^{2}= -1

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.

^{2}

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.