Hello. I'm Professor Von Schmohawk and welcome to Why U.
So far we have seen how complex numbers can be added
subtracted
and multiplied.
In this lecture we will see how complex numbers can also be divided
where the dividend
and the divisor
may each contain a real part
and/or an imaginary part.
The result of this division will be a complex quotient
that may also contain a real part and an imaginary part.
We will also see how the vector representation of the quotient
is geometrically related to the vectors of the dividend
and the divisor.
As a general example let's call the dividend "z-one"
the divisor "z-two"
and the quotient "z-three".
In the previous lecture we introduced two parameters
that describe the direction and length of a complex number's vector representation.
The first parameter called the "argument" of the complex number
describes the vector's direction
and is defined as the angle of the vector from the positive real axis.
The argument of a complex number is written as "arg"
followed by the complex number in parentheses.
The second parameter called the "absolute value" or "modulus" of the complex number
describes the vector's length, defined as the distance from the origin
to the point representing the number on the complex plane.
The modulus of a complex number
is written as that number enclosed in a pair of vertical bars
in the same way that the absolute value of a real number is written.
We also said that when two complex numbers are multiplied
their arguments are added
and their moduli are multiplied.
So by adding the arguments of two complex numbers and multiplying their moduli
we produce the modulus and argument of their product, which we will call "z-one".
So when we multiply two complex numbers "z-two" and "z-three"
producing the product "z-one"
the modulus of "z-one"
will be equal to the product of the moduli of "z-two" and "z-three".
And the argument of "z-one"
will be equal to the sum of the arguments of "z-two" and "z-three".
We can also use these relationships to derive the relationship
between the vectors of divided complex numbers and their quotient.
To derive the vector relationships for complex division
we divide both sides of the first equation by "z-two"
and divide both sides of the second equation by the modulus of "z-two".
Likewise, in the third equation
we subtract the argument of "z-two" from both sides.
So this tells us that when we divide a complex number "z-one"
by another complex number "z-two"
producing the quotient "z-three"
the modulus of the quotient
will be equal to the modulus of "z-one" divided by the modulus of "z-two"
and the argument of the quotient
will be equal to the argument of "z-one" minus the argument of "z-two".
Now that we know how the lengths and angles of the vectors in complex division are related
let's try a simple division example
dividing the complex number "negative 8, plus i"
by 2.
Even though this fraction is a valid representation of the quotient
we would like it to be written as a complex number in standard form.
Standard form complex numbers are written as "a" plus "b i"
where "a" and "b" are real numbers and are called the "real part"
and the "imaginary part".
In this example, the quotient is easily separated into its real and imaginary parts.
Once both parts have been simplified
the result of this division is a complex number in standard form.
So dividing the complex number "negative 8, plus i" by 2
produces the quotient "negative 4, plus one-half i".
Now let's see what this complex division looks like when we view these numbers as vectors.
When we divide "negative 8, plus i"
by 2
we get the quotient "negative 4, plus one-half i".
So let's see how this agrees with the vector relationships we just derived for complex division.
In this example, "z-one" is "negative 8, plus i"
"z-two" is 2
and the quotient "z-three" is "negative 4, plus one-half i".
The equation we derived for the moduli
tells us that the length of the quotient's vector "z-three"
should equal the length of vector "z-one"
divided by the length of vector "z-two".
Since the length of vector "z-two" is 2
dividing vector "z-one" by 2 should produce a vector one-half its length
pointing in the same direction
which was the result that was produced.
Likewise, the equation we derived for the arguments
tells us that the angle of vector "z-three"
should equal the angle of vector "z-one"
minus the angle of vector "z-two" which is zero.
And since the angle of vector "z-two" is zero
the angles of "z-one" and "z-three" should be equal
which is the result that was produced.
In this last example, we divided the complex number "negative 8, plus i" by 2.
Since the divisor "two" was a real number
this fraction was easy to separate into the sum of a real part and an imaginary part.
However, this process can become more complicated
when the divisor is a complex number with both a real part and an imaginary part.
To see why, let's divide "negative 8, plus i"
by the complex number "2 plus 3i".
When we separate this fraction into two fractions with the same denominators
as we did in the previous example
both fractions still contain a mixture of real and imaginary numbers.
However, if we can convert this fraction into an equivalent fraction
whose denominator is a real number
then we will be able to split this fraction into the sum of a real part and an imaginary part
as we did in the previous example.
This can be accomplished by creating the "complex conjugate" of the denominator.
A complex conjugate is easy to create
since it is done by simply changing the sign of its imaginary term.
So the complex conjugate of the denominator "2 plus 3i "
is "2 minus 3i".
Now, an important property of complex conjugates
is that when a complex number is multiplied by its complex conjugate
the product will always be a real number.
For example, let's use the distributive property to multiply "2 plus 3i"
by its complex conjugate "2 minus 3i".
2 times 2
is 4.
2 times "negative 3i"
is "negative 6i".
3i times 2
is 6i
and 3i times "negative three i"
is "negative 9, i-squared".
And since "i-squared"
is "negative 1"
"negative 9, i-squared" is "negative 9" times "negative 1"
or "positive 9".
Adding these results
"negative 6i" plus 6i
is zero
leaving 4 plus 9
or 13.
So as in the example we have just seen
multiplying any complex number by its complex conjugate always produces a real number.
The reason for this is easy to see when we view this process in vector form.
When we take any complex number
and multiply it by its complex conjugate
we add their arguments.
But since the arguments of the complex number
and its complex conjugate
have the same magnitude but opposite signs
their sum is always zero.
Therefore, the product of a complex number and its complex conjugate
always has an argument of zero.
The product's vector will therefore lie on the real axis and represent a real number.
Since multiplying a complex number by its complex conjugate produces a real number
we can use this property to convert a fraction with a complex denominator
into an equivalent fraction with a real denominator.
That fraction can then be easily separated into its real and imaginary parts.
To change this complex fraction into an equivalent fraction with a real denominator
we start by multiplying the denominator by its complex conjugate.
However, we must also multiply the numerator by the same number
so that the value of the fraction remains unchanged.
Starting with the denominator
we have already seen that "2 plus 3i" times its complex conjugate "2 minus 3i"
is 13.
So now let's multiply the terms in the numerator.
"Negative 8" times 2
is "negative 16"
"negative 8" times "negative 3i"
is 24i
"i" times 2
is 2i
and "i" times "negative 3i"
is "negative 3, i-squared".
And since "i-squared"
is "negative 1"
"negative 3, i-squared" is 3.
Adding these results
24i plus 2i
is 26i
and "negative 16" plus 3
is "negative 13".
So the numerator of our result is "negative 13" plus 26i.
Now that we have produced an equivalent fraction whose denominator is a real number
it is easily separated into the sum of a real fraction and an imaginary fraction.
Simplifying the fractions, "negative thirteen-thirteenths"
is "negative 1"
and "26 thirteenths"
is 2.
So "negative 8, plus i" divided by "2 plus 3i"
is "negative 1, plus 2i".
In this lecture, we have seen that when a complex number
containing both a real and an imaginary part is divided by a real number
we first create a fraction with the complex number as the numerator
and the real number as the denominator
and then split this fraction into its real and imaginary parts.
On the other hand, when a complex number is divided by another complex number
containing both a real and an imaginary part
we first multiply the top and bottom of the fraction
by the complex conjugate of the denominator
giving us an equivalent fraction
that can then be separated into its real and imaginary parts.
Now that we can perform all four arithmetic operations
addition, subtraction, multiplication, and division, with complex numbers
we will see how complex numbers can be used in complex functions
in the same way that real numbers are used in real functions.