Why is Pi² so close to 10?(fermatslibrary.com)
fermatslibrary.com
Why is Pi² so close to 10?
http://fermatslibrary.com/s/why-is-pi-squared-so-close-to-10#email-newsletter
21 comments
Why is e^2 so close to 7? Just to highlight how arbitrary pi and 10 are...
The real question should be why integers aren't factors of pi or e
Everything else is quite arbitrary.
Everything else is quite arbitrary.
Because these are irrational numbers, they can't be expressed as a product of fractions. Integers are a special type of fractions.
Ahh I meant the base system we count in. Worded it badly.
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.
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.
"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.
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.
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...
https://en.wikipedia.org/wiki/Monstrous_moonshine
https://www.reddit.com/r/explainlikeimfive/comments/2im1l1/e...
Eulers identity on the other hand...
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.
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.
> A much cooler problem of this nature is: why are musical notes generated by powers of the 12th root of 2?
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.
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.
You’re right; it’s not difficult, but it’s (IMO) a lot cooler than the OP.
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?
So the musical notes thing is by human fiat, which at least pi^2 isn't. Perhaps I'm missing your meaning though?
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.
Some of the character of older music is actually lost because of this. But hey, now all keys work the same.
>which at least pi^2 isn't
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).
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).
The reason that powers of the 12th root of 2 are used is that these numbers are close to simple rational numbers, which the human auditory system can pick up.
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"explained in any number of ways", which involve complex numbers, geometry, trignometry, and calculus, is what makes it profound.
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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)
Is almost a definition.
Once the identity was pointed out to me and I didn't immediately grok, I realized I would never be a first-class mathematician :p
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Why is 3^2 so close to 9?
in node.js with its bad math it's even exactly the same!
Because God carries a slide rule.
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.
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.
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
Or for more details, http://www.math.chalmers.se/~wastlund/Cosmic.pdf
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I like your mindset:)
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...
Here's a more interesting result with the same flavour: https://math.stackexchange.com/questions/4544/why-is-e-pi-sq...
Why is Pi*10^7 seconds so close to one year?
because a year has ~ ½×1×2×3×4×5×τ days - 11 and π:=½τ
Why Pi^2 is so close to g, gravitational acceleration on the Earths surface?
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.
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.
...if you measure in a particular set of units. It's not so close to g = 21 mph/s, for example.
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.
In reality the g constant is not a constant, the value depends on the place of the Earth.
That's his point. It's actually closer to 9.81 than it is to 10.
What are you talking about, g is roughly 7e10 furlong/fortnight^2
You must not be American. They're not that close in the USA.
I'm an American but I have never used g in imperial units, I've only ever known it as -9.8m/s^2.
I love your high school so much better than my high school.
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.
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.
If you make up your own units Pi^2 can be close to anything you want.
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_...
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_...
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.
Is the square root of 163 supposed to be a number I recognize? It looks awfully magical in that equation.
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
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
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.
I heard that this is a standard floating point test, if you get non integer value your floating point implementation is subtly broken ;)
You're using 'complex multiplication' in a pretty specialised sense there. This isn't your grandma's (a + ib)*(c + id)
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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.
Thats a really neat inline pdf viewer! Rendered on the backend with pdf.js or something?
It's pdf2htmlEX:
https://coolwanglu.github.io/pdf2htmlEX/
https://coolwanglu.github.io/pdf2htmlEX/
Sad to see that project isn't updated any more.
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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.
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.
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).
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).
It isn't.
Why is pi^3 so close to 31? Why is pi^2 + pi so close to 13?
What does this symbol ζ mean here? Like ζ(2)...what is that?
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