Hey it’s Professor Dave, I wanna tell you about complex numbers.
We’ve pretty much wrapped up all the algebra we’re going to cover in our study of mathematics,
which means we need to move on to trigonometry, and then the dreaded calculus.
Before we get to those subjects, there are a handful of one-off topics that are important
to understand, but didn’t quite fit into the flow up until this point.
So let’s knock these out before moving on to harder maths.
The first of these concepts is complex numbers.
When we were discussing the types of numbers, we said that there are real numbers, which
are all the rational and irrational numbers, like negative five, or pi, or seven thirds.
But there are also imaginary numbers.
These are the ones we get when we try to take the square root of a negative number.
The simplest imaginary number is I, in lower-case italics, and it is equal to the square root
of negative one.
We need this number, because there is no real number that, when squared, gives negative
one, and that’s why I is imaginary.
But sometimes we can manipulate I to get real numbers.
I squared, for example, is root negative one times root negative one, which is negative one.
I cubed can be expressed as I squared times I, and we just said I squared is negative
one, so I cubed is equal to negative I. I to the fourth is I squared times I squared,
or negative one times negative one, which is one.
This series of four values repeats every four powers of I, so I to the fifth through eighth
will be these values once more, and then we start again.
I can also be found with other numbers.
Take the square root of negative nine, for example.
This can be expressed as root nine times negative one, and therefore root nine times root negative one.
Root nine is three, and root negative one is I, so we get three I.
In this way, the root of negative X will always be I root X. Anytime we have imaginary numbers
together with real numbers, they are called complex numbers, and we write them in the
form of A plus BI.
Complex numbers have a real part, that’s the A, and then an imaginary part, which is BI
In this way, all numbers are technically complex numbers, because if B is zero, the imaginary
part goes away and we just have a real number.
If A is zero, we have a pure imaginary number, as there is nothing real about it.
But if A and B are both nonzero, we have a complex number in its standard form.
We can do operations with complex numbers just like we would if I was some variable,
like X.
If we had the quantity two plus X plus the quantity three minus two X, we would add the
numbers together to get five, and add the X’s together to get negative X.
It’s exactly the same with I, so if we change these X’s to I’s, we get five minus I.
Since we already know how to do this, we won’t need to practice.
Multiplication and division work the same way, with just one extra twist, we have to
remember that I squared equals negative one.
So let’s say we have the quantity two minus I times the quantity four plus two I.
We can just FOIL this exactly the way we would with two binomials.
First gives eight, outer gives four I, inner gives negative four I, and last gives negative
two I squared.
The two I terms cancel each other out, and negative two I squared is negative two times
negative one, which is two, so we get eight plus two, or ten.
This is how complex numbers can sometimes multiply to give real numbers.
This fact will come in handy if we want to divide complex numbers.
Take something like five plus two I over three minus I.
Obviously we can’t do much with this as it is.
But if we had a real number in the denominator, we could do something.
As we just learned, it is possible to multiply complex numbers to get a real number, so we
can multiply the top and bottom of this quotient by the same complex number to do exactly that.
But what will this complex number be?
As it turns out, we will always get a real number when we multiply a complex number by
its complex conjugate.
The complex conjugate of a complex number is just the same number but with the sign
in between the two terms reversed, so in this case, we want to multiply the top and bottom
by three plus I.
If this had been a plus, we would use a minus instead.
But now, we can FOIL the bottom and the top.
Let’s do the top first.
Fifteen, plus five I, plus six I, plus two I squared.
We combine these to get eleven I, two I squared becomes negative two, and we end up with thirteen
plus eleven I.
Then on the bottom, we get nine, plus three I, minus three I, minus I squared.
We notice that the I terms cancel out, which was exactly the plan, and this will always
be the case when we multiply complex conjugates.
The negative I squared becomes positive one, leaving us with ten.
Then we can just express this in standard form by splitting it up into two fractions,
thirteen over ten plus eleven I over ten.
And even though it’s a little messy, that’s the way we divide complex numbers.
Where might we see complex numbers?
Well, remember when we were finding solutions to polynomials, we would sometimes use the
quadratic formula.
But if the number under the radical, or the discriminant, turned out to be negative, we
said that there were no solutions.
Well in truth, there were no real solutions.
We can get imaginary roots if we express them in terms of I.
For quadratics with no real solutions, there will be two solutions in the form of complex
numbers, and they will specifically be complex conjugates.
In this way, a polynomial of degree N will always have N linear factors.
There can be repeats, so they aren’t always all unique.
They aren’t always all real numbers either, some could be pairs of complex conjugates,
as we just saw with the quadratic.
But if we take all of these into account, an Nth degree polynomial will always have
N linear factors.
This truth is known as the linear factorization theorem.
Complex numbers are very important in math, so let’s check comprehension.