What's your favourite irrational number?
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What's your favourite irrational number?
Mine has to be phi aka the Golden Ratio...
edited to remove my lazy grammar.
edited to remove my lazy grammar.
Narsil wrote:For the record, I am not:
b)obsessed with penii, I just have bad luck and they follow me everywh...
SpitValve wrote:And as for Optimus being influenced by Buddhism, I severly doubt it.
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The halting probability, Ω.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
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skeptical scientist wrote:The halting probability, Ω.
That's what I came here to say. I guess I'll beat someone else to the punch and say e^pipi (though I'm not sure it's irrational).
Torn Apart By Dingos wrote:skeptical scientist wrote:The halting probability, Ω.
That's what I came here to say. I guess I'll beat someone else to the punch and say e^pipi (though I'm not sure it's irrational).
I'm sure it must be. Anyway, I side with pi because not even its continued fraction expansion has a pattern to it. (Though I'm sure this is true for some other nice irrationals. Please tell me what they are!)

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 gmalivuk
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Code: Select all
1
3 + 
1
1 + 
1
4 + 
1
1 + 
1
5 + 
1
9 + 
1
2 + 
...
Which is about 3.828656
 crazyjimbo
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gmalivuk wrote:Code: Select all
1
3 + 
1
1 + 
1
4 + 
1
1 + 
1
5 + 
1
9 + 
1
2 + 
...
Which is about 3.828656
That's my favourite now too. Or maybe...
Code: Select all
1
3 + 
1
8 + 
1
2 + 
1
8 + 
1
6 + 
1
5 + 
1
6 + 
...
The recursion possibilities are endless
 3.14159265...
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Why thank youNyarlathotep wrote:I'm afraid I am very fond of pi, even though it's cliche, simply because it gives me an excuse to bring pies to math class.
(6(1/1 + 1/4 + 1/9 + 1/16.....))^(1/2)
I love pi, because of that.
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crazyjimbo wrote:gmalivuk wrote:Code: Select all
1
3 + 
1
1 + 
1
4 + 
1
1 + 
1
5 + 
1
9 + 
1
2 + 
...
Which is about 3.828656
That's my favourite now too. Or maybe...Code: Select all
1
3 + 
1
8 + 
1
2 + 
1
8 + 
1
6 + 
1
5 + 
1
6 + 
...
The recursion possibilities are endless
This is brilliant! What pattern of numbers do you get if you keep doing this? Does it converge? I demand to know!
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jestingrabbit wrote:PaulT wrote:What pattern of numbers do you get if you keep doing this? Does it converge? I demand to know!
Its probably the sort of thing you have to work out for yourself. I don't know how much study there is into this sort of thing, but I wouldn't expect much.
My guess is it doesn't converge. It's clear that the next term of the series is between 28/9 and 25/8, both of which are less than pi. The series looks like it bounces back and forth between two integers, jumping up an integer at certain points, not dropping back. It looks to me like the only possible convergence point is at any positive integer x plus a third  and even then, I don't think it could converge there unless it lands directly on it right away after a jump. (That's just what I get at first glance  very far from a rigorous proof.)
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It seems to end up oscillating between
3.1389926549519825657 and
3.7574762361169873641,
though I'm not entirely certain yet whether or not this is an artifact of only using a finite number of digits each time...
Interestingly, e seems to only sort of oscillate, in that it sticks around two values, but seems to also have a longer cycle of length 8, where the even iterations wiggle around one of those two values and the odd ones wiggle around the other.
Edit: the fractional part of the numbers Pi converges to in this sequence are interesting, because 10/3 converges to numbers with the same fractional part. (Note that, the way I've coded it, when you get, say, 11.34562..., the continued fraction from that starts with 1+1/(1+1/(3+...)), rather than with integer part 11.)
3.1389926549519825657 and
3.7574762361169873641,
though I'm not entirely certain yet whether or not this is an artifact of only using a finite number of digits each time...
Interestingly, e seems to only sort of oscillate, in that it sticks around two values, but seems to also have a longer cycle of length 8, where the even iterations wiggle around one of those two values and the odd ones wiggle around the other.
Edit: the fractional part of the numbers Pi converges to in this sequence are interesting, because 10/3 converges to numbers with the same fractional part. (Note that, the way I've coded it, when you get, say, 11.34562..., the continued fraction from that starts with 1+1/(1+1/(3+...)), rather than with integer part 11.)
Last edited by gmalivuk on Fri Jun 15, 2007 4:25 pm UTC, edited 1 time in total.
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I might have to go for e because it is nice and easy to code a program that quickly gives you a very long decimal approximation to it.
By the way, does anyone want 10,000 digits of e?
Then again, phi is also nice to code since you can do the Fibonacci sequence and divide your last two terms.
Anyone want 1000 digits of phi?
More digits possibly available on request or boot up your text editor and compiler.
By the way, does anyone want 10,000 digits of e?
Then again, phi is also nice to code since you can do the Fibonacci sequence and divide your last two terms.
Anyone want 1000 digits of phi?
More digits possibly available on request or boot up your text editor and compiler.
li te'o te'a vei pai pi'i ka'o ve'o su'i pa du li no
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QED is Latin for small empty box.
Ceci nâ€™est pas une [s]pipe[/s] signature.
Mathematician is a function mapping tea onto theorems. Sadly this function is irreversible.
QED is Latin for small empty box.
Ceci nâ€™est pas une [s]pipe[/s] signature.
cmacis wrote:Anyone want 1000 digits of phi?
Yes please!! I have nothing better to do this weekend....Would this be a good gift for my dad for father's day? (No, he's not a math geek, i'm just too lazy to go get something )
Narsil wrote:For the record, I am not:
b)obsessed with penii, I just have bad luck and they follow me everywh...
SpitValve wrote:And as for Optimus being influenced by Buddhism, I severly doubt it.
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cmacis wrote:I might have to go for e because it is nice and easy to code a program that quickly gives you a very long decimal approximation to it.
By the way, does anyone want 10,000 digits of e?
Then again, phi is also nice to code since you can do the Fibonacci sequence and divide your last two terms.
Anyone want 1000 digits of phi?
More digits possibly available on request or boot up your text editor and compiler.
Psh, e and phi are actually pretty hard to compute digits for, compared to some other irrational numbers I can come up with.
Sum 10^(n!) comes to mind, for instance. You tell me the number of the digit you want, I tell you as soon as I calculate whether that number is a factorial or not. If I have a convenient way of counting zeroes, I could recite the first 10 million digits of that number.
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cmacis wrote:I might have to go for e because it is nice and easy to code a program that quickly gives you a very long decimal approximation to it.
By the way, does anyone want 10,000 digits of e?
Then again, phi is also nice to code since you can do the Fibonacci sequence and divide your last two terms.
Anyone want 1000 digits of phi?
More digits possibly available on request or boot up your text editor and compiler.
Why on earth do you want a number that it's easy to compute digits of? I like my answer much better, because it's impossible to figure out more than the first few digits of it. Everyone else's answer is computable, and therefore trivial.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
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cmacis wrote:I might have to go for e because it is nice and easy to code a program that quickly gives you a very long decimal approximation to it.
By the way, does anyone want 10,000 digits of e?
Then again, phi is also nice to code since you can do the Fibonacci sequence and divide your last two terms.
Anyone want 1000 digits of phi?
More digits possibly available on request or boot up your text editor and compiler.
Well, Here's e, γ, φ, π, 10000!, 10000#, √1 (yes, this is a joke), √2, and √3
http://www.nerdparadise.com/academia/ma ... /digits/e/
http://www.nerdparadise.com/academia/ma ... its/gamma/
http://www.nerdparadise.com/academia/ma ... igits/phi/
http://www.nerdparadise.com/academia/ma ... digits/pi/
http://www.nerdparadise.com/academia/ma ... its/sqrt1/ (haha)
http://www.nerdparadise.com/academia/ma ... its/sqrt2/
http://www.nerdparadise.com/academia/ma ... its/sqrt3/
http://www.nerdparadise.com/academia/ma ... factorial/
http://www.nerdparadise.com/academia/ma ... primorial/
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OmnipotentEntity wrote:http://www.nerdparadise.com/academia/math/reference/digits/e/
http://www.nerdparadise.com/academia/ma ... its/gamma/
http://www.nerdparadise.com/academia/ma ... igits/phi/
http://www.nerdparadise.com/academia/ma ... digits/pi/
http://www.nerdparadise.com/academia/ma ... its/sqrt2/
http://www.nerdparadise.com/academia/ma ... its/sqrt3/
So I went through the links above and did Search and Replace to count the frequency of each integer within the 10,000 places after the decimal. Here's what I found:
That is all. I'm sure if I sat and stared at this long enough, I could come up with some sort of quasiintelligent analysis, but my brain is getting chewed up by this forecast I working on.
**Repeated edits as I tried to put the columns into the message. I finally gave up and put the crappy image here.
Last edited by chrispy1 on Sat Jun 16, 2007 11:59 am UTC, edited 1 time in total.
Narsil wrote:For the record, I am not:
b)obsessed with penii, I just have bad luck and they follow me everywh...
SpitValve wrote:And as for Optimus being influenced by Buddhism, I severly doubt it.
 3.14159265...
 Irrational (?)
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chrispy1 wrote:OmnipotentEntity wrote:http://www.nerdparadise.com/academia/math/reference/digits/e/
http://www.nerdparadise.com/academia/ma ... its/gamma/
http://www.nerdparadise.com/academia/ma ... igits/phi/
http://www.nerdparadise.com/academia/ma ... digits/pi/
http://www.nerdparadise.com/academia/ma ... its/sqrt2/
http://www.nerdparadise.com/academia/ma ... its/sqrt3/
So I went through the links above and did Search and Replace to count the frequency of each integer within the 10,000 places after the decimal. Here's what I found:
That is all. I'm sure if I sat and stared at this long enough, I could come up with some sort of quasiintelligent analysis, but my brain is getting chewed up by this forecast I working on.
**Repeated edits as I tried to put the columns into the message. I finally gave up and put the crappy image here.
The numbers are normal is what I see from that.
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skeptical scientist wrote:The halting probability, Ω.
Which halting probability do you prefer? There are many, you know.
Personally, I am quite fond of φ, though I can't give any rational reason why.
This is not true.
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dumbclown wrote:My favourite number would have to be planck's constant. Sadly as it is a physical constant it can't really be proven to be irrational.
So I will have to go with Pi.
Well being a physical constant means you can give it any value you want, after suitably choosing the units.
Though it's worth pointing out that in any system of units, at least one of h or hbar must be irrational, so you've got that going for you. Or at least, any system where one of those constants can be given an exact value, like, say, 1.
Well being a physical constant means you can give it any value you want, after suitably choosing the units.
In any system where you don't define units in terms of the constants but instead choose the units "arbitrarily," won't constants be "almost guaranteed" to be irrational, because of the uncountability of the reals vs. the countability of the rationals?
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Herman wrote:Well being a physical constant means you can give it any value you want, after suitably choosing the units.
In any system where you don't define units in terms of the constants but instead choose the units "arbitrarily," won't constants be "almost guaranteed" to be irrational, because of the uncountability of the reals vs. the countability of the rationals?
Well... kind of. The problem is that when you define the units beforehand, the constants can only have so much precision to them, and I find it hard to describe a number you only know, say, 10 digits of as rational or irrational...
Regarding the aforementioned normality of those numbers, there is actually what appears to be an exception when you look at the first 10000 digits of phi in base12.
Code: Select all
digit frequency
0 839
1 839
2 820
3 847
4 842
5 856
6 835
7 794
8 851
9 830
10 814
11 833
The likelihood of this particular arrangement happening by chance (assuming a uniform distribution) actually comes out to 0.028, largely becuase there are almost 40 fewer 7s than one would expect. Of course, that really is just a fluke, because when you go out to 100k digits the onesided pvalue is 0.37, which means we definitely don't throw out the hypothesis that the digits are distributed normally.
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