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0.999...= 1

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Re: 0.999...= 1

#121

Post by tom-thistime »

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…

(STATEMENT OF PERSONAL IGNORANCE [SOPI]: Anyone who actually understands this stuff please correct my mistakes below. Thanks.)

In the real numbers, which are not always simple or intuitive, 0.99... = 1. That's true and I seem to understand the proof.

But the real numbers aren't the only system that might be sitting behind "0.99..." and "1" when I write those symbols down and talk intuitively to people in my family. The reals are just the system we're taught first.

I believe there are other systems (I think the surreals are an example) that work just as well for everyday purposes, but where ( my understanding is that) there are numbers that differ from 1 by a value that approaches zero, yet those numbers are not equal to 1. (I've played with the surreals but only as a hobbyist.)

If you do calculus in these other numbers, I think physically meaningful problems will still yield the same answers. (For example Zeno's Paradoxes are still not an excuse for failure to attend school.) But it isn't a law of nature, I think, that all number systems that can hold 0.999... and 1 must make them equal.

Re: 0.999...= 1

#122

Post by qppo »

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)

I think the mistake in your thinking is that infinity is a value or reachable destination. It's a concept and it behaves differently than a number.

Also go read Generatingfunctionology and Concrete Mathematics

Re: 0.999...= 1

#123

Post by topaz0 »

vfinn wrote: post #62

Sorry for my naivety, but why one couldn't prove by induction that adding 9s never close the gap, or let's say, that by definition the operation is such, that it never closes the gap. If you can always halve the pie, then you can continue eating forever. To me it would be much easier to accept that (1/3)*3 is not 1.

By induction, you can prove that no _finite length_ sequence of 9s after the decimal reach one. It has nothing to say about the infinite limit, though.

Re: 0.999...= 1

#124

Post by CogitoCogito »

empath75 wrote: post #101
CogitoCogito wrote: post #96

Earlier quoted context omitted.

> Peano arithmetic only covers nonnegative whole numbers, so one will never exist. Thank you for the pedantism. How about I replace "Peano arithmetic" with the "operations of multiplication/addition/division/etc. expressible upon the rational numbers"?

It's not a rational number. Real numbers are defined as an equivalence class such that if the differences of two infinite sequences of rationals tend toward zero, then they are equal. The difference between 0.999... and 1.000... clearly tends towards zero as it heads of to infinity, and so they are equal. If you want to argue that it doesn't then you have to come up with some other definition for numbers which have a…

> It's not a rational number.

> Real numbers are defined as an equivalence class such that if the differences of two infinite sequences of rationals tend toward zero, then they are equal. The difference between 0.999... and 1.000... clearly tends towards zero as it heads of to infinity, and so they are equal.

> If you want to argue that it doesn't then you have to come up with some other definition for numbers which have an infinite decimal expansion.

> (Technically, of course, 1 is a rational number, but if you're using 0.9999... to represent it, you're using a real number representation, so you're bound by the definition)

I'm not sure why you think I don't know the difference between rational and real numbers, but I assure you I do. What I said was I don't see how a proof involving the standard arithmetic operations found within the rational numbers, but not including any concepts of limits, completeness, etc. is invalid. Let me know if you still don't understand my point.

Re: 0.999...= 1

#125

Post by SpicyLemonZest »

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)

A mathematician would quibble with your notation, but you're basically right. The fact that 0.999... = 1 depends on the fact that 0 is the only number smaller than 1/n for every integer n. This isn't exactly an arbitrary axiom, it encodes some of our natural intuitions about what a number is, but you can construct a system known as the "hyperreal numbers" where it isn't true and 0.999... doesn't converge.

Re: 0.999...= 1

#126

Post by WilliamEdward »

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…

This is essentially just a matter of limits, without which the world wouldn't make any sense.

So you must explain that if you move your hand closer to an object, technically you are halving the distance infinitely many times, but if 0.999... != 1 then your hand would never touch anything.

Re: 0.999...= 1

#127

Post by jmull »

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).

Re: 0.999...= 1

#128

Post by CogitoCogito »

tom-thistime wrote: post #106
CogitoCogito wrote: post #96

Earlier quoted context omitted.

> Peano arithmetic only covers nonnegative whole numbers, so one will never exist. Thank you for the pedantism. How about I replace "Peano arithmetic" with the "operations of multiplication/addition/division/etc. expressible upon the rational numbers"?

You bring the Peano arithmetic, I think people who respond to you can discuss Peano arithmetic without needing to be accused of pedantry. (Disclosure: I have no idea what Peano arithmetic is.)

Given that mcphage understands what Peano arithmetic is, I find it entirely unbelievable that he/she did not understand my point especially with phrases of mine like "Unless you're making use of limits, completeness, or something equivalent". So yes in this case I honestly find the pedantry entirely unhelpful.

Regardless, the point has been clarified, but discussing it further is kind of pointless.

edit: It just occurred to me that the original pedantic comment essentially breaks the following Hacker News guideline:

Please respond to the strongest plausible interpretation of what someone says, not a weaker one that's easier to criticize. Assume good faith.

I just found it a bit interesting considering it is my response complaining about the pedantry and not the pedantry that is receiving downvotes.

Re: 0.999...= 1

#130

Post by gilbetron »

smlckz wrote: post #113

Reminds me of the paradoxes of Zeno [1], especially the paradox of Achilles and the tortoise. At least one can simply prove that 0.999... = 1 without much hard work. Maybe less controversial than the following: 1 + 2 + 3 + ... [somehow] = -1/12 {{Riemann's zeta(-1)?}} 1 + 2 + 4 + 8 + 16 + ... [somehow] = -1 As well as the weird prime product (Product of 1/(1-(p^-2)) for p prime) and the sum of x^-2 from x=1 to [ sigh…

1 + 2 + 3 + ... does not equal -1/12, except if you redefine what you mean by +. https://www.youtube.com/watch?v=YuIIjLr6vUA
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