​My latest video, about the evolution of exponent notation is out!

There’s quite a bit in here that people don’t talk much about these days, so I hope that you learn as much from it as I learned while making it.

I created a poster-style image for this video (with the help of designer Sam Baskin) which chronicles an abridged evolution of the notation. Originally I planned to set up a “pay what you want” for the digital file, so that people could either download it for free or opt to chip in a little bit to support the channel. But it looks like in order to do this through my email list, Kit.com would require people to enter a credit card, even if they’re downloading it for free.

That doesn’t work for me, so instead, here’s a dropbox link to the poster.

If you do want to support my work, you can always “buy me a digital coffee” through Ko-Fi, or subscribe to my Patreon.

When I started researching and writing this video, I expected the main idea to be something to the effect that “modern exponent notation made modern math possible.” The idea being that without Cartesian exponents or something very similar, later innovations, such as the power rule for differentiation or even exponential functions wouldn't be possible. This idea is still in the video, but I came to realize that the real meat was in trying to understand how and why people wrote powers the way they did before the advent of the modern notation.

From our perspective, it seems like an obvious thing that notation for a "repeated multiplication" should just show a)what it is that's being multiplied by itself, and b) the number of times that it is multiplied.

So why was such a notation apparently out of reach for so long?

Understanding this sent me down the path of trying to understand how premodern mathematicians understood numbers, as well as the extent to which geometric analogies were important and not important. In Euclidean geometry, for example, “square” and “cube” literally mean areas and volumes, where the Pythagorean Theorem isn’t actually a² + b² = c² but is instead talking about two areas being added together to equal the third.

But the problem is that in geometry, everything "breaks" after the third degree. Algebra in these days was more about finding the solutions to problems involving numbers, but the geometric terminology stuck around anyway.

All of this makes "the thing that we call exponents" a bit complicated and messy both to do math with, nevermind to teach during this time. And since, for me, this story felt like a story about communicators and educators as much as it was about cutting edge mathematicians, I decided to center Robert Recorde in my story. He was decidedly not a mathematical innovator, even though in his educational writing he invented the = sign. In his final book, Recorde takes a stab at trying to make exponents easier to understand (by making them more visual!), but ultimately, from our modern perspective we would consider him to still be missing the point.

Meanwhile, even once Cartesian notation was able to indicate positive integer exponents, we then had to “break” the idea of repeated multiplication in order to bring negative and fractional exponents into the picture, since multiplying something by itself ½ times doesn’t seem like a meaningful thing to do.

Even though Descartes himself never went beyond positive integer exponents, his notation ultimately made it possible for the syntax to lead the semantics. In other words, once the notation was allowed to lead the way and take on meaning beyond what would be conveyed verbally, new kinds of abstraction -- like Newton using fractional exponents to derive his generalized binomial theorem -- became possible.

That’s all for now. I hope you enjoy the video and the poster!

Ben