
2i  5  2i  1 5  2i  i 5
 2i
11.
2
 3  7  i 3  i 7  i
5  2 5
2
21
 (1) 21  21
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.)
12.
7i  9i  16i
8.)
13.
(5  6i )  (2  11i)  3 5i
14.
9.)
(2  3i )  (4  2i)  2  3i  4  2i
 2  i
Multiplying & Dividing
Complex Numbers
Part of 7.9 in your book
REMEMBER: i² = -1
Multiply
1)
2)
3i  4i  12i  12(1)  12
2
7i 
2
 7 i  49( 1)  49
2 2
You try…
3)
7i  12i  84i  84(1)
2
 84
4)
 11i    11 i   121(1)
2
2
2
 121
Multiply
5)
4  3i 7  2i 
 28 8i 21i 6i
 28  29i  6i
 28  29i  6(1)
2
 28  29i  6
 22  29i
2
You try…
6)
2  i 3 10i 
 6  20i  3i  10i
2
 6  17i  10i
 6  17i 101
 6  17i  10
 16 17i
2
You try…
7)
5  7i 5  7i 
 2535i35i 49i
 25  49(1)
 25  49
 74
2
Conjugate
-The conjugate of a + bi is a – bi
-The conjugate of a – bi is a + bi
Find the conjugate of each
number…
8)
3  4i
9)
4  7i
10)
5i
3  4i
4  7i
5i
6
6
Because 6  0i is the same as 6  0i
11)
Divide…
12)
5  9i 1  i 5  5i  9i  9i

2
1 i  i  i
1 i 1 i
14  4i 14  4i


2
1 i
2
2
 7  2i
You try…
13)
2  3i 3  5i
3  5i 3  5i
6  10i  9i  15i

2
9  15i  15i  25i
9  19i 9  19i

2 
9  25i
34
2