Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the previous lecture, we saw how complex numbers can be added
or subtracted.
In this lecture, we will demonstrate how complex numbers can be multiplied.
We will start with a simple example
multiplying the complex number 3+2i
times 2 .
Since this complex number is a sum of two numbers, 3
and 2i
we can use the distributive property to multiply this sum times two.
2 times 3
is 6
and 2 times 2i
is 4i
resulting in the complex number 6+4i .
This multiplication can be visualized graphically on the complex plane
as we showed in the previous lecture.
For instance, the complex number 3+2i
is represented as a point
whose horizontal position is determined by the value of its real part, 3
and whose vertical position is determined by the value of its imaginary part, 2i .
We also saw that each complex number can be visualized as a "vector"
which is drawn as an arrow extending from the origin to that point.
When working with vectors in the complex plane
the length of a vector is often referred to as the "modulus"
or "absolute value" of the complex number.
When we multiply the complex number 3+2i by 2
giving us 6+4i
we see that the resulting vector points in the same direction as the original vector
but is twice as long.
In other words, the modulus has been multiplied by two.
We can also say that this vector has been "scaled" by a factor of two.
For this reason, when discussing vectors
real numbers are sometimes referred to as "scalars".
So now let's look at a more interesting example
multiplying the same complex number 3+2i
times the imaginary number i .
Just as before, we use the distributive property
to multiply this complex number times i .
i times 3
is 3i
and i times 2i
is two "i squared".
Now since the value of "i squared"
is defined as "negative 1"
two "i squared" is "negative 2"
and the result of this multiplication
is -2+3i.
Visualizing this result graphically
we see that the resulting vector is the same length as the original vector
but rotated counterclockwise by 90 degrees.
In fact, multiplying any complex number by i
will always produce a vector of the same length
rotated counterclockwise by 90 degrees.
For example, let's take the result of this multiplication, -2+3i
and multiply again by i .
Using the distributive property
i times "negative 2"
is "negative 2i"
and i times 3i
is three "i squared".
And since the value of "i squared" is "negative 1"
three "i-squared" is "negative 3" .
The result of this multiplication
is therefore "negative 3" minus 2i .
Once again, the resulting vector is the same length as the original vector
but rotated counterclockwise by 90 degrees.
If we then take that result, "negative 3" minus 2i
and multiply again by i
we get i times "negative 3"
which is "negative 3i"
and i times "negative 2i"
which is negative two "i squared".
And since "i squared"
is "negative 1"
negative two "i squared" is "negative 2" times "negative 1"
or "positive 2" .
So the result of this multiplication
is 2-3i .
Once again, the resulting vector is the same length
rotated counterclockwise by 90 degrees.
If we continue this process by taking that result, 2-3i
and multiply a fourth time by i
we get i times 2
or 2i
and i times "negative 3i"
which is negative three "i squared"
or "negative 3" times "negative 1"
which is "positive 3" .
The result
is therefore 3+2i
bringing us back to the original vector.
We could go on like this forever
with the same rotating cycle repeating every four multiplications by i .
As we saw in this example, repeatedly multiplying any complex number by i
produces a repeating cycle of 90-degree counterclockwise rotations.
Even repeatedly multiplying i times itself
produces a series of 90-degree rotations
that in this case represents i raised to increasing integer powers.
For instance, starting with i
if we multiply i times i we get "i squared" or "negative 1" .
Multiplying again by i gives us "i cubed"
or "negative i" .
A third multiplication by i gives us i to the fourth power
or "positive 1" .
And a fourth multiplication by i gives us i to the fifth power
bringing us back to i .
Just as we saw before
continuing to multiply by i will cause this rotating pattern to repeat forever
with every four multiplications by i
bringing us back to the number we started with.
So let's say that we have the expression " i raised to the fifteenth power".
Since i to the fourth power is "one"
we can eliminate every group of four multiplications by i
without changing the value of that expression.
Therefore, i to the fifteenth power is equal to i to the third power.
And since "i squared" is "negative 1"
i to the third power is "negative i"
and so is i to the fifteenth power.
So any power of i greater than 3 can be simplified
by repeatedly subtracting 4 from the power that i is raised to.
So far we have multiplied a complex number by a real number
and multiplied a complex number by i .
So what do you think will happen
if we multiply a complex number by a real number times i ?
We have seen that multiplying a complex number by a real number
scales the vector, changing its length without changing its direction.
And multiplying a complex number by i
causes its vector to rotate counterclockwise by 90 degrees
changing its direction without changing its length.
Therefore, if a complex number is multiplied both by i
and a real number
the vector will be rotated counterclockwise by 90 degrees
and its length will be scaled by the real number.
So at this point, we should have everything we need to know
in order to multiply two complex numbers together
for example, 2+i times 3+2i.
We will start by showing how the numbers are multiplied algebraically
using the distributed property.
And then we will see how this multiplication can be graphically visualized using vectors.
Multiplying, using the distributive property
2 times 3
is 6
and 2 times 2i
is 4i .
Likewise, i times 3
is 3i
and i times 2i
is two "i squared".
And since "i squared"
is "negative 1"
two "i squared" is "negative 2" .
Combining these results
we get 6 + 4i + 3i + "negative 2" .
We can now combine like terms.
4i + 3i
is 7i .
And 6 + "negative 2"
is 4.
So the product of these two complex numbers
is the complex number 4+7i
As we have just seen, multiplying complex numbers is not difficult.
We simply use the distributive property
and then combine like terms, remembering that "i squared" is "negative 1" .
But how can this multiplication process be visualized using vectors?
We will do this by changing this multiplication into a vector addition.
First we will multiply 3+2i by 2
and draw a vector to represent that result.
Then we will multiply 3+2i by i
and draw a second vector representing that result.
Finally, we will visually represent the sum of these results
by adding those two vectors.
So let's start by drawing the vector for 3+2i .
Next, we multiply 3+2i by 2 .
This produces a vector pointing in the same direction as the vector for 3+2i
but with twice its length.
Now in a second step, we multiply 3+2i by i .
This produces a vector of the same length as 3+2i
rotated 90 degrees.
These two vectors are then added.
This graphically produces the result we obtained algebraically, 4+7i .
So using vectors, we have graphically shown that 2+i
times 3+2i
is 4+7i .
Now, let's do one last example, multiplying 3+2i
by 2+4i .
Just as we did in the previous example, we will start with 3+2i
and multiply this number by 2
and 4i
and then add the two results.
Multiplying 3+2i by 2
produces a vector pointing in the same direction as 3+2i
but twice its length.
Then we multiply 3+2i by 4i .
As we saw, multiplying by i produces a vector rotated 90 degrees counterclockwise.
However, since we are also multiplying by 4
this vector will be four times as long as the original vector.
These two vectors are then added
producing the result -2+16i .
So we have shown graphically that 2+4i
times 3+2i
is -2+16i .
Of course this can be confirmed algebraically
by multiplying using the distributive property.
2 times 3
is 6
and 2 times 2i
is 4i .
Likewise, 4i times 3
is 12i
and 4i times 2i
is 8 "i squared".
And since "i squared"
is "negative 1"
8 "i squared" is "negative 8" .
Combining these results and combining like terms
4i + 12i
is 16i
and 6 + "negative 8"
is "negative 2"
which confirms the result we obtained graphically.
Earlier in this lecture, we said that the length of a vector in the complex plane
is called its "modulus" or "absolute value".
So the modulus or absolute value of a complex number
is its distance from the origin on the complex plane.
If in addition to the complex number's modulus
we know the angle of its vector from the positive real axis
then we can uniquely identify that complex number's position on the plane.
That angle is called the "argument" of the complex number.
Although we would need some knowledge of trigonometry to show why this is true
when two complex numbers are multiplied
their arguments are added
and their moduli are multiplied.
As an example of this property
if two complex numbers had arguments of 40 degrees
and 160 degrees
then the argument of their product would be 40 plus 160
or 200 degrees.
Additionally, if these numbers had moduli of 2
and 4
then the modulus of their product would be 2 times 4
or 8.
So far we have seen how to perform addition, subtraction, and multiplication
using complex numbers.
In the next lecture, we will see how to perform complex division.