Don
This will though:
x = 0.9r
10x = 9.9r
9x = 9 (take away x = 0.9r)
x = 1
Failing that, how do you define 1.
Jokester
Poll: Does 0.99 Recurring = 1
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This will though:
x = 0.9r
10x = 9.9r
9x = 9 (take away x = 0.9r)
x = 1
Failing that, how do you define 1.
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Hmm.. I'm guessing by "infinitely detailed" you mean that there is no finite decimal representation of the numerical value 1/3, in base 10. With this I agree. In base 10 we have a finite representation of 1/10, namely 0.1.
Is there a finite decimal representation of 1/10 in base 3?
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Soldato
Cannot be exactly evaluated, but can easily be defined.
"I define 0.9r to be :
Sum over i from 1 to infinity of 0.9x(0.1)^i"
There, I've defined it. No problem with that "hurdle" in your argument.
A countably infinite number, of which there is a 1-1 correspondence with the Natural Numbers.
Every Cambridge degree (even Theoretical Physics) is a BA, its tradition.
You don't kknow enough about maths. I've defined what 0.9r is using standard notations within maths, which does not produce a contradiction elsewhere, so therefore 0.9r exists in maths. If you know something about the contruction of The Field of Real Numbers I don't, I'd be interested to hear why my definition of 0.9r doesn't make it a number?
Takle W11tho's example of "Sin (pi)". Sin is an infinite power polynomial. Pi has an infinitely long, non-repeating decimal expansion. So sin(pi) is an infinite number of terms of an infinitely long decimal expansion. Guess what sin(pi) is? 0. Bang on zero. You can see this just by looking at the side of a triangle as sin goes to pi. Its exactly zero. Why? Because the definition of pi (which involves using infinite expansions if you wish to evaluate it in decimal form) says its zero. I don't need to know the decimal expansion of pi to know sin(pi) = 0, or cos(2pi) = 1, its exact, because I know the definitions of sin and cos.
By that logic the "=" sign is pointless, because the things on either side of the equation are not the same, and if they were they'd be written the same.
x^2 - 1 = (x+1)(x-1)
Thats true for all x, But by your logic, they aren't, or why would we bother writing one as the other? 0.9r = 1 is a more complicated entity, but its the same principle.
So without using maths, we're to prove something which is a mathematical entity true? Thats like walking into a French oral exam and saying "Without using French, you are to answer all the questions". Kind of stupid isn't it?
That would be pi.
I understand exactly what your getting at, and had made similar points several times, only to get similarly "missed".
Maybe when you said certain numbers couldn't be defined, you should have said they couldn't be exactly evaluated, or couldn't be quantified, rather than "defined". Pi, in my view, can be defined but not accurately and fully quantified. Ditto infinity. You can use an expression to represent it, but can't give an accurate quantification of it, so maths has systemic assumptions built-in on how to handle such situations, and the 'proofs' given use that system so the validity of such proofs is implicit - given the assumptions about such representation and symbology.
I gave up putting this point, Lee, as our board mathematicans seemingly either can't or won't address that point.
Soldato
Of course such numbers are unevaluatable (is that a word?), thats part of their nature, being unrepeating infinite expansions, they contain an infinite amount of "information", which is not physically possible to exist.
Lee's comment was that they are undefinable, a very different thing indeed.
Such is the nature of any such system. Take a dictionary for instance. You look up the definition of a word, and are given it in more words. If you don't understand them, you look up them, only to be given more and more words. The cycle is never ending, unless you know the meaning of a few base words and built yourself up from there. With maths you "define" your few choice "words" (axioms) and move on.
The proofs are only as valid as your assumptions, but in a self contained system, what more do you need? You can discuss the philsophical ideas of the universe as much as people like, but within maths, proofs exist for itself. You can't use maths to prove anything outside itself, but within itself, its structure allows such things.
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It's not a point. I don't mean to sound patronising - but you guys really don't seem to know what you are talking about.
Root(2) is defined as the solution to the equation:
x² - 2 = 0
1/3 = 0.333333..... is defined as the solution to the equation:
3x - 1 = 0
Just like a word is defined in the dictionary. You don't use the definition of the word each time you use it, you just use the word! If we worked in a number system based on Pi, then we could easily represent Pi as 1 or something. People aren't avoiding your argument - it simply doesn't have any brunt to it. This is a maths question, and we have given a maths answer. I don't know that much about Number Fields, but I know it is one extremely big and complicated subject. Some people should be willing to accept facts from those with more knowledge on a subject.
lol, it's nice to see someone who understands my view on this, even though i didn't explain myself brilliantly.
What you just said about numbers not being able to be evaluated in base10, was what i was trying to get at with saying undefinable. Guess i used the wrong word.
In a nutshell though, it seems the rules of maths are so deeply entrenched in some peoples minds that they can't step back and use some common sense.
It doesn't matter how you describe it, whether you say:
0.9 with more 9s than can be accounted for.
or
0.9 with 9s added for infinite time.
or
0.9 with all the 9s in the conceivable universe all being added at once at the speed of light with a whole bunch of 9s squeezed in at the same time.
you always have the same result:
a zero, followed by a dot, followed by 9s. I fail to see how anyone in their right mind with all the maths in the world can ever 'equate' this to 1.
Don
But common sense also suggest that three thirds is also one.
Jokester
Jokester
Soldato
Or rather we've spent so much time thinking about (and in) maths that we know that "common sense" has sweet FA to do with maths at times.
Both of those are to do with physics, time, space and light. Where does
" I define 1/3 to be the solution to 3x-1=0"
matter about time, space and light? Same with the definition of 0.9r. I know how it behaves (exactly like 1) so it exists in maths. Doesn't exist in reality, but then whats reality got to do about the constraints of maths?
Perhaps because the proofs I've posted about 4.45 trillion times say they are equal?
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I don't believe I saw this answered.
1/10 has no finite decimal representation in base 3. It is a recurring decimal:
0.00220022002200...
You quote me and then say that, without 'meaning to sound patronising'? What you've just said is, effectively, "you're talking rubbish, so don't contradict your betters". And that isn't patronising?
I said, right in my first post on this days ago and early on in the first thread on this, that you could use mathematics to prove that proposition, and that the proofs would be valid. I've repeated that assertion several times. I've also said that that isn't the only way you can look at it.
Alpha said
and I agree. Completely. The point I've made, over and over again, is that it is possible to look at this outside the closed system and that as soon as you do, you look at those assumptions.
Let me try one more time. Within maths, 0.9r=1 and it does because because of the assumptions made. Those assumptions are, within the system, valid because they work consistently, and have not been disproven. But, the closed system does make assumptions about what happens in special cases, if for no other reason than that those special cases are only evident at points where we cannot empirically demonstrate what happens, i.e. as we tend towards the limit at infinity. And we can't demonstrate it empirically because of the unquantifiable nature of infinity.
So, within the closed system, 0.9r = 1 and you will get, and never have got, any argument from me on that.
It's the other way of looking at it that I dispute and on that, the only answer I ever get is that we can look at it in terms of maths. Of course we can, but it avoids the point I (and, I suspect, several others) have been making. Only a complete plonker would assert otherwise than that you can look at it in those terms, and, in that context, I also said right up front that I wasn't going to argue with Alpha. I wouldn't even if I disagreed (and I don't) because I regard him as particularly bright and what's more, a specialist (albeit relatively early in his education) in a field I'm far from being a specialist in.
Some of the argument in this thread is coming from the fact that, intuitively, 0.9r=1 looks like it ought to be false and it's an easy trap to fall into. But the mathematical proofs, as explained, are trivial and self-evidently correct.
But there IS another way of looking at it, and that other way involves questioning the rather nebulous nature of infinity. I called this the "philosophical" way and all that I get from the mathematicans (including Alpha) is rather dismissive remarks about philosophy or, more specifically, philosophy of maths.
Am I prepared to accept "facts" from "people that know more about this that me. No. I'll hold a discussion and maybe be convinced that they're right and I'm wrong, but where I've an opinion I'll not just be told "I know better than you, so accept it". I expect someone saying that to be able to demonstrate it and so far, nobody has even tried.
Where did I say you couldn't define root 2? Or 1/3?
Actually, if you read what I said, I said exactly and specifically the opposite - that they COULD be defined. Defined, but not fully quantified.
And like any analogy, the dictionary analogy only stretches so far. Dictionaries are about linguistics, communication of concepts, about the interpretation of ideas and there is, by definition, a degree of imprecision in that process. I have never suggested that you need to use a definition of a word every time you use it, nor have I suggested, or even implied, that you need to prove everything from first principles or axioms, upwards, every time you use maths. You're getting my points confused with some other comments in the past of this thread. Once again - I don't dispute the mathematical proofs. What I do dispute is that maths is the ONLY way of looking at this. And on that point, you can agree or disagree, or decline to get involved, but please don't tell me to accept "facts" on it from my 'betters'. I have seen no evidence here that anyone is such on this subject.
I've said repeatedly that I respect Alpha for his maths ability and knowledge, but that doesn't mean that I either will just accept anything he says outside of that field or, for that matter, accept that he is ultimately aware of all that he could be inside that field. An undergraduate is, however bright and knowledgeable, an undergraduate and, again by definition, still learning. I'm sure Alpha would be the first to acknowledge that.
yes he is an undergrad and still learning, but people with phd's and field's medals and professor's of mathematics will all tell you 0.9r =1 and he has been taught by people of that stature
I was taught that in the first few weeks in an introductory course on the foundations of analysis by Prof Liebeck in my first year, and it is also in his book (introduction to pure mathematics).
if you can put some what of a formal arguement together which proves 0.9r isnt equal to one. I would be happy to try and relay it to a professor at imperial and get a response if you feel the maths undergrads on this board are incapable of a proper reply
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and to add no one has really put down a formal philsophical arguement down, unless I have missed it???
its kind of easy to keep saying philsophically the two aren't equal, but if you could define some philsophical axioms or frame work of what can and can't exist and deduce from them why the two values are not equal then alpha and the other people who think 0.9r=1 might listen and not just dismiss it, but so far no sight of that
its kind of easy to keep saying philsophically the two aren't equal, but if you could define some philsophical axioms or frame work of what can and can't exist and deduce from them why the two values are not equal then alpha and the other people who think 0.9r=1 might listen and not just dismiss it, but so far no sight of that
Soldato
sorry for butting in (and yes, 0.9r does equal 1), just noticed you said you were at imperial.
what course/year? im third year aero
Then it just seems you have a chip on your shoulder. Who would you accept it from? A PhD student, a Prof of Mathematics, a fields Medalist - who? And from my understanding, Alpha is about 7 months away from getting an MA in Mathematics - hardly early on in his education in Maths!
You keep saying things like "assumptions are, within the system, valid because they work consistently, and have not been disproven". It's not about them "working consistently", it's about them being true. And with regards to them being disproven, well how can you disprove something that has been proved, several ways beyond all doubt. All this "I'm looking at it from the outside" doesn't work because you are looking at things within a closed system. And with regards to "we cannot empirically demonstrate what happens, i.e. as we tend towards the limit at infinity", we use limits all the time - it is very basic Analysis. Things like Calculus work because we do know what happens in these limits. There is no 'mathematical wizardry', they are just things that are true within a closed system - we have proved them so.
However, I don't suppose any of this will make the slightest bit of difference to your opinion - so there isn't much point. The fact is, you are talking about things you don't know much about. Now you could take that comment on the chin, and say "fair enough - I don't know much about Number Field Theory", but I doubt you will.
I didn't mean to sound patronising earlier - but there was no other way I could get my point across. I am not trying to get your back up - honestly. This has moved from a discussion into an argument, and I didn't want that to happen. We will simply have to agree to disagree (i.e I say you can only look at this mathematically, you don't).
No hard feelings to anyone else on here that I might have riled - it wasn't my intention!
Hey - carvegio! Sorry to wander off topic, but I wanted to ask you a few questions about STEP. Which papers did you take, and how did you get on? Are some questions easier than others? I.e are the Mechanics ones easier than the Pure etc. What techniques did you use in them?
Thanks!
Soldato
I was told in maths just a few weeks ago about this. If you try and make it simpler first it can be proven quiet easily. First, work out what 0.11r is:
0.11r = 1/9th
Now figure out 0.22r
0.22r = 2/9th's
(cary on same sequence..)
0.33r = 3/9ths
0.44r = 4/9ths
0.55r = 5/9ths
0.66r = 6/9ths
0.77r = 7/9ths
0.88r = 8/9ths
0.99r = 9/9ths (or 1 whole one)
So im saying yes, 0.99r does equal to one.
0.11r = 1/9th
Now figure out 0.22r
0.22r = 2/9th's
(cary on same sequence..)
0.33r = 3/9ths
0.44r = 4/9ths
0.55r = 5/9ths
0.66r = 6/9ths
0.77r = 7/9ths
0.88r = 8/9ths
0.99r = 9/9ths (or 1 whole one)
So im saying yes, 0.99r does equal to one.
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