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.