What Does The Imaginary Unit i Look Like?

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.

The Imaginary Unit

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,

Complex Number

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.

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John Redden

John Redden

Professor and Author

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