the cult of reasoning
ultralifter, here is what I said about it.
Fractions, like 1/3, are not useful (except in estimating and rounding, and for exact divisions of solid, limited numbers) unless they’re used to describe quantities of physical objects, or tangible realities.
1/3 of the pie. 3/4 of 12. Half of the rent.
NOT for describing .9rep
That’s lovely. It did not answer my question at all.
Do you or do you not accept that 1/2 = 2/4? Do you or do you not accept that 1/3 = 3/9?
If you said “yes” to both of those, you have no ground to stand on, because you’ve already admitted that one number can have multiple representations. If you say “no”, you’ve already rejected 5th-grade math, and you’ve got no ground to stand on.
Math should not limit itself to what is useful, or what can describe the real world (but that’s a subject for a great debate).
okay, you know what? Maybe you SHOULD reread these posts, because I never said that there couldn’t be two forms of one number, or three, or infinite others. That’s kinoons opinion. I don’t agree with it. So stop accusing me of these things.
My question was directed at kinoons. Still, I haven’t accused you of anything. All of what I said was based on which answer you gave the question “Is 1/2 = 2/4?” If you felt that was an attack on you or a misrepresentation of what you said, then I apologize.
Look, this discussion is getting long. I have one more question for the anti-equalitarians: What would be sufficient to convince you that you’re wrong?
I realize this is difficult to answer, so I’ll offer up what would work against me. My currency is mathematical proof; in this particularly, proofs drawn from analysis. So if someone can offer up a mathematical proof that .9… != 1, I’ll eat my words.
Let me take a moment to remind you that mathematical proofs do not involve analogies or rhetorical devices. A proof of this statement will be of the following form: The axioms of set theory imply A, A implies B, and so on and so forth, until we reach A[sub]n[/sub], which implies that .9… is not equal to 1. Of course, you would have to cite whatever theorems you use; I’m not willing to accept anything on anyone’s say-so.
There you have it. Now it’s your turn. What would convince you that you are wrong?
I’d do it for a cheeseburger, ultrafilter. But I’m still pondering ‘infinite limit’, which seems contradictory to me.
Maybe I’ll understand when I read MrDeath’s “Axiom of infinity” thread. Haven’t been able to find it, though.
Peace,
mangeorge
I’ve started “my” MPSIMS thread. I’m going a bit further afield than just the Axiom of Infinity; I’m going to try to draw the line between what is math and what isn’t. It’ll take a few days, but I invite you all to go over the first post and start asking me questions on it.
alright, here we go…
I accept your proofs as being mathematically true, I’m not going to argue that. You can sucessfully manipulate the numbers to make 0.999… = 1. At the same time there are other elementary laws of math that seem to be violated by those proofs.
as far as 2/4 and 1/2 – they are the same value, but not the same number. Those fractions are two different representations of the division operation to get to the answer of 0.5. You have to perform the operation to get to the same number.
Where I have the disconnect is how the number 1.0 (more 0’s are not necessary, they are insignificant and have no value) can be equal to 0.999… when the simple law of place value says (0.anything) must be less than 1.0
Also, on a number line, 0.999… implies that you can get infinitely close to 1.0, but the lack of anything beside a 0 in front of the decimal means you never get to one, your always just a little short, even if it is infinitely small, its still not quite one.
That’s what I’ve been trying to say, kinoons. Close, but not quite.
I think they’re fudgeing the facts a little. They seem to claim that infinity ends without stopping. Works ok for math, but it plays hell with my poetic sensibilities.
I’m going to see what MrDeath has to say.
Peace,
mangeorge
If we can derive that .9… = 1 and that .9… != 1, we have successfully shown that we can prove any statement we like. This is called inconsistency, and it represents a major problem for any formal theory. I’ll prove this in another post.
What you said is of ultimate importance, though–the other laws of math only seem to be violated. I am entirely sympathetic for you–a lot of higher math is counterintuitive and can take quite some time to understand.
As Edmund Landau (author of a classic work in the field of analysis (the field which deals with limits, and series, and derivatives, and integrals, and all that)) once said, “Please forget what you have learned in school; you haven’t really learned it.”
Unfortunately, the law of place value isn’t that simple. It really only applies to decimal representations where only a finite number of terms are “unique”. By unique, I mean that they’re not all zero, and there is no pattern.
Once you start with an infinite decimal sequence, all bets are off.
As has been demonstrated, .9… = 1. Once you accept the mathematical proofs, you have to accept the conclusion. Otherwise, you’re positing that mathematics is inherently contradictory. This would invalidate most of human knowledge, so you can see why we’re a little leary of accepting what you say.
I love mathematical hair-splitting. The answer, as explained in the book ‘Where Mathematics Comes From’ by George Lakoff and Rafael E. Nunez, is that in the set of real numbers, or the real number system, there is no difference between 0.9999… and 1.0000… (The reals include such classics as pi, e, and the square root of two, in addition to the rationals (numbers you can represent as finite fractions) and the naturals (the positive integers)). In fact, in his famous diagonal proof, Cantor used 0.9999… as equal to 1.0000… and it was valid, because he was working with the reals. However, in the hyperreals, a completely different number system, there exist such things as infintesimals, represented by a symbol I can’t reproduce here, such that I + I = I, or infintesimal plus infintesimal equals infintesimal. In other words, you cannot add infintesimals and end up with a number. I (as opposed to i, the imaginary number) does not exist in the reals, but in the hyperreals it is the difference between 0.9999… and 1.0000… So, to summarize, it all depends on which number system you are in. Interesting, huh?
I’m not entirely convinced and I don’t see the fabric of the universe falling apart around me 
Seriously though, how important is the idea that 0.999… is equal to one?
I take that back. I prove it in this thread.
It’s not particularly important unless you’re a mathematician. What is important is the notion that any statement with a proof must be true. While the universe wouldn’t fall apart if this were found to be false, our understanding of it would be set back quite a bit.
FWIW, not all true statements have proofs; this was the subject of Godel’s famed incompleteness proof.
a calc one education taken 2 years ago and never used since is a dangerous thing…
I recall (this is going to be really rusty, so hopefully I can get it close enough so you guys have some idea of what I am talking about) that in the process of performing quadratic equations you often came up with answers that were irrational and were thrown out. How does that idea, that the answer does not make sense so it is rejected, applied here?
You don’t believe that any number besides 1 can equal one? What are 3/3, 2/2, 2-1, and 2*.5 equal to?
There’s been some talk about not accepting that an infinite convergent series has a definite sum- that’s bullcrap. Look at volumes with riemann sums:
The volume of a cube 2 units wide is 8, we know that.
And to prove it with calc:
A(x) = 2^2 = 4
integration from 0 to 2 of 4… the indefinite integral
of 4 is 2x^2 so the integration with these bounds is 8-0=8. So the area is 8. But when we consider that the integral can also be expressed as a sum, we’ve got that an infinite series evaluates to 8. Now, if the cube had an length of 1, with your logic applied to riemann sums, the volume of the cube would be .9999rep, which is different from one.
What I’m trying to say is if you reject that .999rep is equal to one, as you can’t “put your finger” on something with so many decimals, then you also can’t put your finger on any volume and therefore basically everything we know about three dimensional objects can be thrown out- volumes aren’t absolute as your riemann sums don’t have an exact volume.
If you reject that .999rep = 1 based on the fact you can’t “define” .999rep on a number line (which you can, it’s at 1), then you reject the entire concept of volumes, and all of my studying of integral calculus has been useless. You really don’t want to tell me that (especially if I’m within attacking distance)!
I’d like to nominate ultrafilter for a silver star in the fight against ignorance.
IMHO, this doesn’t belong in GD unless the anti-equalitarians want to argue the acceptance of the Axiom of Completeness, which I don’t see anyone doing.
Do you realize that the notation 0.5 is just a way of writing 5/10? I mean, we’re not talking values, quantities or numbers here, just simple notation.
Kind of like eins and uno are different ways of saying “one”. 1/2, 0.5, 2/4 and one-half are different ways of identifying , well, one half.
2/2, 1, 0.999…, and cos(0) are different ways of identifying one. I think you have trouble with the concept of equality.
[QUOTE]
*Originally posted by bugg *
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Which answers kinoons earlier question about why this is important. If you deny that .999rep=1 then you jerk a very large block out of the foundation of Calculus.
I guess I wouldn’t say that would take us back to the stone age, but it’s a pretty big deal (whether you appreciate it or not is a matter of mathematical education).
Assuming you started out with a quadratic equation, you wouldn’t throw out any answers unless they don’t make sense in the physical context of the problem. Mathematically, any answer would satisfy the original equation, and would therefore be a valid solution. However, mathematical and physical reality are often not the same thing (as I believe we’ve demonstrated here ;)).
However, you might be given the stipulation that x has to be a rational number. Then would you have to throw out irrational roots. Going into much more detail about that would require a detailed explanation about functions, domains, and all that. I’d be happy to provide it if you want to hear it, but if you don’t, I’d be equally happy not typing it up.
I have a little trouble accepting this, but my concerns are very technical, and I need to read more about the hyperreals before I say anything. If anyone wants to hear and address my concerns, please e-mail me.