Complex numbers

Content of the topic


What are complex numbers?

What are complex numbers?

Operations on complex numbers

Absolute value of complex numbers, sets on the complex plane

Let’ see what complex numbers are.

First, let’s talk a bit about numbers.

This is 3, for example.

And this is 4.

And unfortunately, sometimes we need negative numbers, too.

Then we may need numbers that express ratios.

These are called rational numbers.

Like the solution of this equation:

And then there are equations where the solution is not a rational number.

So, we introduce the irrational numbers that fill the gap between the rational numbers on the number line.

And that takes us to real numbers. At every point of the number line there is a real number.

But in certain cases - especially if physicists are lurking around - we need numbers that have some quite unusual properties.

For example, one like this:

Right off the top of our heads, we cannot find many numbers that would fit here, because

These strange numbers were named imaginary numbers.

Since the real numbers already took up all spots on the number line, we place the imaginary numbers on an axis perpendicular to it.

The unit of the imaginary axis is .

Its most important property is .

Numbers that consist of real and imaginary parts are called complex numbers.

So, complex numbers are in the form of , and they are located on the so called complex plane.

Here are two complex numbers:

and let’s see how we add or even multiply them together.

For addition, we simply add the real parts

and the imaginary parts.

Multiplication is more exciting.

But .

The funniest is division.

Stay tuned...


Operations on complex numbers

The idea of complex numbers originated from the disappointment that the equation of

does not have a real solution.

We could have just shrugged this little problem off, but as it turns out, some solution would be quite useful, especially for physics problems. So, we need to somehow magically create a solution for this.

That’s how we invented our little imaginary friends, the imaginary numbers.

They live on the imaginary axis perpendicular to the real number line...

and their most important property is that

.

Numbers that have a real part and an imaginary part and can be expressed in the form of  are called complex numbers.

And now let’s see what kind of operations we can do on complex numbers.

Then, there is this weird thing called conjugate.

The conjugate of complex number  is .

Geometrically, conjugation is a reflection about the real axis.

Splendid. Now we are ready for multiplication.

Well, division will be interesting.

We will try to remove  from the denominator.

We do this with the help of its conjugate.

This little trick with the conjugate always works.

If we multiply a complex number by its conjugate, we always get a real number:

Same thing if we add them:

Now, it would be nice to derive some benefit from these complex numbers.


Factorization in complex

Here is this thing… well, a polynomial. Let’s try to turn it into a product of linear factors.

Here comes this formula:

And now let’s try to factorize this one:

There is no such identity as

so we try using the previous one, with some hocus-pocus.

Next, let's see a more complicated one.

One significant benefit of complex numbers is that using complex numbers, all polynomials can be factored into linear factors. 

This is called the Fundamental Theorem of Algebra.

Now, let’ solve some quadratic equations that we thought were hopeless.

Here comes the solution formula:

The absolute value of a complex number is its distance from zero.

We can compute this distance using the Pythagorean Theorem.

Let's see one more.

Instead of the formula, here we try factorization.

And now let’s see what else we can do with these complex numbers.


The polar form (trigonometric form)

There is a big problem with the algebraic form of complex numbers. That is, it is almost impossible to raise them to powers.

Let's try to compute the value of

Well, this is it.

But this could only be some sick joke...

There must be a simple way to raise complex numbers to powers.

This is the usual  algebraic form of complex numbers,

and now we replace it with a polar form (trigonometric form).

The main idea of thispolar form (trigonometric form) is that it describes complex numbers using two new attributes: one is the absolute value and the other is the angle.

We denote the absolute value by r,

and the angle... well, the angle by theta. Here it is:

The polar form (trigonometric form) makes it surprisingly simple to multiply complex numbers,

and to divide them.

And now, let’s get back to the issue of powers.

We want to compute the value of .

Here comes the polar form (trigonometric form):

And now we start computing the powers.

The nth power is computed by raising r to the nth power, and multiple the angle by n:

This way

and that, if we feel like it, can be transformed back into algebraic form.

Now let's try to compute this:

Let's see first the polar form (trigonometric form).

But there is a snag.

This equation here:

has another solution.

As to which one we need, we could decide by flipping a coin,

but it is better to draw a figure.

It seems we need the negative solution.

And now we are ready for the multiplication.


Absolute value of complex numbers, sets on the complex plane

The absolute value of a complex number is its distance from zero.

We can compute this distance using the Pythagorean Theorem.

Let's see one more.

Instead of the formula, here we try factorization.

And now let’s see what else we can do with these complex numbers.

Let’s try to graph on the complex plane those complex numbers where:

We use the algebraic form,

that is  

Next come some coordinate geometry horror stories.

The equation of

is the equation of a circle centered at the origin, with a radius of r.

Based on this,  is also a circle centered at the origin, with a radius of r=2.

And  means the circle and its inside.

Coordinate geometry horror stories:

The equation of a line:

The equation of a circle:

Let’s find on the complex plane complex numbers such that:

We use the algebraic form, so we replace z 

with  everywhere.

The inequality means one side of the line.

Let's see which side.

It is always a good idea to try a=0 and b=0.

This seems to fit, so we need this side of the line.

Next, let's see what is going on with this one:

The inequality means one side of the circle.

Either the inside or the outside of the circle.

Again, it is a good idea to try a=0 and b=0.

It seems we need the outside.

And because equality is not allowed,

the circle itself is not part of the region.

Finally, let’s see what this is about:

We will need to complete the square.


The complex radicals

The differences between real and complex radicals.

Here comes some magic:

The question is: where is the trick?

The fact is: there is no trick.

For example, a while ago we defined what  means. We said .

This is in spite of the fact that there is another number whose square is 4: that number is negative 2.

For complex numbers, the situation is much more entertaining.

For example

Yes, but

furthermore:

So, there are four numbers whose fourth power is 1.

This minor inconvenience prompts us to define radicals differently for complex numbers than we did for real numbers.

The nth root of a real number always meant exactly one number. The nth root of a complex number, on the other hand, means all numbers whose nth power is the original number.

For example

real                      complex

The nth roots of complex number  are complex numbers  

where the following holds:

 and

Here, r denotes the absolute value of the complex number, which is a real number.

So, this is an ordinary real radical - just like in the old days.

RADICALS

Here is this complex number:

Let's see what happens if we search its 5th root.

First, we need the trigonometric form.

And then we are ready for the radical.

This means five complex numbers.

Case k=5 doesn’t matter. We return to case k=0.

So, that’s it for radicals.