Spinning Numbers refers to a special type of number engineers use to represent signals. Numbers that spin! The yellow dot is a spinning number.

Notation

Spinning numbers exist in the world of complex numbers. They are represented with this unusual math notation,

$\large \quad e^{j\omega t}$

This is pretty strange looking, but for the moment, don’t let that bother you. Just appreciate that the expression as a whole represents a number that spins.

In the animation above,

  • The coordinate system is the complex number plane. The horizontal axis is real, the vertical axis is imaginary.
  • The spinning number is the yellow dot. The green stick isn’t part of the number, it’s just there to help us see it.
  • The white circle has a radius of $1$.

A spinning number is made of,

  • $e$ is the base of the natural logarithm, the special number $e = 2.7182818\ldots$
  • $j$ is the imaginary unit. $j = \sqrt{-1}\qquad$
    • Engineers use $j$ instead of $i$ because we use $i$ for something else (current).
  • $\omega$ (omega) is how fast the number spins. It is the angular frequency, measured in radians per second. $2\pi$ radians is a full circle. $1$ radian is about $57\degree.$ If you make $\omega$ bigger, the number spins faster.
  • $t$ is time. You know what time is.

Any time you see $Ae^{\,\pm\,j\,\text{something}\,t}$ you are looking at a spinning number.

A spinning number has $e$ with an imaginary exponent $j$. The exponent includes time $t$. Time is the engine that drives the number forward. The $\pm$ sign controls the direction of rotation. The $\text{something}$ term controls the rate of rotation. $A$ is the amplitude, it determines the radius of the circle.

Signals

Spinning numbers are one of the most beautiful and useful ideas in mathematics and engineering. We use them to represent signals. Spinning numbers are the heart of Fourier and Laplace Transforms.

If you combine two spinning numbers you can represent a sine wave or cosine wave. We do this using Euler’s formula,

$\sin \omega t = \dfrac{1}{2} (e^{+j\omega t} - e^{-j\omega t}) \qquad \cos \omega t = \dfrac{1}{2} (e^{+j\omega t} + e^{-j\omega t})$

With some algebra fiddling we can turn these around,

$\sin \omega t = \dfrac{e^{+j\omega t} - e^{-j\omega t}}{2j}$ $\qquad\cos \omega t = \dfrac{e^{+j\omega t} + e^{-j\omega t}}{2}$

For now, don’t worry about the fancy math notation, there’s plenty of time to study that. Chill, enjoy the videos,

Euler's sine wave animation Euler’s sine wave$\qquad$Euler's cosine wave animation Euler’s cosine wave

If you want to know more and you can’t stand to wait, jump to AC analysis introduction where the theory of spinning numbers is developed and applied.

The animated logo is created in D3.js. The source code is logo-d3.html.

The unique properties of imaginary numbers Veritasium