What's your favourite irrational number?

For the discussion of math. Duh.

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liza
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Postby liza » Sun Jun 17, 2007 5:43 am UTC

__Kit wrote:Does 0.9 reoccurring count?

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


Yeah, not irrational. Even if you didn't count that one outright (I won't start that discussion again), it IS equal to 9/9 undeniably, and is therefore rational.

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Postby Maseiken » Sun Jun 17, 2007 6:09 am UTC

I like multiplying Pi by e in the Algebraic form (Although using the letters Pi and not the symbol is cheating i guess)
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Postby skeptical scientist » Sun Jun 17, 2007 6:47 am UTC

Stig Hemmer wrote:
skeptical scientist wrote:The halting probability, ?.

Which halting probability do you prefer? There are many, you know.

Personally, I am quite fond of ?, though I can't give any rational reason why.

I'm equally fond of all of them, really, since they are all random c.e. reals which solve the halting problen, and the fact that this characterizes them and that they are dense in [0,1] is even cooler.
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Postby Dobblesworth » Sun Jun 17, 2007 2:17 pm UTC

The formula used for calculating values in the Normal distribution function. I just remember reading the formula in Stats and thinking "W T F?!" Even our maths teacher with an Oxford degree doesn't understand it.

http://en.wikipedia.org/wiki/Normal_distribution

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Postby gmalivuk » Sun Jun 17, 2007 3:09 pm UTC

Dobblesworth wrote:The formula used for calculating values in the Normal distribution function. I just remember reading the formula in Stats and thinking "W T F?!" Even our maths teacher with an Oxford degree doesn't understand it.

http://en.wikipedia.org/wiki/Normal_distribution


How much did your teacher pay for that Oxford degree? And is it a degree in math? It's just an exponential multiplied by a constant term to make the total area 1.

Somnia wrote:
__Kit wrote:Does 0.9 reoccurring count?

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


Yeah, not irrational. Even if you didn't count that one outright (I won't start that discussion again), it IS equal to 9/9 undeniably, and is therefore rational.


Well being equal to 9/9 (which is obviously 1) is not what's undeniable, since that's what's always argued. But it's a repeating decimal, undeniably, and therefore rational.
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Postby Spivak » Mon Jun 18, 2007 4:29 am UTC

I'd choose a transcendental number. But one of the obvious ones. One of the uncountable many that it's impossible to construct by an algorithm. Sadly, it's impossible to say anything about my number :(
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Postby EradicateIV » Mon Jun 18, 2007 4:38 am UTC

I love the fact that you can say there is any irrational number between two numbers.
(2)^1/2 + C... :-)
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Postby skeptical scientist » Mon Jun 18, 2007 12:01 pm UTC

Spivak wrote:I'd choose a transcendental number. But one of the obvious ones. One of the uncountable many that it's impossible to construct by an algorithm. Sadly, it's impossible to say anything about my number :(

The number I gave (Ω) is transcendental, and one of the uncountably many that it's impossible to construct by an algorithm, but one can, nevertheless, say a great deal about it.
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Postby gmalivuk » Mon Jun 18, 2007 3:11 pm UTC

skeptical scientist wrote:
Spivak wrote:I'd choose a transcendental number. But one of the obvious ones. One of the uncountable many that it's impossible to construct by an algorithm. Sadly, it's impossible to say anything about my number :(

The number I gave (Ω) is transcendental, and one of the uncountably many that it's impossible to construct by an algorithm, but one can, nevertheless, say a great deal about it.


Maybe Spivak really meant one of the uncountably many that aren't even possible to define in a finite way.
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Postby apeman5291 » Mon Jun 18, 2007 7:53 pm UTC

I'd have to go with something completely random, like sin(1), in radians.

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Postby cmacis » Mon Jun 18, 2007 8:06 pm UTC

Right now, the sum from n=1 to infinity of 1/n^2. Comes to (pi^2)/6 iirc.

Sigma 1/n^2 is so nice in analysis. :)
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Postby bcoblentz » Mon Jun 18, 2007 8:13 pm UTC

j;)
Last edited by bcoblentz on Fri Aug 20, 2010 12:46 am UTC, edited 1 time in total.

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Postby Taejo » Mon Jun 25, 2007 1:21 am UTC

bcoblentz wrote:as for my favorite irrational, i'd have to go with liouville's constant since i like to pretend it was created to annoy other mathematicians (which is also why i like weierstrass's nowhere-differentiable curves)


I <3 Weierstrass. But I'd go for Ω_BrainFuck as my favourite irrational.
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Postby skeptical scientist » Mon Jun 25, 2007 2:40 am UTC

Taejo wrote:I <3 Weierstrass. But I'd go for Ω_BrainFuck as my favourite irrational.

"Ω_BrainFuck"? What do you mean?
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Postby hotaru » Mon Jun 25, 2007 1:08 pm UTC


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Postby jestingrabbit » Mon Jun 25, 2007 2:12 pm UTC

skeptical scientist wrote:
Taejo wrote:I <3 Weierstrass. But I'd go for Ω_BrainFuck as my favourite irrational.

"Ω_BrainFuck"? What do you mean?
Chaitin's constant calculated wrt the computer language brainfuck probably.

edit: and to hotaru and several others who've said similar numbers- can you actually prove that its irrational? I bet you can't. Sorry, that's overly confrontational, but its been under my skin bugging me for a while.

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Postby antonfire » Mon Jun 25, 2007 6:54 pm UTC

If it were rational, wouldn't it be computable?

(No, I don't know how to prove that it's not computable.)

Edit: now I do. Yayfor Wikipedia.

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Postby gmalivuk » Mon Jun 25, 2007 7:19 pm UTC

antonfire wrote:If it were rational, wouldn't it be computable?


Yes, which is why people listing uncomputable numbers weren't included. The admonition was more for people like hotaru (hence listing hotaru's name instead of Taejo), who post things like (e(pi+sqrt(pi)(sqrt(pi+1)+1)))^(1/pi).

That number, while most likely irrational, isn't obviously so. To be valid in a "what's your favorite irrational number" thread, it seems you should first be sure your numbers are irrational.
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Postby skeptical scientist » Mon Jun 25, 2007 8:44 pm UTC

jestingrabbit wrote:
skeptical scientist wrote:
Taejo wrote:I <3 Weierstrass. But I'd go for Ω_BrainFuck as my favourite irrational.

"Ω_BrainFuck"? What do you mean?
Chaitin's constant calculated wrt the computer language brainfuck probably.

edit: and to hotaru and several others who've said similar numbers- can you actually prove that its irrational? I bet you can't. Sorry, that's overly confrontational, but its been under my skin bugging me for a while.

Ah, that makes sense. I didn't know there was a programming language called "brainfuck". Although technically speaking, Chaitin's constant should be with respect to a universal prefix-free machine, which isn't exactly the same as a programming language.
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Postby necroforest » Mon Jun 25, 2007 9:56 pm UTC

The one generated by this:

Code: Select all

print "0."
while true:
       print random.randint(0, 9),

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Postby hotaru » Mon Jun 25, 2007 10:40 pm UTC

necroforest wrote:The one generated by this:

Code: Select all

print "0."
while true:
       print random.randint(0, 9),

that number is almost certainly rational.
unless you have some sort of computer that actually can run that program forever...

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Postby gmalivuk » Tue Jun 26, 2007 1:22 am UTC

hotaru wrote:
necroforest wrote:The one generated by this:

Code: Select all

print "0."
while true:
       print random.randint(0, 9),

that number is almost certainly rational.
unless you have some sort of computer that actually can run that program forever...


Especially since, as we all know, it's just going to come out 0.4444444..., which converges to 4/9.

In my opinion, one of the easiest ways to get a new irrational number is a nonterminating continued fraction. (1;2,3,4,5,6...), for instance.
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Postby necroforest » Tue Jun 26, 2007 1:39 am UTC

hotaru wrote:
necroforest wrote:The one generated by this:

Code: Select all

print "0."
while true:
       print random.randint(0, 9),

that number is almost certainly rational.
unless you have some sort of computer that actually can run that program forever...


I figure it's like pi or e... you can enumerate digits for as long as you want, but you'll never actually enumerate them all (since it's irrational of course!), but because you arbitrarily hit Ctrl-C doesn't mean the number's rational.

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Postby antonfire » Tue Jun 26, 2007 3:25 am UTC

gmalivuk wrote:Yes, which is why people listing uncomputable numbers weren't included.

Ah, right. I misunderstood.

necroforest wrote:The one generated by this:

Code: Select all

print "0."
while true:
       print random.randint(0, 9),

Either the digits eventually repeat (since a computer only has a finite number of states), or the computer's random number generator uses (directly or indirectly) data from outside your computer (reminds me of this bash quote), in which case that program will yield a different number every time you run it. So either it's rational, or it's not well-defined.

(And I use parentheses too much, it seems.)


Edit: also, people think of numbers in terms of their binary/decimal representations too much. agree/disagree?

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Postby Spivak » Tue Jun 26, 2007 5:12 am UTC

Maybe Spivak really meant one of the uncountably many that aren't even possible to define in a finite way.


Thanks, that sounded a lot better. I didn't know anything about this omega probability :oops: . I haven't done analisys or algorithm analisys in my course. I'll study more before I start giving fancy examples.
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Postby necroforest » Tue Jun 26, 2007 1:06 pm UTC

antonfire wrote: (since a computer only has a finite number of states)

Good point, I didn't think of that. I was thinking more along the lines of a probabilistic Turing machine that just copied random bits to the output tape. Although, I believe that wouldn't be an issue if you got your random bits from, say, /dev/[u]random, since those come from a non-algorithmic source, namely interrupt timings and such. And yes, I admit it's not well-defined :wink:

antonfire wrote:Edit: also, people think of numbers in terms of their binary/decimal representations too much. agree/disagree?


Agreed.

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Postby gmalivuk » Tue Jun 26, 2007 3:29 pm UTC

necroforest wrote:
hotaru wrote:
necroforest wrote:The one generated by this:

Code: Select all

print "0."
while true:
       print random.randint(0, 9),

that number is almost certainly rational.
unless you have some sort of computer that actually can run that program forever...


I figure it's like pi or e... you can enumerate digits for as long as you want, but you'll never actually enumerate them all (since it's irrational of course!), but because you arbitrarily hit Ctrl-C doesn't mean the number's rational.


Yes, but pi and e are proven to be irrational, and there are algorithms you can use to deterministically find out what any given digit is ahead of time. Unless you specify the pseudorandom algorithm being used (and change it a bit to ensure it won't actually repeat), I feel like the one you've listed isn't any particular number to begin with.
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Re: What's your favourite irrational number?

Postby Monox D. I-Fly » Wed Jun 08, 2016 4:28 am UTC

My favorite irrational number is -½ √2, since it contains a negative, a fraction, and an irrational itself.

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Re:

Postby Bloopy » Tue Sep 06, 2016 4:36 am UTC

themandotcom wrote:The Euler-Masheroni constant (Yes, it is irrational!) :P

This... will be mine too. Thanks to playing around with it a bit in this thread.

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Re: What's your favourite irrational number?

Postby Eebster the Great » Tue Sep 06, 2016 5:09 pm UTC

There are plenty of dimensionless physical constants you could choose, like the fine structure constant, but again there is no way to really be sure they are irrational (and for at least some of these quantities, quantum information theorists might say they probably are not irrational).

What about the number x defined such that 0 < x < 1 and the ith digit of x is the ith element of 0 (mod 10), ordered lexicographically, where 0 is zero dagger?

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Re: What's your favourite irrational number?

Postby LucasBrown » Thu Sep 08, 2016 3:52 am UTC

Eebster the Great wrote:What about the number x defined such that 0 < x < 1 and the ith digit of x is the ith element of 0 (mod 10), ordered lexicographically, where 0 is zero dagger?


The problem with that is that it might not exist.

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Re: What's your favourite irrational number?

Postby Eebster the Great » Thu Sep 08, 2016 4:29 am UTC

It certainly exists in ZFC + 0. Sure, ZFC + 0 isn't proven consistent yet, but as I understand it, everyone seems pretty sure it is.

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Re: What's your favourite irrational number?

Postby Mike Rosoft » Wed Oct 19, 2016 7:27 pm UTC

There are some irrational numbers which can be expressed as a solution of a polynomial equation, but can't be expressed as an explicit formula of addition, subtraction, multiplication, division, and n-th roots. (Of course, in both cases it is assumed that we start with integers.) The canonical example is the unique real solution of:
x^5-x-1=0

(I like this subtle difference.)

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Re: What's your favourite irrational number?

Postby Millydelev » Thu Dec 01, 2016 6:13 am UTC

Square root of pi

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Re: What's your favourite irrational number?

Postby Paradoxica » Tue Dec 06, 2016 12:53 pm UTC

Brjuno number(s)
GENERATION -705 - 992 i: The first time you see this, copy it into your sig on any forum. Square it, and then add i to the generation.

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Re: What's your favourite irrational number?

Postby Lothario O'Leary » Tue Dec 20, 2016 4:03 am UTC

My current favorite is sqrt(9 3/4), which is basically the proven-irrational version of e^pi-pi.
(Check out the first 20 digits or so. No, I still have no idea why it behaves that way.)

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Re: What's your favourite irrational number?

Postby zukenft » Wed Dec 21, 2016 12:30 am UTC

this might be a good place to ask, as it has something to do with irrational numbers. what do you call a type of irrational number that can be 'defined'?
for example, 0.101001000100001... the number of zeroes increase every iteration. this sequence certainly do not repeat, but it is very easy to predict the nth decimal.
other example would be 0.122333444455555... (repeat the number by its value) 0.11235813... (fibonacci sequence)

maybe the term is 'programmable irrational'? I think once you define the number you can write a program to generate it.

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Re: What's your favourite irrational number?

Postby Lothario O'Leary » Wed Dec 21, 2016 12:58 am UTC

zukenft wrote:this might be a good place to ask, as it has something to do with irrational numbers. what do you call a type of irrational number that can be 'defined'?
for example, 0.101001000100001... the number of zeroes increase every iteration. this sequence certainly do not repeat, but it is very easy to predict the nth decimal.
other example would be 0.122333444455555... (repeat the number by its value) 0.11235813... (fibonacci sequence)

maybe the term is 'programmable irrational'? I think once you define the number you can write a program to generate it.
The term is "computable", but that includes the traditional irrationals too - if you want you can write a program to generate sqrt(2) or pi (people have actually done that), it would just be a bit longer.
(Note that there is a slight complication there - in some cases, you might not be able to be sure that your next digits won't end up being a ludicrous amount of nines (or zeroes) in a row, perhaps much longer than the rest of your current digits, leaving you unable to easily determine what the actual next digit is if you're working approximately. Thpugh this is not a problem for sqrt(2) and pi (easily proven in the former case, known theorem in the latter), and is of course astronomically unlikely to ever happen anywhere else but hadn't been proven definitely.)

As it happens, due to some complicated math stuff (Goedel's theorems, mostly), it's actually possible to define a number that it's mathematically impossible to write a program for. (And there are of course many other such numbers that can't even be defined - at least not finitely.) The assorted omega numbers discussed here previously mostly work that way.

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Re: What's your favourite irrational number?

Postby Eebster the Great » Wed Dec 21, 2016 4:06 am UTC

Lothario O'Leary wrote:My current favorite is sqrt(9 3/4), which is basically the proven-irrational version of e^pi-pi.
(Check out the first 20 digits or so. No, I still have no idea why it behaves that way.)

You mean √9.75 = 3.122498999199199102923446560469897230536479988995828154226... ? There are a lot of 9s in there, but otherwise no obvious pattern.You mean it looks kind of like 3.12249900?


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