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Nothing New About the Square Root of Two

In November and December 2023, I was partially forced out of programming by school and university applications. There are, unfortunately, too not enough hours in a day. Now that I have time to do fun things again, I spend this month programming slightly less, and reading slightly more.

Imagining Numbers by Barry Mazur is a truly poetic (math) book that takes the reader on a journey to discover an intuition for thinking about imaginary numbers. The figures in the book are stunning and elegant.

The most beautiful in the book figure displays the initial digits of the square root of two. I tried to recreate it in Inkscape, but my rudimentary knowledge of Inkscape failed me, and so I gave up. Here is the figure from the book:

The square root of two rounded to 99 decimal places

I cannot pinpoint why I like this figure so much, but I know I do.

My interest was also peaked by the fact that some early Italian mathematicians called the roots of numbers “sides” (“lato” in Italian). It’s pretty trivial, but I love the geometric interpretation of roots that this term implies. Calling a root a side makes the following core relationship obvious (substitute any number of $2$):

Thinking of a square root as the length of side of a square with the some area

In contrast to “lato” (or alternatively the Latin “latus”), I find that root or radical do a very poor job at painting an image on my mental canvas. Maybe that’s my fault, but it seems that radical and root both refer to some “basis”, whereas side clearly refers to a geometric property. In addition, the term lato can be extended into higher dimensions. For example, a cube has the volume $V = a^3$, so the side of a cube of volume $2$ has the length $\sqrt[3]{2}$. I cannot imagine a tesseract, but I can certainly imagine that it has sides!

As an aside, do you think that the title of this post rhymes?