Radical Notation

There was one day in my life when I got a standing ovation in a calculus class. I’ll admit, it was an extra special group of students who were prone to spontaneous outbursts of enthusiasm. Business Calc, amiright? But it was a day that stands out in my memory. That was the day I went on a long notation based tangent and told them, among other things, the story of the radical symbol. One short version of the story, per Leonard Euler, has √ being modeled after the letter “r”, which is the first letter of the Latin word “radix” which means root. Conveniently, it is also the first letter of the english word root. Other versions of the story say that the shape is inherited from the Arabic letter “ج” and the Arab mathematician Al-Qalaṣādī. But the more interesting substory, is how often notation is arrived at in a totally roundabout or random way.

Folklore abounds, and notations evolve, and the origin of mathematical notation is an endless source of fascinating speculation.

As far as I’ve seen, the most frequently cited text on the subject is A History of Mathematical Notations by Florian Cajori. There’s a really entertaining Math Overflow thread dedicated to notation that makes people “uncomfortable.” It includes some favorites like why is

sin2(x)=sin(x)·sin(x)

while

sin-1(x)=arcsin(x).

An inverse function, not a reciprocal, as you would expect if we were playing fair. I can’t blame students for feeling like we’re trying to Numberwang them.

On this blog Division by Zero, Dave Richeson gives a great account of the day the division symbol went viral. I remember that day fondly. Richeson reveals the real story behind that symbol that definitely looks like a fraction with dots in the place of the numerator and deniminator but is actually so much deeper and historically rich.

The notation for division in general is pretty fraught. I always notice my students struggle with the notation a | b for “a divides b” which means that b/a is an integer. It is a bit confounding. As was pointed out on Math Overflow, one should never use a symmetric symbol for an asymmetric relation.

Jeff Miller, a retired high school math teacher, maintains a nice page about first-uses and attributions of various mathematical notation, like matrices, relations and delimiters. For example, did you know the use of the Greek π for that number 3.14159… didn’t show up until 1706 when William Jones just offhandedly threw it into the mix? One guy, without preamble, forever altered baked good consumption in the month of march.

There’s a great post on the Wolfram blog all about the notebooks of Leibniz. It’s a long post, but it gives a great historical account of Leibniz and his relationship to notation and computation — specifically how Leibniz’ calculus ratiocinator is like a proto-wolfram Alpha — with great pictures of his notebooks. Nothing says living on the edge of human innovation like using alchemy symbols in your mathematical notation, while simultaneously laying out the schematic for a universal arithmetic machine!

In case you need to brush up on some of your fancy (non-alchemy) notation, and get that fraktur “g” just right, I am always happy to recommend Old Pappus’ Book of Mathematical Calligraphy.

And then there’s this, my favorite notation themed short story of all time.

Many thanks to everyone on Twitter who send me interesting notation links and anecdotes. Feel free to send along more @extremefriday.

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