Mathematical coincidences
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Mathematical coincidences
http://en.wikipedia.org/wiki/Mathematical_coincidences
I was reading this article and it inspired me to try to come up with my own. If you have any more or can explain these feel free to do so. The equals sign means approximately equal to. Here they are:
(pi)^3 = 2^5  1
e * pi = 2e + pi
e^3 = 4 * 5
I was reading this article and it inspired me to try to come up with my own. If you have any more or can explain these feel free to do so. The equals sign means approximately equal to. Here they are:
(pi)^3 = 2^5  1
e * pi = 2e + pi
e^3 = 4 * 5
Re: Mathematical coincidences
Well, theres these.
I'd be concerned about any inclination to try to "explain" approximately equal things, since those are simply coincidences based on the magnitudes of the numbers involved. Cases with strict equality are more likely to have actual explanations, but things being approximately equal doesn't mean much.
I'd be concerned about any inclination to try to "explain" approximately equal things, since those are simply coincidences based on the magnitudes of the numbers involved. Cases with strict equality are more likely to have actual explanations, but things being approximately equal doesn't mean much.
Last edited by Dopefish on Mon Jun 25, 2012 6:22 pm UTC, edited 1 time in total.
 gmalivuk
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Re: Mathematical coincidences
Yeah, asking for an explanation indicates a misunderstanding of the word "coincidence", I think.
 jestingrabbit
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Re: Mathematical coincidences
One kind of explanation could be pointing out an infinite family of approximations that a particular approximation belongs to. For instance 355/113 ~ pi because its one of the terminated continued fraction representations of pi.
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Re: Mathematical coincidences
Another "explanation" (though admittedly these sorts of explanations can sometimes be a little vague or subjective), which is also linked to in the footnotes to the Wikipedia article linked above, is in this brief note by Harvard number theorist Noam Elkies. It's about the fact that pi squared is slightly less than 10.
http://www.math.harvard.edu/~elkies/Misc/pi10.pdf
http://www.math.harvard.edu/~elkies/Misc/pi10.pdf
Re: Mathematical coincidences
The continued fraction with prime partial denominators
1/(2+1/(3+1/(5+1/(7+1/(11+...) ~= 9/(2π)  1.
1/(2+1/(3+1/(5+1/(7+1/(11+...) ~= 9/(2π)  1.
Re: Mathematical coincidences
Coincidently,
i=n
I ΣΩ(i!)  (n2)^2 I <= 3 (2 <= n <= 23),
i=2
where Ω is the PrimeOmega function.
i=n
I ΣΩ(i!)  (n2)^2 I <= 3 (2 <= n <= 23),
i=2
where Ω is the PrimeOmega function.

 Posts: 236
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Re: Mathematical coincidences
I found an interesting coincidence involving h2o, continued fractions, and the math constant e. First reduce h2o to (2, 1, 1, (half the numbers are in the chemical formula and the other half are atomic numbers). Then type "continued fraction (2, 1, 1, " into wolfram alpha to get 43/17 = 2.52941. One of the simplest formulas that has e as its limit is (1 + (1/n))^n. When you plug in 6 for n you get 2.52162 which is right to 3 decimal places.
 gmalivuk
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Re: Mathematical coincidences
Being right to 3 decimal places is completely irrelevant, and plugging in 6 for n has no relationship whatsoever to e, because e is the limit as n goes to infinity, which is rather a lot larger than six.scratch123 wrote:I found an interesting coincidence involving h2o, continued fractions, and the math constant e. First reduce h2o to (2, 1, 1, 8) (half the numbers are in the chemical formula and the other half are atomic numbers). Then type "continued fraction (2, 1, 1, 8)" into wolfram alpha to get 43/17 = 2.52941. One of the simplest formulas that has e as its limit is (1 + (1/n))^n. When you plug in 6 for n you get 2.52162 which is right to 3 decimal places.
You have failed to find any mathematical coincidences, apart from the entirely uninteresting fact that 43/17 is kinda close to 117649/46656.
 t1mm01994
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Re: Mathematical coincidences
I couldn't help but view scratch's post as a joke, but I don't know whether that's true..
 gmalivuk
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Re: Mathematical coincidences
Given the nonsense posted in the same user's "representing chemical formulas" thread, as well as in PMs to me, I'm unfortunately pretty sure it wasn't meant as a joke.
Re: Mathematical coincidences
Putting some meat on the bones of my last message, the PrimeOmega function is,
of course, the number of not necessarily distinct prime factors of the positive integers.
So starting from 2, the sequence begins 1,1,2,1,2,1,3,2,2,1.....
Summing these figures term by term produces 1,2,4,5,7,8,11,13,15,16.....,
which is the PrimeOmega function of the factorials.
Summing these figures term by term produces 1,3,7,12,19,27,38,51,66,82.....
For each of the first 23 terms in this last sequence, the absolute difference from the
sequence of squares 0,1,4,9,16,25,36,49,64,81..... respectively, is never more than 3,
which is what the inequality is stating.
of course, the number of not necessarily distinct prime factors of the positive integers.
So starting from 2, the sequence begins 1,1,2,1,2,1,3,2,2,1.....
Summing these figures term by term produces 1,2,4,5,7,8,11,13,15,16.....,
which is the PrimeOmega function of the factorials.
Summing these figures term by term produces 1,3,7,12,19,27,38,51,66,82.....
For each of the first 23 terms in this last sequence, the absolute difference from the
sequence of squares 0,1,4,9,16,25,36,49,64,81..... respectively, is never more than 3,
which is what the inequality is stating.

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Re: Mathematical coincidences
gmalivuk wrote:Being right to 3 decimal places is completely irrelevant, and plugging in 6 for n has no relationship whatsoever to e, because e is the limit as n goes to infinity, which is rather a lot larger than six.scratch123 wrote:I found an interesting coincidence involving h2o, continued fractions, and the math constant e. First reduce h2o to (2, 1, 1, (half the numbers are in the chemical formula and the other half are atomic numbers). Then type "continued fraction (2, 1, 1, " into wolfram alpha to get 43/17 = 2.52941. One of the simplest formulas that has e as its limit is (1 + (1/n))^n. When you plug in 6 for n you get 2.52162 which is right to 3 decimal places.
You have failed to find any mathematical coincidences, apart from the entirely uninteresting fact that 43/17 is kinda close to 117649/46656.
I think 3 decimal places is pretty good for such a simple formula. The number 6 may not be related to e but it is related to carbon and most carbon based life needs water to survive.
I have been experimenting with putting the numbers from proton/neutron and proton/electron ratios into polynomials and have been finding some interesting things. Many of the roots of the polynomial contain numbers that when put in base 2 form chains of palindromic sequences of 0's and 1's. For example lets take the polynomial (1/1.001378)x^2 + x + (1/1836.1). Its roots are 0.50068942.8763i and 0.500689+42.8763i. When the real part is converted to base 2 it is equal to 0.1000000000101101001001111000000001110111100010011010010001011. As you can see it has many more palindromic sequences than would be expected by chance. The imaginary part is equal to 101010.1110000001010101001100100110000101111100000110111101101 which again contains many palindromic sequences. You can find similar things if you choose to do other things to the polynomial such as not dividing with one or changing the positions of the x's (except for the middle x).
 gmalivuk
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Re: Mathematical coincidences
No, you still haven't. And I told you not to make a thread about such nonsense again.scratch123 wrote:I have been experimenting with putting the numbers from proton/neutron and proton/electron ratios into polynomials and have been finding some interesting things.
How many would you expect by chance? How would you go about computing this? How many more such sequences would be needed for statistical significance?As you can see it has many more palindromic sequences than would be expected by chance.
Re: Mathematical coincidences
scratch123 wrote:gmalivuk wrote:Being right to 3 decimal places is completely irrelevant, and plugging in 6 for n has no relationship whatsoever to e, because e is the limit as n goes to infinity, which is rather a lot larger than six.scratch123 wrote:I found an interesting coincidence involving h2o, continued fractions, and the math constant e. First reduce h2o to (2, 1, 1, (half the numbers are in the chemical formula and the other half are atomic numbers). Then type "continued fraction (2, 1, 1, " into wolfram alpha to get 43/17 = 2.52941. One of the simplest formulas that has e as its limit is (1 + (1/n))^n. When you plug in 6 for n you get 2.52162 which is right to 3 decimal places.
You have failed to find any mathematical coincidences, apart from the entirely uninteresting fact that 43/17 is kinda close to 117649/46656.
I think 3 decimal places is pretty good for such a simple formula. The number 6 may not be related to e but it is related to carbon and most carbon based life needs water to survive.
I have been experimenting with putting the numbers from proton/neutron and proton/electron ratios into polynomials and have been finding some interesting things. Many of the roots of the polynomial contain numbers that when put in base 2 form chains of palindromic sequences of 0's and 1's. For example lets take the polynomial (1/1.001378)x^2 + x + (1/1836.1). Its roots are 0.50068942.8763i and 0.500689+42.8763i. When the real part is converted to base 2 it is equal to 0.1000000000101101001001111000000001110111100010011010010001011. As you can see it has many more palindromic sequences than would be expected by chance. The imaginary part is equal to 101010.1110000001010101001100100110000101111100000110111101101 which again contains many palindromic sequences. You can find similar things if you choose to do other things to the polynomial such as not dividing with one or changing the positions of the x's (except for the middle x).
Oh my god.
When you take the reciprocal of that relationship, you get approximately .397, which is just .001 less than the San Diego Padres record on the 4th of July!
WOAHHH
Re: Mathematical coincidences
scratch123 wrote:I have been experimenting with putting the numbers from proton/neutron and proton/electron ratios into polynomials and have been finding some interesting things. Many of the roots of the polynomial contain numbers that when put in base 2 form chains of palindromic sequences of 0's and 1's. For example lets take the polynomial (1/1.001378)x^2 + x + (1/1836.1). Its roots are 0.50068942.8763i and 0.500689+42.8763i. When the real part is converted to base 2 it is equal to 0.1000000000101101001001111000000001110111100010011010010001011. As you can see it has many more palindromic sequences than would be expected by chance. The imaginary part is equal to 101010.1110000001010101001100100110000101111100000110111101101 which again contains many palindromic sequences. You can find similar things if you choose to do other things to the polynomial such as not dividing with one or changing the positions of the x's (except for the middle x).
In base 2, every number is entirely a string of palindromes.
If there are more palindromes than you'd expect for an infinite nonrepeating binimal (think about whether this is a meaningful statement anyway) then all that could mean is that the palindromic sequences are shorter than you'd expect. This would actually be evidence of the lack of a pattern.
Of course, as the number of palindromic sequences for an infinite nonrepeating binimal is always going to be countably infinite, there can only ever be exactly the same number as you'd expect.
Last edited by eSOANEM on Fri Jul 20, 2012 10:44 am UTC, edited 2 times in total.
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 chridd
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Re: Mathematical coincidences
If you want palindromes within numbers, the beginning of the decimal expansion of pi has plenty of 3digit palindromes:
3.1415926535897932384626433832795...
It makes it easier to memorize.
(Somewhat related, e has a repeat right at the beginning (2.71828182845...), making it appear to be rational on many calculators.)
3.1415926535897932384626433832795...
It makes it easier to memorize.
(Somewhat related, e has a repeat right at the beginning (2.71828182845...), making it appear to be rational on many calculators.)
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Re: Mathematical coincidences
chridd wrote:If you want palindromes within numbers, the beginning of the decimal expansion of pi has plenty of 3digit palindromes:
3.1415926535897932384626433832795...
It makes it easier to memorize.
(Somewhat related, e has a repeat right at the beginning (2.71828182845...), making it appear to be rational on many calculators.)
The irrational number song is far easier.
http://www.youtube.com/watch?v=iqVyjOv8X0
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