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0.999...= 1

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Re: 0.999...= 1

#601

Post by andrewprock »

DavidVoid wrote: post #463
tom-thistime wrote: post #205

Earlier quoted context omitted.

Then the question is, is 0.00....1 equal to zero? We use the definition of equality above and we say yes. EDIT: The number above seems well defined. It's lim n->inf (10^-n). That's zero.

It is not. The decimal representation of a Real number has to be indexed by Natural numbers, i.e. every decimal digit[n] has a well-defined index n which is a Natural number. Infinity is not a Natural number, so 0.00...1 is not a Real number either.

This is usually described mathematically by saying that the representation of a decimal has to be countable, whereas the number 0.000...1 is not a countable representation.

I will say though that if your explanation of 0.999... = 1.0 requires that you explain the distinction between countable and uncountable infinities, that's a big ask for most lay people.

Re: 0.999...= 1

#603

Post by tom-thistime »

pasquinelli wrote: post #356
tom-thistime wrote: post #190

Earlier quoted context omitted.

Possibly people are looking at two different symbols and asking "can you show me logically why those are equal." If they're given a definition of "equal" and they still object, that's a different problem. I have this problem every time I play with group theory again. You get the axioms for a group, which say there is some identity but don't explicity require the identity to be unique. You can easily prove that the id…

I think a lot of people don't think of math in terms of definitions and proof. Math was just something they were taught as kids. And even if they've gotten into more advanced math, i think the 1 = .9... question hits their kindergarten brain and they just say "no" to it the same way they'd say "no" to someone singing the alphabet song in the wrong order.

I imagine that's true sometimes.

On the other hand, hypothetically, if it were the case that we were missing a key definition that's needed for some proof, and someone didn't believe that proof, then maybe we would't quite know enough yet to decide that the person is in mental kindergarten. A better first step for us might be to supply the missing definition.

Re: 0.999...= 1

#604

Post by jl2718 »

dvt wrote: post #552
jturpin wrote: post #528

Earlier quoted context omitted.

I do think it is technically wrong to call it dogma - the decimal is a geometric series with a limit, right? And limits have an unambiguous definition, it's the smallest value that the series approaches but never exceeds as it tends to infinity. I think the part that is admittedly weird is that the notation "0.999..." refers to the limit as the series tends to infinity, and it kind of hides that fact from you. Even j…

> the decimal is a geometric series with a limit, right.. Right, but that's not really the crux of the matter. Hint: look at how the supremum is defined [1]. The definition of the supremum is how we end up with 0.999... = 1. [1] https://math.stackexchange.com/questions/1977204/limit-of-mo...

I suppose my point is that you could turn the repeating decimal into an infinite series, and a student might accept that, and you could define the suprememum and they might agree that it is 1, but then you ask them if the series is equal to the supremum, so they don't know what to do with the series, so they turn it into a sequence. Now they ask whether the last item in the sequence is equal to the supremum. Of course not! This is by definition.

And now you realize that you and the student have been operating by different rules. Their rules of equality are based on symbolic equality, so you actually have to relax the rules a bit to make limit equality work. And then, more importantly, you have to show that all the other rules are still intact. Actually, in this case, they aren't. Symbolic equality involving infinity is now horribly broken, and you have to express all equality in terms of limits to maintain consistency. Explore this further and you keep finding more inconsistencies that have to be settled by new rules that define new areas of mathematics.

So who is right? The natural world appears to be much more permissive than limit equality, preferring epsilon-equality. Symbolic equality is the only purely self-consistent system, but you can't do much with it. It's also possible that the natural world works with symbolic rules (quantum) but the complexity is great enough to resemble epsilon equality (continuum).

So, .999... == 1 by tautology. It's not some brilliant mathematical insight. The interesting part is the consequence of defining it as so.

Re: 0.999...= 1

#605

Post by [deleted] »

dvt wrote: post #552
jturpin wrote: post #528

Earlier quoted context omitted.

I do think it is technically wrong to call it dogma - the decimal is a geometric series with a limit, right? And limits have an unambiguous definition, it's the smallest value that the series approaches but never exceeds as it tends to infinity. I think the part that is admittedly weird is that the notation "0.999..." refers to the limit as the series tends to infinity, and it kind of hides that fact from you. Even j…

> the decimal is a geometric series with a limit, right.. Right, but that's not really the crux of the matter. Hint: look at how the supremum is defined [1]. The definition of the supremum is how we end up with 0.999... = 1. [1] https://math.stackexchange.com/questions/1977204/limit-of-mo...

[deleted]

Re: 0.999...= 1

#606

Post by tom-thistime »

andrewprock wrote: post #601
DavidVoid wrote: post #463

Earlier quoted context omitted.

It is not. The decimal representation of a Real number has to be indexed by Natural numbers, i.e. every decimal digit[n] has a well-defined index n which is a Natural number. Infinity is not a Natural number, so 0.00...1 is not a Real number either.

This is usually described mathematically by saying that the representation of a decimal has to be countable, whereas the number 0.000...1 is not a countable representation. I will say though that if your explanation of 0.999... = 1.0 requires that you explain the distinction between countable and uncountable infinities, that's a big ask for most lay people.

Could you please explain a little more what "not a countable representation" means?

Re: 0.999...= 1

#607

Post by aw1621107 »

tom-thistime wrote: post #606
andrewprock wrote: post #601

Earlier quoted context omitted.

This is usually described mathematically by saying that the representation of a decimal has to be countable, whereas the number 0.000...1 is not a countable representation. I will say though that if your explanation of 0.999... = 1.0 requires that you explain the distinction between countable and uncountable infinities, that's a big ask for most lay people.

Could you please explain a little more what "not a countable representation" means?

I think it's just another way to word DavidVoid's explanation a few comments up.

Re: 0.999...= 1

#608

Post by hota_mazi »

effie wrote: post #579
hota_mazi wrote: post #247

Earlier quoted context omitted.

Not only can numbers have more than one representation, but they can also have zero! Looking at you, irrational numbers.

Irrationals have a unique decimal representation in the mathematical sense: given a definition such as $x^2 = 2 $, any digit of the decimal expansion of $x$ can be determined.

> Irrationals have a unique decimal representation in the mathematical sense

Not all of them do. Actually, so many don't that mathematically, the number of them that do is zero.

Sure, there are exceptions like sqrt(2) and sqrt(3), but there is an uncountable infinity of irrationals between these two numbers that just don't have a representation.

Re: 0.999...= 1

#609

Post by tom-thistime »

esmi wrote: post #559
tom-thistime wrote: post #468

Earlier quoted context omitted.

We seem to agree on this: you don't think there's any need for a way to determine whether two real numbers are equal. For ordinary math, though, using some criterion for equality (for example x>=y and y>=x) is basic and not controversial. So it seems unconvincing (to me) when you seem to imply the opposite.

There is an easy way to prove two numbers are equal. Typically in the reals there are three possibilities: a > b, a b and a < b then you are left to conclude a = b. And this is exactly what is done in Apostol's Calculus Vol 1 (IMHO the greatest calc book ever written) chapter 1 when he proves that the area under n^2 is EXACTLY (n^3)/3, with no "calculus". You would be shocked how far into calculus the author gets wit…

Thanks, I'll take a look. I like that kind of thing very much.

I use applied math. I haven't taken a class in real analysis. But it's fun how often grinding out the solution to a "real world," practical PDE turns out not to actually be the nicest (simplest and/or clearest and/or sufficiently insight-producing) way to understand the (hopefully) corresponding physical problem in the lab.

Stripping off the "calculus" and replacing it by limits sometimes seems to help highlight alternate perspectives that the magic "integrals" and "derivatives" kind of conceal.

Even when it's not more effective, it's definitely more fun.

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