What is the square root of -1?
What times itself is less than zero?

Let's call the square root of -1 i.
All we know about i is that i⨯i=-1.

There's no i on our left-right number line,
so it goes on a new up-down number line.

Our two number lines make an XY plane.

Each point is X+Yi. We can add and subtract points.

2+2=4, and 2i+2i=4i, so (2+2i)+(2+2i)=4+4i

Now we can move around the plane!

We can also multiply points.

Remember that i⨯i=-1, so 2i⨯3i=-6.

Multiplication rotates and scales complex numbers.

Now we can stretch and spin!

The Complex Plane provides a simple and direct
connection between algebra and geometry.

To learn about the beautiful world of Complex Analysis,
I recommend Indra's Pearls and Visual Complex Analysis.

Not looking to read a whole book? Here's the 3-minute version:

Thanks for reading! Stay tuned for more explorables like this.