# What Does The Imaginary Unit i Look Like?

## TL;DR — An arrow in the complex plane!

Before we can lay our eyes on the beautiful imaginary unit *i*, we have to quickly review complex numbers. Alright, we know from a basic math class that *i* is defined as the square root of −1.

And to express a square root of a negative number in terms of the imaginary unit *i*, we use the following property (*a is a non-negative real number*):

Now a **complex number** is any number of the form,

Here *a* and *b* are real numbers, where *a* is called the **real part** and *b* is called the **imaginary part**. For example,

In this case, the complex number *z* has a real part of 3 and an imaginary part of −4.

It’s interesting to note that 7, a real number, is also a complex number. This is because it can be written as 7+0i

with a real part of 7 and an imaginary part of 0. Hence, the set of real numbers is a subset of the set of complex numbers.

Next, we need to define the **complex plane**. This looks similar to the rectangular coordinate plane where the *x*-axis represents the real part of the complex number — the **real axis** (**Re**), and the *y*-axis represents the imaginary part of the complex number — the **imaginary axis** (**Im**).

Furthermore, every complex number can be viewed as a **position vector** extending from the origin in this complex plane. In other words, a complex number *z* = *a*+*bi* = <*a*, *b*> is viewed as follows:

Wow, that’s fascinating — *WTF*! Here are some complex numbers and their corresponding graphs in the complex plane.

Alright, we are ready to look at the beautiful imaginary unit *i.* Now think of the imaginary unit as a complex number:

That is, 0 units on the real axis and 1 unit on the imaginary axis. And without further ado, behold *i *written as a vector in the complex plane:

There you go, that’s what *i* looks like. Exciting! Please be sure to clap and share this beauty with your friends.

**Extra: **Here is my YouTube playlist on complex number operations.