Example 1: i: Powers:
Information on Complex Numbers
Simplify i2 by replacing i by its definition.
What result do we get for the square of
(the square root of something)?
We get the something.
Apply this to our example.
Note: We'll use this often.
Simplify i3 by using the previous result and clever factoring.
i3 = i2 * i.
Replace i2.
We get i3 = (-1) * i
We don't usually write the (-1). Rewrite this without (-1).
Finally, i3 = -i.
Simplify i4 in a similar way (using factors of i2).
i4 = i2 * i2
Replace i2.
i4 = i2 * i2 = (-1) * (-1) = 1
Summarize these 4 powers.
Example 2: Powers of i > 4:
Information on Complex Numbers
Use the above results to obtain simplifications for
i5, i6, i7, and i8, by factoring the powers of i.
i5 = i4 * i
i6 = i4 * i2
i7 = i4 * i3
i8 = i4 * i4
Use the above summary to replace i4.
i5 = (1) * i = i
i6 = (1) * i2 = i2 = -1
i7 = (1) * i3 = i3 = -i
i8 = (1) * i4 = i4 = 1
This pattern repeats for each group of 4 powers of i.
Use the idea of this pattern to simplify i17.
i17 = i16 * i, or
i17 = (i4)4 * i, or
i17 = (1)4 * i = i
Simplify i23 in a similar way.
Here we can group as follows:
i23 = i20 * i3, or
i23 = (i4)5 * i3, or
i23 = (1)5 * i3 = i3
In summary, we can always reduce a
power of i to one of the first 4 powers.
Information on Complex Numbers
Simplify 
.
First factor out the 
.
Rewrite this in terms of i.
Information on Complex Numbers
Simplify 
.
First rewrite this in terms of i.
What do we call this type of number?
It is called a complex number.
What are the two parts called?
The "4" is the real part, and the coefficient of i is called the imaginary part. So here the imaginary part is 
.
Will the imaginary part always have a square root sign in it?
This problem would make you think that, but either part can be any value, such as 3, -4, 2.56.
Information on Complex Numbers
Simplify 
Do you see any common factors here?
Yes.
Factor them out.
Divide the common factors.
Express the real part of this result.
2/3
Express the imaginary part of this result.
. As a reminder, this part is the coefficient of "i", and doesn't include it.
Example 6: Adding Complex Numbers:
Information on Complex Numbers
Simplify q = (4 - 3i) + (5 - 7i)
How do we calculate the real part of q?
By adding the real parts of the two complex numbers.
How do we calculate the imaginary part of q?
By adding the imaginary parts of the two complex numbers.
Find the real part.
4 + 5 = 9
Find the imaginary part.
-3 + (-7) = -10
Use these results to express q.
q = (4 - 3i) + (5 - 7i) = 9 - 10i
Example 7: Subtracting Complex
Numbers:
Information on Complex Numbers
Simplify w = (4 - 3i) - (5 - 7i)
How do we calculate the real part of w?
By subtracting the real parts of the two complex numbers.
How do we calculate the imaginary part of w?
By subtracting the imaginary parts of the two complex numbers.
Find the real part.
4 - 5 = -1
Find the imaginary part.
-3 - (-7) = 4
Use these results to express w.
w = (4 - 3i) - (5 - 7i) = -1 + 4i
Multiplying Complex Numbers:
Information on Complex Numbers
Simplify (2i) * (7i)
Find the product.
(2i) * (7i) = 14 i2
Replace the square of "i".
14i2 = 14(-1) = -14
Information on Complex Numbers
Simplify (-5i)2
Does the squaring apply to 5i or to -5i?
To -5i, that is to the whole quantity in the parentheses.
Do the squaring.
(-5i)2 = 25i2
Replace the square of "i".
(-5i)2 = 25(-1) = -25
Information on Complex Numbers
Simplify q = -3 i * (4 - 5 i ) Can we distribute the multiplication with complex numbers?
Yes.
Do it.
q = -12 i + 15 i2
Can we replace i2?
Yes.
With what?
Now rewrite the expression for q.
q = -12 i + 15 (-1) = -12 i - 15
Information on Complex Numbers
Simplify q = (5 - 2i) * (3 + 4i)
We have the product of two binomials. How do we usually get the product when we have real, not complex, numbers?
We multiply each term of the first binomial by each term of the second.
Is that what we do here?
Yes.
Can we use FOIL to set up the multiplication of these complex numbers?
Yes.
Do it.
q = 5*3 + 5*(4i) + (-2i)*3 + (-2i)*(4i)
Do the multiplication.
q = 15 + 20i - 6i + (-8)i2
Combine the like terms.
q = 15 + 14i -8i2
Replace i2.
q = 15 + 14i -8(-1), or
q = 15 + 14i + 8, or
q = 23 + 14i
Information on Complex Numbers
Simplify w = (2 + i)*(2 - i)
This looks like (a + b)*(a - b) = a2 - b2.
What did we call this special product?
We called it "the difference of squares" because of the nature of the result.
Can we consider w = (2 + i)*(2 - i) in the same way?
Yes.
Evaluate w.
w = 22 -( i )2 = 4 - i2
Replace i2.
w = 4 -(-1) = 5.
Information on Complex Numbers
Simplify q = (3 + i )2 Rewrite this as a product of factors.
q = (3 + i ) * (3 + i )
Can we use FOIL on this?
Yes.
Do it.
q = 9 + 3 i + i * 3 + i2
Can we add the two middle terms?
Yes.
Do it.
q = 9 + 6 i + i2
Can we replace i2 ?
Yes.
Do it.
Since
we get q = 9 + 6 i - 1 = 8 + 6 i
How is the result different if we
evaluate (3 - i )2 ?
(3 - i )2 = 9 -3 i - i * 3 + i2, or
(3 - i )2 = 9 -6 i + (-1) = 8 -6 i
How is the result different if we
evaluate (3 + i ) * (3 - i )?
(3 + i ) * (3 - i ) = 9 - 3 i + i * 3 - i2
= 9 + 0 i - (-1) = 10.
Information on Complex Numbers
Simplify u = (2 - 3i)2 + (2 - 3i)
The "square" term looks like
(a - b)2 = a2 - 2ab = b2.
What did we call this special product?
We called it "the perfect square of a difference".
Can we use that here?
Yes.
Evaluate the square term using this special product.
(2 - 3i)2 = 22 - 2(2)(3)i + 32(i)2, or
(2 - 3i)2 = 4 - 12i +9i2, or
(2 - 3i)2 = 4 - 12i -9
(2 - 3i)2 = -5 - 12i
Use this to evaluate u (defined above).
u = -5 -12i + (2 - 3i)
Combine the like terms.
u = -5 + 2 -12i -3i, or
u = -3 -15i.
Quotients involving Complex Numbers:
Example 15:
Information on Complex Numbers
Simplify 
Can we do any factoring here?
Yes.
Do it.
Cancel the common factor.
q = 5 - 2i
Information on Complex Numbers
Simplify 
How do we proceed?
We need to rationalize the denominator.
How do we do that?
We need to multiply the denominator by its complex conjugate.
Express the complex conjugate of 4 - i.
The conjugate of 4 - i is 4 + i.
Now use this result to set up the rationalization of the denominator.
Why do we also multiply
the numerator by (4 + i)?
We want to obtain an equivalent fraction, which we accomplish by multiplying the original fraction by 1 in the form
Do the multiplication.
Substitute for i2.
Information on Complex Numbers
Simplify What should we do first?
We need to rationalize the denominator.
How?
We need to multiply by the conjugate.
Set that up.
Do the multiplication.
Replace i2 and combine like terms.
Information on Complex Numbers
Simplify Rewrite in terms of i.
Are we done?
No.
What else must we do?
Rationalize the denominator.
How do we do that?
Multiply the numerator and denominator by the conjugate.
What is the conjugate here?
2 + 6i
Set up the rationalization process.
Do the multiplication.
Combine like terms.
Replace i2.
Divide out the common factor.
