Advanced math and probability

[Kurald Galain]

Well, this came up in a recent thread and turned out to be more interesting than the thread's topic, so I'm forking it out. The way I figure, math and probability has a bearing on game mechanics, right?

Incorrect.

.999 repeating and 1.000 repeating are different numbers that are infinitely far apart. The fact that we can't represent the difference between 1/3 and .999 repeating/3 is irrelevant, the difference exists.

I'm afraid that http://en.wikipedia.org/wiki/.999 proves you wrong.

[Tengu]

0.(9) is indeed the same number as 1. That's not really advanced math. Well, maybe for people who don't get xkcd jokes and other non-educated folk.

[Dan_Hemmens]

Not done any advanced maths in years, but I take it that the proof is something like: 0.(9) is effectively the sum to infinity of 9/10 + 9/100 etc, which you can work out the same way you work out 1/2 + 1/4 ... ?

[nagora]

Not done any advanced maths in years, but I take it that the proof is something like: 0.(9) is effectively the sum to infinity of 9/10 + 9/100 etc, which you can work out the same way you work out 1/2 + 1/4 ... ?

You could argue, I suppose, that the number .999... represents the difference between 1 and .9 divided infinitely, and the difference is therefore <anything>/infinity=0. But, yes summing the series would probably be the direct route for a mathematician.

[Jack_Simth]

Not done any advanced maths in years, but I take it that the proof is something like: 0.(9) is effectively the sum to infinity of 9/10 + 9/100 etc, which you can work out the same way you work out 1/2 + 1/4 ... ?

That's one way of doing it, yes.

Another is the conversion of infinite repeating decimals to fractions - results in 9/9 for .9999....

Another is subtraction; 1 - 0.9999... = 0.0000.... forever. There is no termination on the zeros.

Of course, the simplest is the practicalist view - if there is any difference at all, it is literally infinitely small of a difference, and doesn't make a difference. And a difference that makes no difference isn't really a difference, is it?

[Emperor Tippy]

Well, this came up in a recent thread and turned out to be more interesting than the thread's topic, so I'm forking it out. The way I figure, math and probability has a bearing on game mechanics, right?

I'm afraid that http://en.wikipedia.org/wiki/.999 proves you wrong.

From your own link

Although the real numbers form an extremely useful number system, the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in Mathematics: A Very Short Introduction that the resulting identity 0.999… = 1 is a convention as well:

However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.

0.(9) = 1 is necessary for some forms of math but in the end it is arbitrary.

[Dan_Hemmens]

That's one way of doing it, yes.

Another is the conversion of infinite repeating decimals to fractions - results in 9/9 for .9999....

Problem is that the fraction method kind of feels like a trick (like the 1 + 1 = 3 thing)

Another is subtraction; 1 - 0.9999... = 0.0000.... forever. There is no termination on the zeros.

That's rather elegant, however.

[Kurald Galain]

From your own link

0.(9) = 1 is necessary for some forms of math but in the end it is arbitrary.

And in your own sentence, "not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic".

Lots of conventions are arbitrary (e.g. that 0! = 1) but they are nevertheless well-accepted conventions. You might as well be arguing that you didn't believe in relativity because "it's only a theory" - that belies only a lack of understanding of the word "theory".

[Dan_Hemmens]

Lots of conventions are arbitrary (e.g. that 0! = 1) but they are nevertheless well-accepted conventions. You might as well be arguing that you didn't believe in relativity because "it's only a theory" - that belies only a lack of understanding of the word "theory".

But it's just a *theory* that means it's *made up* like *fairies* and *Jesus*.

Also, I think your last sentence betrays a lack of understanding of the word "belie".

[Pinnacle]

1/3 is 0.333... and 2/3 is 0.666...
1/3 + 2/3 is, of course, 1. 0.333... + 0.666... is 0.999...
If A+B=C and A+B=D is also true, C=D must be true.

[Jack_Simth]

Problem is that the fraction method kind of feels like a trick (like the 1 + 1 = 3 thing)

You've seen the method before, right?

That's rather elegant, however.

Thanks.

[Dan_Hemmens]

You've seen the method before, right?

1 + 1 = 3? Yes (a couple of ways). The 0.(9) = 1 thing I actually never saw until today, I was always rubbish at maths - I was a physicist.

[Emperor Tippy]

And in your own sentence, "not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic".

Lots of conventions are arbitrary (e.g. that 0! = 1) but they are nevertheless well-accepted conventions. You might as well be arguing that you didn't believe in relativity because "it's only a theory" - that belies only a lack of understanding of the word "theory".

Just because an arbitrary convention is well accepted doesn't make it any less arbitrary. 0!=1 is useful, just like 0.(9)=1 is useful. So both are kept and used in theoretical math.

Physics theories are different from math theories. One is based on measurable, quantifiable, objective fact and the other is based on arbitrary conventions.

[averagejoe]

I knew some neat proofs for this, but it's been so long that I can't remember them at this point. It's just one of those wierd mathematical things that completely blows people's minds. Like how if you take the graph f(x)=1/x from x=1 to infinity and make a solid by rotating the line around the x-axis, then you get an object with a finite volume but infinite surface area. That totally blew my mind, anyways. This isn't really advanced math, though.

From your own link


0.(9) = 1 is necessary for some forms of math but in the end it is arbitrary.

I would like to add my voice to Kurald in saying that, in the end, math is pretty arbitrary, and it's only how it is because of convenience. Anyways, achedemics like to be contrary, especially when it comes to well established things, because that's how you get ahead in achedamia. .(9)=1 is, however, a well-proven and useful result, and it would pretty much always be correct to say that it is the case that .(9)=1.

Edit:

Just because an arbitrary convention is well accepted doesn't make it any less arbitrary. 0!=1 is useful, just like 0.(9)=1 is useful. So both are kept and used in theoretical math.

Physics theories are different from math theories. One is based on measurable, quantifiable, objective fact and the other is based on arbitrary conventions.

Well, yeah. Mathematics is basically philisophical, it just happens to be a philosophy that models our world really well, so physics makes good use of it. .(9) does not in any way represent a physical object, it represents a mathematical one. However, it's silly to dismiss this solution as just an arbitrary convention, since that could be said for enough of math that it renders the statement basically meaningless. To use a better metaphor, you might as well say, "Einsteinien reletivity isn't proven, it assumes that the universe is observable and quantifiable."

[Jack_Simth]

1 + 1 = 3? Yes (a couple of ways). The 0.(9) = 1 thing I actually never saw until today, I was always rubbish at maths - I was a physicist.

I'm referring to the general method of converting infinite repeating decimals to fractions, actually.

You take your infinite repeating decimal - let's go with .234234234234234... just for grins. And we'll say it is equal to b. I like b. We'll use b.

So:
b = 0.234234234234234...
I'm just assigning b here. Nothing magical - strictly mundane stuff.

Well, that also means:
1000*b= 234.234234234234...
Again - I'm just using the value I've assigned to b.

The magic starts to happen after this - see, I've now got a system of equations
b = 0.234234234234234...
1000*b= 234.234234234234...

If both sides are equal, we can subtract one from the other - lets subtract the first line from the second:
(1000*b)-b = 234.234234234234... - 0.234234234234234...
->
999*b = 234
Basically, I've just subtracted 1 from the multiplier on b, and canceled out everything after the first iteration (which I now have expressed as an integer).

Now we isolate b, by dividing both sides by 999:
b = 234/999

And now we have an integer over an integer. This works for basically any repeating decimal (you just choose 10^x, where x is the size of your iteration, rather than using 1000). All steps are nice and straightforward, no trickery beyond standard equation manipulations.

When you do that with .999999...:
b=0.99999...
->10*b=9.99999...
->(10*b)-b=9.99999...-0.99999...
->9*b=9
->b=9/9
->b=1
->0.9999....=1; QED.

[Guildorn Tanaleth]

I would like to add my voice to Kurald in saying that, in the end, math is pretty arbitrary, and it's only how it is because of convenience.

Correction: Conventional Mathematics is the way it is in order to reflect our perceived reality. There are plenty of branches of Math dealing with theoretical number systems and other things based on different axioms, so you can choose almost any set of (internally consistent) conventions & still be doing Math.

[Ka'ladun]

Here's the most thorough proof I've seen.

0.(9) can be represented of the sum of 9*.1^n, where n = 1 to infinity, which comes out to .9 + .09 + .009 + and so on. Now, this particular arrangement is a geometric series. The sum of a geometric series is a/(1-r), where a is the first term in the series, and r is the exponential factor of the series, in this case .1. Plugging in these variables yields .9/(1-.1) = .9/.9 = 1.

[averagejoe]

Correction: Conventional Mathematics is the way it is in order to reflect our perceived reality. There are plenty of branches of Math dealing with theoretical number systems and other things based on different axioms, so you can choose almost any set of (internally consistent) conventions & still be doing Math.

Okay, fine, convenience or human interest. My point was that whatever rules are there for math are basically because someone chose to use those rules. I was just assuming that we were referring to conventional math, or there is no way of determining the truth of the statement .(9)=1 without different rules being presented.

[Prophaniti]

Physics theories are different from math theories. One is based on measurable, quantifiable, objective fact and the other is based on arbitrary conventions.

Sorry, but isn't math kinda the poster child for measurable, quantifiable, objective fact? Seriously, every advanced science keeps coming back to math. It's like the universal science that everything answers to. Damn math.

[Dan_Hemmens]

I'm referring to the general method of converting infinite repeating decimals to fractions, actually.

I don't think I'd seen it explicitly actually.

Cool.

[averagejoe]

Sorry, but isn't math kinda the poster child for measurable, quantifiable, objective fact? Seriously, every advanced science keeps coming back to math. It's like the universal science that everything answers to. Damn math.

No. Mathematics is not, in and of itself, measurable or quantifiable (though it is fairly objective). It's an interesting coincidence that science so often can be represented by math, and there is a lot that you can use math for in science, but in the end the two are entirely different animals. In science you can't really call something true until it has been observed. In math you can call something true when you've made a convincing proof. In other words, science says, "Let's try something and see what happens." Math says, "Let's assume these things, and see what logically pops out." Math is inherently philosophical, but with a definite structure and specific rules. If you cannot take something and say, "Look, this is true because we're seeing it," then it really isn't science.

[Jack_Simth]

I don't think I'd seen it explicitly actually.

Cool.

Mind you, once you've seen it once, you can short-cut it as much as you like:

0.1234123412341234.... = 1234/9999
0.57893456789578934567895789345678957893456789... = 57893456789 / 99999999999
0.11111111....=1/9

... and so on, because you as a human you can look-ahead, so to speak, and recognize the pattern.

[Emperor Tippy]

Sorry, but isn't math kinda the poster child for measurable, quantifiable, objective fact? Seriously, every advanced science keeps coming back to math. It's like the universal science that everything answers to. Damn math.

No, math isn't objective fact. It uses arbitrary postulates in a lot of the higher end stuff.

EDIT: It is objective in that if two people start off with the same base premise they will always get the same answer. It's not objective in that it doesn't exist independent of humans.
END EDIT


Are those arbitrary postulates useful and even necessary? Yes. But they are still arbitrary.

I understand full well all of the proofs of 0.(9)=1, I even understand the applications of 0.(9)=1 and why it is used.

I don't even disagree with its use in said math.

[averagejoe]

*snip*

I don't get it then. In what way is it not true?

[Jack_Simth]

I understand full well all of the proofs of 0.(9)=1, I even understand the applications of 0.(9)=1 and why it is used.

I don't even disagree with its use in said math.

...
Just to make sure I have this straight:

1) You agree with the proofs, that basically just rely on the number system as defined, and fairly basic mathematical manipulation.
2) You disagree with their conclusions despite (1), because everything involved is strictly arbitrary.

If both (1) and (2), there is no point in further debate, because you cannot be convinced. This is at the point of 2+2=5, because the meaning of the symbols 2, +, =, and 5 are arbitrary.

[Prophaniti]

Isn't at least basic math pretty much universal? I mean, even if you called the numbers something else, when you have two *something* and you gain two more, you have four *something* now. I don't know much advanced math, but it's all based on the same basic principles, most of which are not only irrefutable but demonstratable, and definitely not dependant on humans to exist. Sure, nothing else would be around to appreciate or study this phenomenon of doubling, but it would still happen.

Does this basis in observable reality really decline that much in high level math? Math has always struck me as more of an explanation of science than anything else. Example: We see that objects fall to the earth unless acted upon. Math explains how, why, how fast and far, ect. All of which can be verified by observing falling things.

Somebody tell me if I'm way off base here, before I further demonstrate how utterly ignorant I am about mathamatical theory...

[averagejoe]

Isn't at least basic math pretty much universal? I mean, even if you called the numbers something else, when you have two *something* and you gain two more, you have four *something* now. I don't know much advanced math, but it's all based on the same basic principles, most of which are not only irrefutable but demonstratable, and definitely not dependant on humans to exist. Sure, nothing else would be around to appreciate or study this phenomenon of doubling, but it would still happen.

Does this basis in observable reality really decline that much in high level math? Math has always struck me as more of an explanation of science than anything else. Example: We see that objects fall to the earth unless acted upon. Math explains how, why, how fast and far, ect. All of which can be verified by observing falling things.

Somebody tell me if I'm way off base here, before I further demonstrate how utterly ignorant I am about mathamatical theory...

Well, I can't answer for the underlying philosophy (that is, your first paragraph), but higher level math basically doesn't have any basis in the observable world, or, at least, is meant to generalize to things that aren't, for example n-dimensional space (or even infinite dimensional space.) Now it turns out that some of the weird things, like using functions as vector spaces, actually has use in quantum mechanics, which is part of what makes QM so weird. However, there are other things that don't have any analog in the real world than we imagine. Unfortunately these tend to be hard to explain, and some of these may even have application in unknown sciences. The thing is, we really don't know how closely math and science are connected, since we don't know enough about the universe to say. However, I will say that, for the most part, math and science evolved separately, and in fact it came as a surprise to many mathematicians that their field had any application beyond arithmetic and basic algebra.

[kirbsys]

I had to do this for my freshman algebra two. Basically my arguement was that since .333R was the same as 1/3 and .333R times 3 was .999R and 1/3 times 3 was 1, then .999R was equal to 1.

How does the 1+1=3 thing work out?

[Emperor Tippy]

...
Just to make sure I have this straight:

1) You agree with the proofs, that basically just rely on the number system as defined, and fairly basic mathematical manipulation.
2) You disagree with their conclusions despite (1), because everything involved is strictly arbitrary.

If both (1) and (2), there is no point in further debate, because you cannot be convinced. This is at the point of 2+2=5, because the meaning of the symbols 2, +, =, and 5 are arbitrary.

No. 2+2=4 is true in most math and in the real world. 0.(9)=1 is only true in math.

If you have '1' apple and then gain another identical apple you have '2' apples. Replace '1' and '2' with whatever random designation you feel like, it doesn't change the objective fact that 2 is twice the amount of 1.

0.(9)=1 isn't an objective fact or derived from objective fact.

[Jack_Simth]

I had to do this for my freshman algebra two. Basically my arguement was that since .333R was the same as 1/3 and .333R times 3 was .999R and 1/3 times 3 was 1, then .999R was equal to 1.

How does the 1+1=3 thing work out?

Well, I suppose I could build one as an extension of the "proof" that 1=0 (which can thereafter be used to "prove" that any number equals any other number).

In general, though, proofs of that type tend to point out "holes" in the system used (1=0 proof generally makes use of canceling infinities, for instance - demonstrating that either there's some kind of order to infinities, or that infinity - infinity is undefined, or similar).

No. 2+2=4 is true in most math and in the real world. 0.(9)=1 is only true in math.

If you have '1' apple and then gain another identical apple you have '2' apples. Replace '1' and '2' with whatever random designation you feel like, it doesn't change the objective fact that 2 is twice the amount of 1.

0.(9)=1 isn't an objective fact or derived from objective fact.

Pour me 0.(9) cups of water. Then measure the difference between that and 1 cup of water. 0.(9) doesn't have a meaning outside of math anyway.

Or, alternately, 2 + 2 = 5, 2 + 2 != 4. I'm just not using the meaning of the symbol set [2, +, =, 5, !=, 4] that you're used to. In order to distinguish the "value" of 0.(9) and 1, you need to have some form of symbol set and symbol manipulation rules in use. In order to discuss the validity of 0.(9)=1, you also need to have some symbol set and symbol manipulation ruleset in use. What rules and symbol-set are you using such that 0.(9)=1 is not true?

You did not disagree with (1), and you did not disagree with (2). The disconnect is at the level of symbol-meaning, which fundamentally cannot be argued outside of empirical testing. Pour me 0.(9) cups of water, pour me 1 cup of water, then measure the difference between them. What is it?