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0.999...= 1

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Re: 0.999...= 1

#461

Post by ristos »

karatinversion wrote: post #136
ristos wrote: post #112

I don't think that 0.999... = 1 is actually provable. I think this and all of calculus is actually axiomatic, which has the following axiom: Given ε = 1/∞ then: ε = 0 Am I wrong in thinking this way? It seems as though there's no way to actually truly prove that an infinite series converging towards zero actually hits zero (from a constructivist pov)

That's not an axiom we take; the study of real numbers takes some algebraic and order axioms, and a completeness axiom: Any non-empty set of real numbers with an upper bound has a least upper bound. We can't prove your statement Given ε = 1/∞ then: ε = 0 because it is not well-defined, but we can prove this: If ε ≥ 0 and, for every natural number n, ε For suppose there exists an ε which is a counter-example, i.e. ε >…

Why is your proof for every natural number n and not infinity (ω)? also isn't the law of excluded middle axiomatic?

Re: 0.999...= 1

#462

Post by DavidVoid »

_bxg1 wrote: post #204
Okx wrote: post #182

Earlier quoted context omitted.

I usually say "If and only if two numbers are different, then you can find a number between them". People often accept this axiom. Then, I offer them to find a number between 0.999... and 1.

Why not 0.00...1?

Because unlike 0.99..., 0.00...1 is not a Real number.

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.

Re: 0.999...= 1

#463

Post by DavidVoid »

tom-thistime wrote: post #205
_bxg1 wrote: post #204

Earlier quoted context omitted.

Why not 0.00...1?

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.

Re: 0.999...= 1

#464

Post by BurningFrog »

lmkg wrote: post #137
undecisive wrote: post #115

There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…

It's more fundamental: People seem to have the intuition that the decimal representation of a number is a number . I don't know if it's because decimals resemble the ntural numbers or what, but decimals seem to have a primacy for people that fractions do not. The idea that there's a gap between the symbol for a thing and the thing itself is the stumbling block.

Similarly, perhaps, many take for granted that the written version of a word, in current standard spelling is the word.

Re: 0.999...= 1

#465

Post by ristos »

jmull wrote: post #127
ristos wrote: post #112

I don't think that 0.999... = 1 is actually provable. I think this and all of calculus is actually axiomatic, which has the following axiom: Given ε = 1/∞ then: ε = 0 Am I wrong in thinking this way? It seems as though there's no way to actually truly prove that an infinite series converging towards zero actually hits zero (from a constructivist pov)

Well, that wiki page has several proofs using various approaches. There's also a section which addresses common objections. I think you'd have to dismiss them all to make your claim (to do it well, that is).

Afaict all of these rely on the axiom of Archimedes which is what I'm talking about, some of the proofs explicitly stating it and others implicitly. Unless I missed something.

Re: 0.999...= 1

#467

Post by Gehinnn »

TuringTest wrote: post #294
lukifer wrote: post #228

Earlier quoted context omitted.

I didn't grok infinity until I started thinking in terms of verbs rather than nouns. As a static number, the concept of infinity makes no sense; but once reimagined as a process (start counting up from 1, and never stop), all apparent paradoxes disappear. This is the inverse problem: it could just as easily be reframed as 0.000...0001 = 0. Defined as static nouns (does such a thing exist in nature?), it's seemingly p…

I find that "infinite as an endless process" concept intuitively very heplful as well. However, reading Gödel, Escher, Bach [1] showed me that there's another, more static logical interpretation of infinity which also comes handy. In an infinite process, you can always take "one more step" to create the item after that. Let's assume there exists a "final" mathematical object that goes after every finite item in the g…

This thought can be made much more concrete when talking about sets. Clearly we can understand a collection of things as a set. Clearly numbers are things, so we can talk about the set of all numbers. But how many elements are in this set? A clever answer could be: It has as many elements as there are natural numbers! As subsets are also a thing, we could ask next how many subsets the set of all numbers has. A clever, but very problematic answer could be "as many as there are natural numbers!"

Re: 0.999...= 1

#468

Post by tom-thistime »

0xff00ffee wrote: post #444
tom-thistime wrote: post #423

Earlier quoted context omitted.

Thanks to a commenter who pointed out that my sum above should be sum(n=1 ... N)(9*10^-n). I can't, uh ... fully endorse that comment, which is not entirely accurate and doesn't answer my question. But I sure did miss the '9'.

You asked for a "law" of equality (whatever that is) and provided an answer that proves it converges to 1. What more could you possibly want?

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.

Re: 0.999...= 1

#469

Post by foobuzzHN »

mjd wrote: post #280
lmkg wrote: post #137

Earlier quoted context omitted.

It's more fundamental: People seem to have the intuition that the decimal representation of a number is a number . I don't know if it's because decimals resemble the ntural numbers or what, but decimals seem to have a primacy for people that fractions do not. The idea that there's a gap between the symbol for a thing and the thing itself is the stumbling block.

I think this is a very insightful remark. People think that numerals _are_ numbers, and it's hard to explain why this is not the case, because we have no way to talk about specific numbers _except_ by using numerals. But many frequently-asked questions are based in a confusion between numbers and numerals. For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that a…

> A particularly interesting example I remember is the guy who asked what were the digits that made up the number ∞. It's a number, so it must have digits, right?

I don't think there are many areas of math where ∞ is a number. In my experience people have a whole other problem with ∞, thinking that it is some sort of huge concept defined globally in math, where it is just a notation shared by various non-mystical definitions across subjects (e.g. bijection-based definition of infinite set, epsilon-based definition of convergence, etc)

Re: 0.999...= 1

#470

Post by gspr »

brutt wrote: post #432
gspr wrote: post #428

Earlier quoted context omitted.

That's not a definition. Neither 10^∞ nor 1/∞ is defined in any standard system, so you'll have to define those too if you want to use them to define 0.000…1.

See https://en.wikipedia.org/wiki/Surreal_number .

You're working with surreal numbers? This is not what people would expect unless it's explicitly stated. In addition, you're likely going to have a hard time explaining surreal numbers to someone who struggles to grasp that 0.999... = 1 in the ordinary reals.
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