Download Complex Numbers

Sec. 2.5
Complex
Numbers
First, try these…
Evaluate:
3
i  1 i   1  i
2
4
2 2
i  1 i  i i  1
Definition:
Complex Number
A complex number is any number that can be
written in the form
a  bi
where a and b are real numbers. The real number a
is the real part, the real number b is the imaginary
part, and a + bi is the standard form.
Definition:
Complex Number
a  bi
If b = 0, we have the real number a…
 All real numbers are also complex numbers!!!
If a = 0, the number
a  bi
becomes
bi
Imaginary Number
Two complex numbers are equal if and only if their
real and imaginary parts are equal.
Operations with Complex
Numbers
If a + bi and c + di are two complex numbers, then
Sum:
 a  bi    c  di    a  c   b  d  i
Difference:
 a  bi    c  di    a  c   b  d  i
Let’s Practice!!!
Write the given expressions in the form bi, where b is a
real number.
81 
81 1  9 1  9i
20 
 20 1  2 5 1
 2 5i
Let’s Practice!!!
Find the real numbers x and y that make the given
equation true.
x  yi  2  5i
x = 2, y = 5
Let’s Practice!!!
Write the sum or difference in standard form.
 7  3i    4  5i   11  2i
 2  i   8  3i   6  4i
Let’s Practice!!!
Write the product in standard form.
 2  3i  5  i   10  2i  15i  3i
2
Foil!!!
 10  13i  3 1
 13  13i
Can we use our calculator to evaluate
operations with complex numbers?
Let’s Practice!!!
If
1
3
z 
i , find
2 2
z
2
and
z
3
.
1
3  1
3  1
3
3
3 2
z   
i 

i
i
i i
  

4
4
4
 2 2  2 2  4
2
1 2 3
3
1
3
 
i   1   
i
4
4
4
2 2
Check with calculator!!!
Let’s Practice!!!
1
3
z 
i , find
2 2
If
z
2
and
z
3
.
 1
3  1
3 
z  z z    
i   
i 
 2 2  2 2 
3
2
1
3
3
3 2
1
3
 
i
i  i    0i   1  1
4 4
4
4
4
4
Check with calculator!!!