Download Imaginary Number

7-7 Imaginary and Complex
Numbers
Why Imaginary Numbers?
 What
is the square root of 9?
9  3 because 3 3  9
 What
is the square root of -9?
9  ?
no real number
New type of number was defined for this purpose.
It is called an Imaginary Number
Imaginary numbers are NOT in the Real Set.
 The
constant, i, is defined as the
square root of negative 1:
i  1
 Multiples
Numbers
of i are called Imaginary
 The
square root of -9 is an imaginary
number...
9  9  1  3 i  3i

To simplify a square root with negative
coefficient inside radical, write it as an
imaginary number.
 Powers
of i:
i
i  1
2
i  i  i  1 i  i
3
2
i  i i  1 1  1
4
2
2
 This
5
pattern repeats:
4
i  i  i  1i  i
i  i i  1 1  1
6
4
2
i  i  i  1 i  i
7
4
3
i  i  i  11  1
8
4
4
Multiples of i
 We
can find higher powers of i using
this repeating pattern: i, -1, -i, 1
85
i ?
What is the highest number less than or equal to 85
that is divisible by 4? 84
i 
4 21
i
So the answer is:
85
1
i 1 i  i
Powers of i - Practice
28
i
1
75
i
-i
113
i
i
86
i
-1
1089
i
i
Negative Exponents
Ex:
Ex:
i
6
1
1
1


1 1
 
i  i
6 1
 
 i i
4
2 1
1i

 i
1
1 1
 (i)  2   1
i 1
Odd negative powers are opposite
Even negative powers are the same!
2
Simplify:
Ex 1:
Ex 2:
 36  6i
 20
 2i 5
Multiply
 10i

Ex 3 5i  ( 2)

Ex 4

Ex 5   2  (  3 )  i
6  3
i
2
18  3 2
2
6
 6
Complex Numbers
 Complex
Number : a + bi ,
Where a and b are real #s and i is
imaginary part
 real and imaginary numbers are
not like terms,
 Examples: 3 - 7i, -2 + 8i, -4i, 5 + 2i
Complex #s
Imaginary #s
Real #s
Rational #s
Irrational #s
Add and Subtract
 Combine
Like Terms
(the real & imaginary parts).
 Example:
(3 + 4i) + (-5 - 2i) = -2 + 2i
Practice
Add these Complex Numbers:
 (4
+ 7i) - (2 - 3i)
 (3 - i) + (7i)
 (-3 + 2i) - (-3 + i)
= 2 +10i
= 3 + 6i
=i
Assignment

7-7/323/1-41, 43-48, 56-64