Transcript Complex Numbers

Complex Numbers
Add and Subtract complex numbers
Multiply and divide complex numbers
Lets solve x 2 = -1
To solve these problem, French
mathematician René Descartes
proposed that a number i be defined such
that i 2 = -1. So 1  i
Simplify
No Decimals
 28
Simplify
No Decimals
 28   1  4  7
 i  2 7  2i 7
Simplify
No Decimals
 32 y
i 32 y
3
3
Simplify
No Decimals
 32 y
i 32 y
3
3
4i y 2 y
Powers of i values go a in a circle
Power of i are a cyclical function
i  1
i  1
2
i 3  i
i4  1
i 5  i , i 9  1  i 13
i 6  1 ; i 10  1  i 14
i 7  i ; i 11  i  i 15
i 8  1 ; i 12  1  i 16


Powers of i values go a in a circle
Find the value the expression
i  1
  i
i  i
38
i 2  1
i  i
3
i4  1
i
158

4 9
2
 1
Powers of i values go a in a circle
Find the value the expression
i  1
i
38
i
158
 
 i
4 9
 i  1
2
i 2  1
i  i
3
i4  1
 
 i
4 39
 i  1
2
Adding
Pure imaginary numbers are like terms.
12i  5i  17i
Multiply Imaginary Numbers
Remember i2 equals -1
 3i  2i
 6i
2
 6(1)
6
Multiply Imaginary Numbers
Pull the i out first
 12   2
 i 12  i 2



Multiply Imaginary Numbers
Pull the i out first
 12   2
 i 12  i 2
 i2 2  2  2 3
 
 (1) 2 6
 2 6
Solve
Remember when you take the square root
of a number you must remember its
positive and negative answer.
5 y 2  20  0
5 y 2  20
y 2  4
y   4
y  2i
Complex number
Have two parts: Real and Imaginary
a  bi
Re al
Im aginary
When adding complex numbers,
add the real numbers together and then the
imaginary numbers.
3  5i   2  4i 


When adding complex numbers,
add the real numbers together and then the
imaginary numbers.
3  5i   2  4i 
 3  2  5  4i
 5i
When subtracting complex numbers,
add the real numbers together and then the
imaginary numbers.
4  6i   3  7i 
When subtracting complex numbers,
add the real numbers together and then the
imaginary numbers.
4  6i   3  7i 
4  6i    3  7i 

When subtracting complex numbers,
add the real numbers together and then the
imaginary numbers.
4  6i   3  7i 
4  6i    3  7i 
 1 i
Multiplying Complex numbers
Its time to remember how to FOIL again.
4  3i 5  i 
20  4i  15i  3i 2
Multiplying Complex numbers
Its time to remember how to FOIL again.
4  3i 5  i 
20  4i  15i  3i 2
20  4i  15i  3 1
Multiplying Complex numbers
Its time to remember how to FOIL again.
4  3i 5  i 
20  4i  15i  3i 2
20  4i  15i  3 1
20  11i  3
23  11i
Divide Complex numbers
Complex Conjugate work as with real
numbers conjugates. Important must
5i
3  2i
break into fractions
5i 3  2i 

3  2i 3  2i 
15i  10i 2

9  4i 2

 10 1  15i 10  15i

9  4 1
13
10 15
 i
13 13
Homework
Page 274
# 19 – 41 odd,
49 – 55 odd
Homework
Page 274
# 18 – 40 even,
48 – 54 even