A Most Amazing Number

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Peterotooled
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A Most Amazing Number

Postby Peterotooled » Thu Feb 16, 2012 12:16 pm UTC

This is a link to a YouTube video by Vihart, a self-styled mathemusician, who publishes many charming mathematical videos in her channel there. It describes a number called "Wau" which is the pronunciation of the Greek letter digamma. Since I am not a mathematician, I cannot adequately describe the derivation of the number, except to say that it is so mysterious and counter-intuitive that I completely fail to understand how it is constructed. I urge you all to go and see the video, and examine her evidence. It may well be a hoax. But if it isn't, it is the strangest number I have ever encountered.


http://www.youtube.com/watch?v=GFLkou8NvJo

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Re: A Most Amazing Number

Postby gfauxpas » Thu Feb 16, 2012 12:30 pm UTC

That's not nice!

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t1mm01994
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Re: A Most Amazing Number

Postby t1mm01994 » Thu Feb 16, 2012 12:50 pm UTC

Hoax. e^(i*2pi)=*wau*? YEeaaaaahh no.

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eSOANEM
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Re: A Most Amazing Number

Postby eSOANEM » Thu Feb 16, 2012 1:01 pm UTC

The thing that irritates me most. Digamma was 6.
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Sizik
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Re: A Most Amazing Number

Postby Sizik » Thu Feb 16, 2012 2:12 pm UTC

Spoiler:
Wau = 1
gmalivuk wrote:
King Author wrote:If space (rather, distance) is an illusion, it'd be possible for one meta-me to experience both body's sensory inputs.
Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.

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z4lis
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Re: A Most Amazing Number

Postby z4lis » Thu Feb 16, 2012 5:11 pm UTC

That number is as mysterious as dihydrogen monoxide is dangerous!
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

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Re: A Most Amazing Number

Postby gfauxpas » Thu Feb 16, 2012 5:54 pm UTC

z4lis wrote:That number is as mysterious as dihydrogen monoxide is dangerous!

I would say that Ϝ is as mysterious as H₂O is mysterious. You may take both for granted, but they're both amazing.

Nyktos
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Re: A Most Amazing Number

Postby Nyktos » Thu Feb 16, 2012 8:40 pm UTC

Saw this a few days ago. I figured it out pretty fast, but it was amusing.

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Proginoskes
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Re: A Most Amazing Number

Postby Proginoskes » Fri Feb 17, 2012 6:35 am UTC

She made a mistake, though ... the derivative of e^Wau with respect to x is not Wau*e; it's ln(Wau) * e.

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t1mm01994
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Re: A Most Amazing Number

Postby t1mm01994 » Fri Feb 17, 2012 10:10 am UTC

She made a youtube note saying "it's Wau*e - e^wau", which works as well..
Gosh, why didn't I figure out that it was a number instead of a huge hoax >.>

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PM 2Ring
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Re: A Most Amazing Number

Postby PM 2Ring » Fri Feb 17, 2012 1:46 pm UTC

t1mm01994 wrote:She made a youtube note saying "it's Wau*e - e^wau", which works as well..
Gosh, why didn't I figure out that it was a number instead of a huge hoax >.>


I wouldn't call it a hoax. It's a cute way to remind people how important Wau is. There are plenty of interesting numbers, but where would they be without Wau?

gfauxpas wrote:
z4lis wrote:That number is as mysterious as dihydrogen monoxide is dangerous!

I would say that Ϝ is as mysterious as H₂O is mysterious. You may take both for granted, but they're both amazing.

This.

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t1mm01994
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Re: A Most Amazing Number

Postby t1mm01994 » Fri Feb 17, 2012 2:31 pm UTC

Yeah, but at first I thought it was a hoax.. Now that I know the true identity of Wau, I know it's not a hoax, but I didn't look at the identities close enough to recognize that Wau fits into all of them.
And e to the i to the e i o equals e to the tau to the tau wau wau sounded strange :P and more than one way to represent as decimal got me at first too. Now that I'm much wiser, it all makes sense...

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Re: A Most Amazing Number

Postby Nyktos » Fri Feb 17, 2012 8:14 pm UTC

All nonzero real numbers can be represented as decimals in two ways. Wau is just the most famous example.

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Sizik
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Re: A Most Amazing Number

Postby Sizik » Fri Feb 17, 2012 8:20 pm UTC

Nyktos wrote:All nonzero real numbers can be represented as decimals in two ways. Wau is just the most famous example.


I don't know about reals, but rationals definitely (as they terminate).
gmalivuk wrote:
King Author wrote:If space (rather, distance) is an illusion, it'd be possible for one meta-me to experience both body's sensory inputs.
Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.

mfb
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Re: A Most Amazing Number

Postby mfb » Fri Feb 17, 2012 8:27 pm UTC

After watching the video exactly wau times, I noticed an error at ~2:50: There is an x missing at both sides, d/dx e^wau = 0.
And 4:35 uses a relativistic mass :(.


Not even all rational numbers terminate, just look at 1/3.
You can write a number with two different decimal representations if and only if it is a multiple of 10^(-n) for some natural n and not 0.
In other words: All rational numbers p/(2^n 5^m) with p!=0 integer and n,m natural numbers.

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Qaanol
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Re: A Most Amazing Number

Postby Qaanol » Fri Feb 17, 2012 9:15 pm UTC

Zero has two distinct decimal representations as well.
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Dason
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Re: A Most Amazing Number

Postby Dason » Fri Feb 17, 2012 9:15 pm UTC

mfb wrote:After watching the video exactly wau times, I noticed an error at ~2:50: There is an x missing at both sides, d/dx e^wau = 0.
And 4:35 uses a relativistic mass :(.


I believe there was an annotation added that corrected that part.
double epsilon = -.0000001;

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Sizik
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Re: A Most Amazing Number

Postby Sizik » Fri Feb 17, 2012 10:06 pm UTC

mfb wrote:Not even all rational numbers terminate, just look at 1/3.
You can write a number with two different decimal representations if and only if it is a multiple of 10^(-n) for some natural n and not 0.
In other words: All rational numbers p/(2^n 5^m) with p!=0 integer and n,m natural numbers.


Ah whoops, I mixed up termination with repeating.
gmalivuk wrote:
King Author wrote:If space (rather, distance) is an illusion, it'd be possible for one meta-me to experience both body's sensory inputs.
Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.

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eSOANEM
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Re: A Most Amazing Number

Postby eSOANEM » Fri Feb 17, 2012 10:59 pm UTC

mfb wrote:And 4:35 uses a relativistic mass :(.


Or a stationary particle.
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moiraemachy
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Re: A Most Amazing Number

Postby moiraemachy » Fri Feb 17, 2012 11:48 pm UTC

Qaanol wrote:Zero has two distinct decimal representations as well.

Nyktos wrote:All nonzero real numbers can be represented as decimals in two ways. Wau is just the most famous example.


Could you guys elaborate? I thought that the only way to get distinct decimal representations of a number is by converting it's finite representation into one that ends in "999...". Which seems impossible to do with zero.

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Talith
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Re: A Most Amazing Number

Postby Talith » Sat Feb 18, 2012 12:24 am UTC

Nyktos wrote:All nonzero real numbers can be represented as decimals in two ways. Wau is just the most famous example.
Maybe true for wau, but you definitely can't represent every nonzero real number as decimals in two ways (try giving the two decimal representations of 1/3).

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Qaanol
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Re: A Most Amazing Number

Postby Qaanol » Sat Feb 18, 2012 3:17 am UTC

0.000… = -0.000…
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moiraemachy
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Re: A Most Amazing Number

Postby moiraemachy » Sat Feb 18, 2012 4:06 am UTC

Aargh. That was cheap, and you know it :x

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Qaanol
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Re: A Most Amazing Number

Postby Qaanol » Sat Feb 18, 2012 4:27 am UTC

How so? Do you contend that 1.000… and −1.000… are identical decimal representations? They clearly differ at the ‘sign’ position of the representation, and that is exactly the same representational difference as distinguishes 0.000… from −0.000…
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moiraemachy
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Re: A Most Amazing Number

Postby moiraemachy » Sat Feb 18, 2012 6:03 am UTC

I'm not pissed because I disagree, I'm pissed because I thought there was more to it than this. I never really gave the notion of "decimal representation" a rigorous look :oops:

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NathanielJ
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Re: A Most Amazing Number

Postby NathanielJ » Sat Feb 18, 2012 6:35 am UTC

Where are we drawing the line here? Maybe zero has an infinite number of decimal representations: 0, 0.0, 0.00, 0.000, ...
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Re: A Most Amazing Number

Postby t1mm01994 » Sat Feb 18, 2012 11:21 am UTC

I believe that consensus says that any trailing zeros may be ignored, i.e. 3.0 and 3 are the exact same number in the exact same representation.. For maths goals, that is.

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Re: A Most Amazing Number

Postby Nyktos » Sat Feb 18, 2012 9:50 pm UTC

Talith wrote:
Nyktos wrote:All nonzero real numbers can be represented as decimals in two ways. Wau is just the most famous example.
Maybe true for wau, but you definitely can't represent every nonzero real number as decimals in two ways (try giving the two decimal representations of 1/3).

Yeah, clearly I wasn't really thinking when I said that. Only rationals that can be expressed with the denominator as a power of ten, yes.

lalop
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Re: A Most Amazing Number

Postby lalop » Sun Feb 19, 2012 1:32 am UTC

1/3 definitely has many decimal representations. 0.33333...., 00.33333..., 000.33333,...

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Talith
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Re: A Most Amazing Number

Postby Talith » Sun Feb 19, 2012 2:27 am UTC

Traditionally a decimal representation is the sum of a sequence (ai/10i)i≥0, where a0 is an integer and the other ais are in the set {0,1,...,9}. So, your representations other than the first are not representations in the traditional sense (either that, or we would say they are the same as the first representation).

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Re: A Most Amazing Number

Postby Yesila » Sun Feb 19, 2012 4:21 am UTC

Qaanol wrote:How so? Do you contend that 1.000… and −1.000… are identical decimal representations? They clearly differ at the ‘sign’ position of the representation, and that is exactly the same representational difference as distinguishes 0.000… from −0.000…


I think that they are identical decimal representations if any truncation (say at the i-th decimal place) of the two results in equality. In the case of -1.000... and 1.000... every truncation results in two different values so they are not identical decimal representations. With 0.000... and -0.000... every truncation results in the same value so they are identical decimal representations.

In the same manner 1.00 and 1.0000 and 1.00000... are all identical decimal representations of 1, yet 1 and .9999.. are not identical decimal representations, they are in fact two (non-identical) decimal representations of 1.

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Re: A Most Amazing Number

Postby lalop » Sun Feb 19, 2012 11:57 am UTC

Talith wrote:Traditionally a decimal representation is the sum of a sequence (ai/10i)i≥0, where a0 is an integer and the other ais are in the set {0,1,...,9}. So, your representations other than the first are not representations in the traditional sense (either that, or we would say they are the same as the first representation).


0 is an integer, and the sum of that sequence is a number, not any sort of representation. And honestly, if you're arguing for these representations being the same, you may as well argue for 0.999... being the same representation as 1. After all, the only grounds by which they are the same is that they in fact represent the same number. Otherwise, they are represented quite differently.

Yesila wrote:I think that they are identical decimal representations if any truncation (say at the i-th decimal place) of the two results in equality.


Not the worst criterion, but it seems rather arbitrary. A more natural way to look at it is to ask how a decimal number would be rigorously defined. The most obvious candidate is as a function f : Z -> {0,...,9} such that for some N, all f(n) for n > N are zero. Equality would then be, of course, function equality (and the numbers themselves equivalence classes of these functions). This way, to specify a decimal you clearly have to specify each decimal place, and equality happens iff two numbers agree at all decimal places.

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Talith
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Re: A Most Amazing Number

Postby Talith » Sun Feb 19, 2012 1:26 pm UTC

0 is an integer sure, but are 00 and 000 really different integers to 0? I don't know any definition of integer that would say they are different (if even 00 is a well defined integer - which I would say it is not)).

Also, in your definition, I assume you're mapping the integer n to the 10n place of the real number. The other option would be to stick with n->10-n but have for some N, for all n<N, f(n)=0 which is also fairly natural because the mapping then 'reads left to right' in the usual way that we view the integers and the decimal places of a real number.

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Re: A Most Amazing Number

Postby lalop » Sun Feb 19, 2012 8:50 pm UTC

Talith wrote:0 is an integer sure, but are 00 and 000 really different integers to 0? I don't know any definition of integer that would say they are different (if even 00 is a well defined integer - which I would say it is not)).


Perhaps I misread last night. The real argument against your definition, I think, is that it's not representative of how we write decimal notation of integers. For example, 345 is written that way not because it's an "integer", but because it's 3*10^2 + 4*10^1 + 5*10^0 . (This is, in fact, the same thing that happens to the sequence's tail, so why are we stopping before we get to the head?) In other words, your sequence's cutoff point at i = 0 seems unrepresentative of how decimal notation is actually used.


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