## What's your favourite irrational number?

For the discussion of math. Duh.

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liza
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__Kit wrote:Does 0.9 reoccurring count?

Only because none of my friends understand the concept of it

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.

Maseiken
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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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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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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

gmalivuk
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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

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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Spivak
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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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I love the fact that you can say there is any irrational number between two numbers.
(2)^1/2 + C...
1010011010

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

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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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Taejo
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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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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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hotaru
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jestingrabbit
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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.

antonfire
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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.

gmalivuk
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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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skeptical scientist
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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.
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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necroforest
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The one generated by this:

Code: Select all

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

hotaru
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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...

gmalivuk
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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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necroforest
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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.

antonfire
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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?

Spivak
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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 . 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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necroforest
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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

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

Agreed.

gmalivuk
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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?

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

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### Re:

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

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?

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?

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?

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.

Mike Rosoft
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### Re: What's your favourite irrational number?

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.)

Millydelev
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### Re: What's your favourite irrational number?

Square root of pi

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### Re: What's your favourite irrational number?

Brjuno number(s)
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Lothario O'Leary
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### Re: What's your favourite irrational number?

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.)

zukenft
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### Re: What's your favourite irrational number?

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?

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?

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?