The Useless Number

In this exciting video, the “Useless Number” refers to complex numbers, which were first thought to be useless when first discovered. However, nowadays everyone knows that complex numbers are very useful in engineering, physics and science!

Also, check out these related posts on complex numbers:

An interesting book about complex numbers is An Imaginary Tale: The Story of [the Square Root of Minus One] (Princeton Science Library). Styled like a storybook, this book tells the history of the imaginary numbers, which was discovered as early as during ancient Egypt. However, people didn’t realize the immense usefulness of complex numbers until much later. Click the image below to read more!

What is i to the power of i?

When you first learnt that $\boxed{i=\sqrt{-1}}$, you have entered the mysterious world of complex numbers.

A mystifying question would be to ask, what is i to the power of i? Is it a complex number?

The surprising answer is that $i^i$ is a real number! Let us explain it here:

The key step is to use Euler’s formula: $\boxed{e^{i\pi}=-1}$. This has been voted as the most beautiful equation in mathematics by many people.

Then, $i=\sqrt{-1}=(-1)^{1/2}={(e^{i\pi})}^{1/2}=e^{i\pi /2}$

Hence, $i^i=e^{i^2\pi /2}=e^{-\pi /2}\approx 0.208$

It is really amazing that an imaginary number to the power of an imaginary number gives a real number, isn’t it? Leave your comments below!