I love math (and have recently gotten into the beauty of the Golden Ratio) but this seems remarkably non-profound.<p>Squaring a number that’s a little higher than 3 will get you close to 10.<p>If it were 9.999999 then it’s another story.
It's... not? It's 9.87. If you throw a dart at the number line, the average distance it will be from the nearest integer is 0.25. The distance from pi^2 to 10 is 0.13. That's below average, but in the second quintile. There's nothing special about 10, other than it happens to be the number of fingers on two human hands.<p>However, deriving approximations for pi from the Basel series is sort of interesting. Except summing the Basel series requires at a bare minimum the theory of Taylor series, so it is not an accessible theorem for a primitive geometer.
A much, much more interesting observation (attributed to Ramanujan) is that e^(pi * sqrt(163)) is extremely close to an integer. In fact it equals 262537412640768743.99999999999925...<p>The most remarkable thing is that this is actually <i>not</i> a coincidence, but a consequence of complex multiplication:
<a href="https://en.wikipedia.org/wiki/Complex_multiplication#Sample_consequence" rel="nofollow">https://en.wikipedia.org/wiki/Complex_multiplication#Sample_...</a>
I think this would get much better responses if the title was something like "You can use the zeta function to get an upper bound for Pi".<p>Here's a more interesting result with the same flavour:
<a href="https://math.stackexchange.com/questions/4544/why-is-e-pi-sqrt163-almost-an-integer" rel="nofollow">https://math.stackexchange.com/questions/4544/why-is-e-pi-sq...</a>
Profound or not, these kinds of (near) identities are a lot of fun to explore. Here are some great lists:<p><a href="http://mathworld.wolfram.com/AlmostInteger.html" rel="nofollow">http://mathworld.wolfram.com/AlmostInteger.html</a><p><a href="https://en.wikipedia.org/wiki/Mathematical_coincidence" rel="nofollow">https://en.wikipedia.org/wiki/Mathematical_coincidence</a><p>Some favourites:<p>- 5 phi e / 7 pi = 1 (to 5dp)<p>- pi^4 + pi^5 = e^6 (to 4dp)<p>- e^pi - pi = 20 (to 3dp)<p>Also, remember that many very profound discoveries come about from noticing fun little oddities.
Related, and slightly more interesting than the linked article (I think), is you can get similar 'almost equal to an integer'-results by using recursive relations.<p>Take, for example, a Fibonacci-like sequence, defined by f(k + 2) = f(k) + f(k + 1) and f(0) = 2, f(1) = 1. Then you get the solution f(n) = a^n + b^n, where a = (1 - sqrt(5))/2 and b = (1 + sqrt(5))/2. So |a| < 1 and |b| > 1. If the recursive relation has integer coefficients, then a^n + b^n will be an integer. For large n, |a^n| will get very small, so b^n will be very close to an integer.<p>For example ((1 + sqrt(5)) / 2) ^ 20 = 15126.9999<p>It should be possible to find a recursive relation where |a| is smaller, so that b^n is closer to an integer, but hey, I'm supposed to be working now.<p>Edit: Coincidentally, this result is also presented in the math stackexchange post that soVeryTired linked to (actually, this is not a coincidence, since this is the most basic recursive relation, and has easily memorable forms for a and b).
While the mathematics here is completely inane, I really enjoyed the presentation whereby the side concepts that the author is building on top of become pop-up margin notes. Very nice.
<i>”and the error is reasonably small because […], a sum whose first term is 1/60 and whose further terms are much smaller yet”</i><p>That’s sloppy for a math paper. “Much smaller yet” doesn’t even guarantee that the series converges.