Why is Pi² so close to 10?

http://fermatslibrary.com/s/why-is-pi-squared-so-close-to-10#email-newsletter

by slbenfica • 8 years ago

41 66 8 years ago

fermatslibrary.com

mathattack• 8 years ago

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.

NegativeLatency• 8 years ago

Eulers identity on the other hand...

andrepd• 8 years ago

Is almost a definition.

wyager3• 8 years ago

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.

kerkeslager• 8 years ago

I think what makes Euler's identity profound to me is that it is a tool for doing so many things. For example, if you want to calculate the log of a negative number:

    -1 = e^(i * pi)
    ln(-1) = i * pi
    ln(-x) = i * pi + ln(x)
    log_b(-x) = (i * pi + ln(x)) / ln(b)

Now you have a formula for calculating the log of a negative number (-x) in any base.

jerf• 8 years ago

"seems remarkably non-profound."

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.

montalbano• 8 years ago

This one's a goodun though, the significance of the closeness of 196883 and 196884:

https://en.wikipedia.org/wiki/Monstrous_moonshine

https://www.reddit.com/r/explainlikeimfive/comments/2im1l1/e...

mxwsn• 8 years ago

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

ShorsHammer2• 8 years ago

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

Everything else is quite arbitrary.

thanatropism3• 8 years ago

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

chasing• 8 years ago

Why is 3^2 so close to 9?

singularity20012• 8 years ago

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

SamReidHughes• 8 years ago

Because God carries a slide rule.

scythe2• 8 years ago

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.

alphydan• 8 years ago

But it is accessible to the primitive geometer... https://www.youtube.com/watch?v=d-o3eB9sfls

Or for more details, http://www.math.chalmers.se/~wastlund/Cosmic.pdf

hazeii• 8 years ago

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

meggar• 8 years ago

Isn't that the gravitational acceleration of the earth?

scythe2• 8 years ago

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

jandrese2• 8 years ago

Or as my dad used to say: Close only counts in horseshoes and hand grenades.

dwighttk• 8 years ago

atom bombs

formula1• 8 years ago

I like your mindset:)

soVeryTired3• 8 years ago

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...

pps43• 8 years ago

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

singularity20012• 8 years ago

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

lostmsu• 8 years ago

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

pleasecalllater• 8 years ago

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_cavalier2• 8 years ago

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

bunderbunder2• 8 years ago

You must not be American. They're not that close in the USA.

fenwick67• 8 years ago

I'm an American but I have never used g in imperial units, I've only ever known it as -9.8m/s^2.

evanb• 8 years ago

...if you measure in a particular set of units. It's not so close to g = 21 mph/s, for example.

davrosthedalek• 8 years ago

What are you talking about, g is roughly 7e10 furlong/fortnight^2

thanatropism3• 8 years ago

g is 9.8whatever in m/s^2. If you choose inches/week^2 instead you'll get a different constant.

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.

nkrisc• 8 years ago

If you make up your own units Pi^2 can be close to anything you want.

madcaptenor2• 8 years ago

Actually not a coincidence. The meter was originally defined as the length of a "seconds pendulum" (a pendulum with period two seconds). The half-period of a pendulum of length L is pi * sqrt(L/g). Set L = 1 and solve for g, and you get g = pi^2.

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.

dmitrybrant• 8 years ago

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_...

jandrese2• 8 years ago

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

soVeryTired3• 8 years ago

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

ChuckMcM• 8 years ago

No, 163 is a number substituted into Euler's equation for the value of -1. When you use the square root of -1 in Euler's equation you get back -1. As far as I know there is no other square root of an integer that gives you an integer result. 163 comes close.

soVeryTired3• 8 years ago

You're using 'complex multiplication' in a pretty specialised sense there. This isn't your grandma's (a + ib)*(c + id)

TekMol• 8 years ago

   extremely close to an integer
   it equals 262537412640768743.99999999999925...

Is that really surprising?

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.

jbfoo• 8 years ago

I heard that this is a standard floating point test, if you get non integer value your floating point implementation is subtly broken ;)

bdamm• 8 years ago

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.

PokemonNoGo2• 8 years ago

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

jwilk• 8 years ago

It's pdf2htmlEX:

https://coolwanglu.github.io/pdf2htmlEX/

PokemonNoGo2• 8 years ago

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

n4r9• 8 years ago

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

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.

mnl• 8 years ago

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

dmurray• 8 years ago

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

gowld2• 8 years ago

ObXkcd:

https://xkcd.com/1047/ and https://mrob.com/pub/ries/index.html

meuk• 8 years ago

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).

jovial_cavalier2• 8 years ago

It isn't.

madcaptenor2• 8 years ago

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

brador• 8 years ago

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

detaro• 8 years ago

https://en.wikipedia.org/wiki/Riemann_zeta_function

Someone• 8 years ago

”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.

IshKebab• 8 years ago

Erm, this isn't a maths paper.

Why is Pi² so close to 10?

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