What's your favourite irrational number?

For the discussion of math. Duh.

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chrispy1
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What's your favourite irrational number?

Postby chrispy1 » Thu Jun 14, 2007 4:32 pm UTC

Mine has to be phi aka the Golden Ratio...

edited to remove my lazy grammar.
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Postby skeptical scientist » Thu Jun 14, 2007 5:16 pm UTC

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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Postby Blatm » Thu Jun 14, 2007 5:36 pm UTC

Double post. Sorry.
Last edited by Blatm on Thu Jun 14, 2007 5:37 pm UTC, edited 1 time in total.

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Postby Blatm » Thu Jun 14, 2007 5:36 pm UTC

Square root of 13, for some odd reason. Boring old e is a close second.

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Postby OmnipotentEntity » Thu Jun 14, 2007 6:42 pm UTC

Area of the Mandelbrot set, assuming it's irrational.
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Postby miles01110 » Thu Jun 14, 2007 6:45 pm UTC

oops, never mind.

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Postby Nyarlathotep » Thu Jun 14, 2007 7:04 pm UTC

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.
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Postby Torn Apart By Dingos » Thu Jun 14, 2007 10:48 pm UTC

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^pi-pi (though I'm not sure it's irrational).

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Postby PaulT » Thu Jun 14, 2007 11:08 pm UTC

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^pi-pi (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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Postby themandotcom » Thu Jun 14, 2007 11:16 pm UTC

The Euler-Masheroni constant (Yes, it is irrational!) :P
e^pi*i=WHAT??

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Postby arbivark » Thu Jun 14, 2007 11:51 pm UTC

pi, because it's my kinsey number.
i don't have a bacon-erdos number.

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Postby gmalivuk » Fri Jun 15, 2007 1:15 am UTC

Code: Select all

                 1
3 + -----------------------------
                   1
    1 + -------------------------
                     1
        4 + ---------------------
                       1
            1 + -----------------
                         1
                5 + -------------
                            1
                    9 + ---------
                              1
                        2 + -----
                             ...


Which is about 3.828656
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Postby crazyjimbo » Fri Jun 15, 2007 1:24 am UTC

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 :)

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Postby 3.14159265... » Fri Jun 15, 2007 2:46 am UTC

Nyarlathotep 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.
Why thank you

(6(1/1 + 1/4 + 1/9 + 1/16.....))^(1/2)

I love pi, because of that.
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Postby antonfire » Fri Jun 15, 2007 4:17 am UTC


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Postby shill » Fri Jun 15, 2007 5:20 am UTC

2^(1/12), or Image. It gives us the lovely equal temperament scale.

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Postby PaulT » Fri Jun 15, 2007 12:29 pm UTC

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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Postby parallax » Fri Jun 15, 2007 1:55 pm UTC

OmnipotentEntity wrote:Area of the Mandelbrot set, assuming it's irrational.


The area of the boundary of the Mandelbrot set, assuming it's nonzero and irrational.

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Postby jestingrabbit » Fri Jun 15, 2007 3:06 pm UTC

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.

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Postby Poker » Fri Jun 15, 2007 3:32 pm UTC

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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Postby gmalivuk » Fri Jun 15, 2007 3:57 pm UTC

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.
Image

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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Postby cmacis » Fri Jun 15, 2007 4:11 pm UTC

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.
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Postby chrispy1 » Fri Jun 15, 2007 4:27 pm UTC

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 :lol: )
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Postby gmalivuk » Fri Jun 15, 2007 4:29 pm UTC

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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Postby skeptical scientist » Fri Jun 15, 2007 4:45 pm UTC

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. :P
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Postby Shadowfish » Fri Jun 15, 2007 6:46 pm UTC

I can't believe no one has mentioned the Feigenbaum constants.

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Postby OmnipotentEntity » Fri Jun 15, 2007 7:24 pm UTC

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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Postby liza » Sat Jun 16, 2007 2:07 am UTC

Gotta go with e. I mean, it's so cool! Any number that is defined as being "the base of logarithm that results from taking the antiderivative of 1/X" is okay in my book (mostly because that only includes e).

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Postby Marbas » Sat Jun 16, 2007 4:24 am UTC

The omega constant!

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Postby chrispy1 » Sat Jun 16, 2007 11:50 am UTC



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:

Image

That is all. I'm sure if I sat and stared at this long enough, I could come up with some sort of quasi-intelligent 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.
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Postby __Kit » Sat Jun 16, 2007 11:56 am UTC

Does 0.9 reoccurring count?

Only because none of my friends understand the concept of it 8)
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Postby PaulT » Sat Jun 16, 2007 12:49 pm UTC

__Kit wrote:Does 0.9 reoccurring count?

Only because none of my friends understand the concept of it 8)

As an irrational number? 1? No, I don't think so.

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Postby 3.14159265... » Sat Jun 16, 2007 2:03 pm UTC

chrispy1 wrote:


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:

Image
That is all. I'm sure if I sat and stared at this long enough, I could come up with some sort of quasi-intelligent 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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Postby Stig Hemmer » Sat Jun 16, 2007 5:38 pm UTC

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.
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Postby une see » Sat Jun 16, 2007 5:48 pm UTC

Rad 2. Although I am very fond of the Golden Ratio...

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Postby gmalivuk » Sat Jun 16, 2007 6:51 pm UTC

3.14159265... wrote:The numbers are normal is what I see from that.


Well, base-10 at least suggests they might be normal. You gotta check every base to be sure. :-)
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Postby dumbclown » Sat Jun 16, 2007 7:26 pm UTC

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.

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Postby gmalivuk » Sat Jun 16, 2007 8:03 pm UTC

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.
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Postby Herman » Sat Jun 16, 2007 9:46 pm UTC

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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Postby gmalivuk » Sat Jun 16, 2007 10:07 pm UTC

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 base-12.

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 one-sided p-value is 0.37, which means we definitely don't throw out the hypothesis that the digits are distributed normally.
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