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0.999...= 1

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Re: 0.999...= 1

#101

Post by empath75 »

CogitoCogito wrote: post #96
mcphage wrote: post #95

Earlier quoted context omitted.

> I've never seen a proof of 0.9... = 1 using Peano arithmetic which made any sense to me. I doubt one actually exists in any true logical meaning. Peano arithmetic only covers nonnegative whole numbers, so one will never exist.

> 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 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)

Re: 0.999...= 1

#102

Post by upofadown »

> ...infinitely many 9s...

How about we prove that an infinite number of 9s is impossible?

Assume that we have a finite number of 9s. Add a 9. The result is not infinite. Add another 9. The result is still not infinite. We can repeat this process for an infinite amount of time and still not have an infinite number of nines.

Any process that can not be completed in a finite amount of time can not complete and can not have a valid result based on that completion. Any process that can not be completed in an infinite amount of time is also bogus, but is in a sense even more bogus.

Added: Note that this is different than the case where we are asked to contemplate infinity with respect to continuous functions. By defining the number of 9s as a discrete (integer) value it opens things up to a discrete argument. These pointless navel gazing exercises always end up as a war of what everyone things things are defined as.

Re: 0.999...= 1

#103

Post by JoeAltmaier »

CogitoCogito wrote: post #94
knzhou wrote: post #44

Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you never define "...". Until you've defined "...", it's just a meaningless sequence of marks on paper. You can't prove anything about it using arithmetic, or otherwise. What the "a…

> Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. I've never considered them right at all. By saying something like 0.9... x 10 = 9.9... and then saying that 9.9... - 0.9... = 9 you're basically just a priori defining 0.9... to be 1. In other words you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 9. So you'r…

Yes that proof depends upon the representation in text of rational numbers (a dot and a series of digits). Try it in hexadecimal - it becomes opaque nonsense. Without some mathematical basis for 0.9... X 10 being something, there's a dangerous dependency on the representation that makes many folks uneasy.

Re: 0.999...= 1

#104

Post by eof »

upofadown wrote: post #102

> ...infinitely many 9s... How about we prove that an infinite number of 9s is impossible? Assume that we have a finite number of 9s. Add a 9. The result is not infinite. Add another 9. The result is still not infinite. We can repeat this process for an infinite amount of time and still not have an infinite number of nines. Any process that can not be completed in a finite amount of time can not complete and can not…

I tried this route as a counter to Cantor's diagonal argument, and got chastised by my then professor. I hope you have better luck, as I was never able to convince myself otherwise.

Re: 0.999...= 1

#105

Post by Faaak »

upofadown wrote: post #102

> ...infinitely many 9s... How about we prove that an infinite number of 9s is impossible? Assume that we have a finite number of 9s. Add a 9. The result is not infinite. Add another 9. The result is still not infinite. We can repeat this process for an infinite amount of time and still not have an infinite number of nines. Any process that can not be completed in a finite amount of time can not complete and can not…

> We can repeat this process for an infinite amount of time

Well, no, you cant. How about I prove it to you:

Assume that we have a finite number of processes. Repeat the process. The number of processes is still not infinite.

/s

Re: 0.999...= 1

#106

Post by tom-thistime »

CogitoCogito wrote: post #96
mcphage wrote: post #95

Earlier quoted context omitted.

> I've never seen a proof of 0.9... = 1 using Peano arithmetic which made any sense to me. I doubt one actually exists in any true logical meaning. Peano arithmetic only covers nonnegative whole numbers, so one will never exist.

> 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.)

Re: 0.999...= 1

#107

Post by [deleted] »

CogitoCogito wrote: post #96
mcphage wrote: post #95

Earlier quoted context omitted.

> I've never seen a proof of 0.9... = 1 using Peano arithmetic which made any sense to me. I doubt one actually exists in any true logical meaning. Peano arithmetic only covers nonnegative whole numbers, so one will never exist.

> 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"?

[deleted]

Re: 0.999...= 1

#108

Post by judofyr »

CogitoCogito wrote: post #94
knzhou wrote: post #44

Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you never define "...". Until you've defined "...", it's just a meaningless sequence of marks on paper. You can't prove anything about it using arithmetic, or otherwise. What the "a…

> Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. I've never considered them right at all. By saying something like 0.9... x 10 = 9.9... and then saying that 9.9... - 0.9... = 9 you're basically just a priori defining 0.9... to be 1. In other words you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 9. So you'r…

You only need to define 0.9… as 9/10 + 9/100 + 9/1000 + …. Without knowing how that series converges you can then use the two expressions mentioned to conclude that it has to be equivalent to 1.

Re: 0.999...= 1

#109

Post by pgtan »

iso1631 wrote: post #12
ginko wrote: post #9

I remember being doubtful when being presented with this in middle school, but after being shown this as fractions makes it obvious: 1/3 = 0.333.. 3 * 1/3 = 3 * 0.333.. 3/3 = 0.999.. 1 = 0.999..

Another secondary school 'proof' x = 0.9999..... 10x = 9.9999..... (10x -x) = 9x = (9.9999.... - 0.9999....) = 9 x = 9/9 = 1

> 10x -x

subtracting infinities is dangerous, you can achieve any result from it

https://www.youtube.com/watch?v=-EtHF5ND3_s

Re: 0.999...= 1

#110

Post by LudwigNagasena »

tomtomtom1 wrote: post #68
knzhou wrote: post #44

Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you never define "...". Until you've defined "...", it's just a meaningless sequence of marks on paper. You can't prove anything about it using arithmetic, or otherwise. What the "a…

if we say that infinitesimals exist. that 1/3 != 0.33.. and 1 != 0.9999... and the probability of possible events is never 0. what are the properties that we would lose?

Completeness is one of the most important properties of real numbers. Basically, you will have to completely throw away real analysis.
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