Many of you are familiar with the great equation Euler's identity eiπ+1=0. It’s an incredible equality that unifies multiple transcendental numbers along with some fundamental numerical units. And these are the kinds of elegant relationships that provide us with a sense that the universe is explainable and amenable to order and reason, using the language of mathematics.
But all is not so simple. And we can see this in almost integers. Almost integers are mathematical equations that are almost equal to a whole number, but not quite. They are the numerical equivalent of a near miss.
Take the number twenty, for example. A nice and round small integer. Well, unlike the equation above, we can’t get to it exactly. But we can get close:
eπ-π = 19.999099979…
It’s far less neat than the above equality but seems just as interesting, in its own way. And there are many other examples, such as π9/e8 = 9.9998387… and sin(11) = −0.999990206…
Want to play at home? There are software packages that help find these almost integers, or more generally, equations for specific numbers—from π to physical constants—that are good, but inexact, approximations. One well-known program is called ries (you might know it from this xkcd comic) and was developed by Robert Munafo. Using this, I was able to find my own almost integer for the number 3, using a combination of the golden ratio, π, and e (equation visualized by Wolfram|Alpha):
= 2.999988924…
Mathematics is generally considered to be elegant, but even when it's not, it's certainly interesting.
Top image:Clay Shonkwiler/Flickr/CC
