
Why is Pi² so close to 10? - slbenfica
http://fermatslibrary.com/s/why-is-pi-squared-so-close-to-10#email-newsletter
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mathattack
I love math (and have recently gotten into the beauty of the Golden Ratio) but
this seems remarkably non-profound.

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.

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NegativeLatency
Eulers identity on the other hand...

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wyager
That’s not really “weird” though - it’s can be straightforwardly explained in
any number of ways. You can think about it in terms of the circle group and
its Lie algebra, in terms of Taylor expansions, etc.

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.

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jagthebeetle
Isn't this just by definition of equal temperament? If you want a pitch to
double after 12 steps (multiplicatively), you choose 12-TET. Other musical
systems exist (e.g. quarter-tone scales), and have existed.

So the musical notes thing is by human fiat, which at least pi^2 isn't.
Perhaps I'm missing your meaning though?

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thanatropism
Before the current "equal temperament" system there was a "just temperament
system" that basically harks back to Pythagoras. The main intervals in the C
scale are very close to whole fractions of low numerator and denominator. The
problem is that this doesn't work very well in every key. So the emergence of
modern music came with a few tries at averaging these things out until equal
temperament in the log scale arose.

Some of the character of older music is actually lost because of this. But
hey, now all keys work the same.

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scythe
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.

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.

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hazeii
It's not 9.81, it's more like 9.87 (you got the distance correct at 0.13 so I
guess a typo?).

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scythe
Edited, thanks, and yes, 9.81 m/s^2 is the gravitational field at Earth's
surface.

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dmitrybrant
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...

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_...](https://en.wikipedia.org/wiki/Complex_multiplication#Sample_consequence)

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jandrese
Is the square root of 163 supposed to be a number I recognize? It looks
awfully magical in that equation.

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soVeryTired
Nope. It's one of a few very special numbers though.

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!

[https://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem](https://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem)

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soVeryTired
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".

Here's a more interesting result with the same flavour:
[https://math.stackexchange.com/questions/4544/why-is-e-pi-
sq...](https://math.stackexchange.com/questions/4544/why-is-e-pi-
sqrt163-almost-an-integer)

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n4r9
Profound or not, these kinds of (near) identities are a lot of fun to explore.
Here are some great lists:

[http://mathworld.wolfram.com/AlmostInteger.html](http://mathworld.wolfram.com/AlmostInteger.html)

[https://en.wikipedia.org/wiki/Mathematical_coincidence](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.

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lostmsu
Why Pi^2 is so close to g, gravitational acceleration on the Earths surface?

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pleasecalllater
Just a coincidence. The g depends on the mass of the planet, while pi is
rather universal for a perfect math circle.

In reality the g constant is not a constant, the value depends on the place of
the Earth.

~~~
jovial_cavalier
That's his point. It's actually closer to 9.81 than it is to 10.

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meuk
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.

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).

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PokemonNoGo
Thats a really neat inline pdf viewer! Rendered on the backend with pdf.js or
something?

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jwilk
It's pdf2htmlEX:

[https://coolwanglu.github.io/pdf2htmlEX/](https://coolwanglu.github.io/pdf2htmlEX/)

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PokemonNoGo
Sad to see that project isn't updated any more.

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mxwsn
Why is e^2 so close to 7? Just to highlight how arbitrary pi and 10 are...

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ShorsHammer
The real question should be why integers aren't factors of pi or e

Everything else is quite arbitrary.

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thanatropism
Because these are irrational numbers, they can't be expressed as a product of
fractions. Integers are a special type of fractions.

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ShorsHammer
Ahh I meant the base system we count in. Worded it badly.

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bdamm
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.

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Someone
_”and the error is reasonably small because […], a sum whose ﬁrst term is 1
/60 and whose further terms are much smaller yet”_

That’s sloppy for a math paper. “Much smaller yet” doesn’t even guarantee that
the series converges.

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IshKebab
Erm, this isn't a maths paper.

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chasing
Why is 3^2 so close to 9?

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singularity2001
in node.js with its bad math it's even _exactly_ the same!

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brador
What does this symbol ζ mean here? Like ζ(2)...what is that?

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detaro
[https://en.wikipedia.org/wiki/Riemann_zeta_function](https://en.wikipedia.org/wiki/Riemann_zeta_function)

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madcaptenor
Why is pi^3 so close to 31? Why is pi^2 + pi so close to 13?

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mnl
More interestingly, why is 355/113 so close to Pi?

~~~
dmurray
There's a nice explanation for this one based on expressing Pi as an infinite
continued fraction.

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SamReidHughes
Because God carries a slide rule.

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jovial_cavalier
It isn't.

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pps43
Why is Pi*10^7 seconds so close to one year?

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singularity2001
because a year has ~ ½×1×2×3×4×5×τ days - 11 and π:=½τ

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gowld
ObXkcd:

[https://xkcd.com/1047/](https://xkcd.com/1047/) and
[https://mrob.com/pub/ries/index.html](https://mrob.com/pub/ries/index.html)

