Why is Pi² so close to 10?
fermatslibrary.com
fermatslibrary.com
Squaring a number that’s a little higher than 3 will get you close to 10.
If it were 9.999999 then it’s another story.
The OP is just not very interesting. Pi^2 isn’t close enough to 10 to trigger my not-a-coincidence detector.
A much cooler problem of this nature is: why are musical notes generated by powers of the 12th root of 2? I remember seeing a good YouTube video about this on one of those math channels.
So the musical notes thing is by human fiat, which at least pi^2 isn't. Perhaps I'm missing your meaning though?
Some of the character of older music is actually lost because of this. But hey, now all keys work the same.
pi^2 is 9.87. If that counts as close to 10, then half of all numbers are close to some integer. And the significance of 10 is entirely cultural. Nothing in math favours base 10, we use it because the right civilisations were dominant at the right time. Other civilisations counted using base 4, base 5, base 6 or base 12 (e.g. using the five fingers and a closed fist to count up to 6 for each hand).
That one isn't especially difficult. Humans like the combination of fundamental tone and first harmonic (I'm not aware that there's a culture with a musical system that doesn't respect the octave). In the western culture, we found the similarity between those tones so notable that we classify them as harmonically indistinguishable; thus, A3 (220 Hz) is harmonically tied to A4 (440 Hz). Many cultures, our western predecessors included, also appreciated the 2nd harmonic (f3, or, since octaves are now identical, f1.5) as consonant with the fundamental.
Following on to that, they developed a pattern of notes at f(1.5^2), f(1.5^3) etc. Here it can be noted that the first 12 exponents of 1.5 are (renormalized to the [1:2] space as necessary): 1 , 1.5, 1.125, 1.6875, 1.265625, 1.8984375, 1.423828125, 1.06787109375, 1.601806640625, 1.20135498046875, 1.802032470703125, 1.3515243530273438 and 1.0136432647705078.
Note how infuriatingly close that last is to 1; on a violin string (~.328m) that would be a 4mm difference. Human societies have wrestled with that discrepancy as far back as the Pythagoreans. Musicians and composers in western society didn't come to a compromise for the issue until ~late 16th century, when they DEFINED a semitone as the 12th root of two. This adds a little bit of error to every interval besides the octave in favor of consistency across every key.
-1 = e^(i * pi)
ln(-1) = i * pi
ln(-x) = i * pi + ln(x)
log_b(-x) = (i * pi + ln(x)) / ln(b)
I'd point out the linked article doesn't particularly claim otherwise. It just provides a slightly deeper explanation but doesn't claim it's an explanation of anything particularly meaningful.
In other news, if you'd like to see the Golden Ratio strut its stuff in a context not related to the supposed aesthetics of the ratio, this relatively recent (and relatively in-depth, for the channel) video and linked proof may be interesting to you: https://www.youtube.com/watch?v=FtNWzlfEQgY You'll seriously be asking "what was jerf talking about?" for like a third of the proof video before foom in comes phi, and it takes quite a bit of abuse before the end.
https://en.wikipedia.org/wiki/Monstrous_moonshine
https://www.reddit.com/r/explainlikeimfive/comments/2im1l1/e...
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.
Or for more details, http://www.math.chalmers.se/~wastlund/Cosmic.pdf
The most remarkable thing is that this is actually not a coincidence, but a consequence of complex multiplication: https://en.wikipedia.org/wiki/Complex_multiplication#Sample_...
Every integer has a unique prime factorisation: that's something kids learn in primary school. The idea of factorisation goes a lot deeper though. Define the gaussian integers as integers of the form a + ib, where a and b are intgers. Unique factorisation holds in the Gaussian integers too, but some of the numbers we recognise as prime are no longer prime. For example, 2 = (1 + i)(1 – i).
It turns out unique factorisation is a rare property of number systems. Consider the set of numbers whose ‘integers’ are of the form a + b sqrt(-5), with a and b standard integers. In this system, 6 = 2x3 = (1 + sqrt(-5)) x (1 - sqrt(-5)). But none of those factors can be factorised further, so 6 has two distinct factorisations.
It turns out there are only nine positive numbers n so that unique factorisation holds in the system a + b*sqrt(-n). 163 is the largest!
extremely close to an integer
it equals 262537412640768743.99999999999925...
The chance that a random number is that close to an integer is about 1:10^12.
So by trying 10^12 crazy formulas you will have one hit. 10^12 possibilites can be covered in about 40 bits.
If we look at "e^(pi*sqrt(163))" as 16 ascii characters, that's 112 bits.
Here's a more interesting result with the same flavour: https://math.stackexchange.com/questions/4544/why-is-e-pi-sq...
http://mathworld.wolfram.com/AlmostInteger.html
https://en.wikipedia.org/wiki/Mathematical_coincidence
Some favourites:
- 5 phi e / 7 pi = 1 (to 5dp)
- pi^4 + pi^5 = e^6 (to 4dp)
- e^pi - pi = 20 (to 3dp)
Also, remember that many very profound discoveries come about from noticing fun little oddities.
In reality the g constant is not a constant, the value depends on the place of the Earth.
Many fields use geometric mass units 1 gm = G (where G is the grav constant in m/s^2) so you have g = 1 gm/s^2. Then you can drop a lot of constants for calculations.
This doesn't work exactly because the meter eventually wasn't defined as the length of the seconds pendulum but rather as one ten-millionth of the distance from the equator to the North Pole - which turns out to be a few millimeters longer - but the idea for using something meter-sized as a unit of length came from the seconds pendulum.
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.
For example ((1 + sqrt(5)) / 2) ^ 20 = 15126.9999
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.
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).
Everything else is quite arbitrary.
That’s sloppy for a math paper. “Much smaller yet” doesn’t even guarantee that the series converges.