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0.999...= 1

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Re: 0.999...= 1

#191

Post by solotronics »

TomGullen wrote: post #167
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…

I find the algebraic way convinces most people: x = 0.9999.. 10x = 9.9999... 10x - x = 9 9x = 9 x = 1

black magic! I wonder if there are any programming languages that are able to handle this properly?

Re: 0.999...= 1

#192

Post by 317070 »

dwheeler wrote: post #153

A formally rigorous proof of this (in Metamath) is here: http://us.metamath.org/mpeuni/0.999....html Unlike typical math proofs, which hint at the underlying steps, every step in this proof only uses precisely an axiom or previously-proven theorem, and you can click on the step to see it. The same is true for all the other theorems. In the end it only depends on predicate logic and ZFC set theory. All the proofs have…

It depends on more than just ZFC, also on the definitions of the real/complex numbers. The crux of the proof is that 0.99999... is being constructed within the real/complex numbers, and in that system it is equal to 1.

And at the point where students see this, the whole concept of real numbers and infinity is usually ill-defined. I actually understand the scepsis for this theorem and where it comes from. The proof relies on the existence of a supremum, which is non-trivial.

Re: 0.999...= 1

#193

Post by CogitoCogito »

dahart wrote: post #172
CogitoCogito wrote: post #165

Earlier quoted context omitted.

> I think the point is not defining 0.9... to be 1, the point is that “...” means an infinite number of 9s. If you shift the decimal point by 1, then nothing changes, there are still an infinite number of 9s. If you shift the decimal point by 5 places, there are still an infinite number of 9s to the right. And here is the logical (induction) step: if you shift the decimal point by an infinite number of places, then t…

> why can’t you have an infinite amount of 9s and then just put a 7 after that? You can. The proof still works if you do that. What you’re refusing to accept here is the definition of infinity.

> You can. The proof still works if you do that.

> What you’re refusing to accept here is the definition of infinity.

I'm refusing to accept the definition of infinity? I have no idea what you mean by that. Would you make the same statement were you aware that I do in fact have a PhD in mathematics in the field of analysis? That I have in fact studied logic? Just as a hypothetical scenario.

Re: 0.999...= 1

#194

Post by mywittyname »

russellbeattie wrote: post #41

I'll just chime in with my completely ignorant theory that 1 - 0.999... = the infinitely smallest number, but is still, in my mind, regardless of any logic, reason, or educated calculations, greater than 0. I understand and accept this is wrong. However, somewhere in my brain I still believe it. Sort of like +0 and -0, which are also different in my head.

Well, if you perform the same calculation in base12, then you'll get a whole number representation, because 12 / 3 is 4. Thus, in base 12, 1 / 3 = 0.4

The problem here is our language for mathematics. Just like you have to accept the silent "k" on the word "knife", even when it doesn't make sense, in math, you have to understand that rational numbers can't always be expressed accurately as decimals.

Re: 0.999...= 1

#195

Post by amelius »

gspr wrote: post #176
amelius wrote: post #149

Earlier quoted context omitted.

Yes, because someone defined it that way. It is because the "limit" in 0.999... = lim[eps->0] 1-eps is implicit and defined as being applied before anything else. But you might as well define that implicit limit as applying over the entire expression. UPDATE: So instead of interpreting the expression as: (lim[eps->0] 1-eps) which is indeed false, you can also interpret the expression as: lim[eps->0] ((1-eps) which is…

> which is indeed false, you can also interpret the expression as: > lim[eps->0] ((1-eps) Assuming you don't mean some special notion of limit, I would guess that by `(1-eps) If so, `f` does indeed have a limit from above that is `false` and a limit from below that is `true`. Where do you wanna go from here? Edit: Corrected stupid wrong assertion about limit from below, d'oh.

Have a look here:

https://www.wolframalpha.com/input/?i=Limit%5BSign%5B1-x-1%5...

This shows that the limit does exist from both sides (but is different from both sides).

Re: 0.999...= 1

#196

Post by CogitoCogito »

ColinWright wrote: post #178
CogitoCogito wrote: post #124

Earlier quoted context omitted.

> 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…

I think you have mis-stated yourself. Either that, or I don't understand what you're saying. Trying to rearrange it and remove as many negatives as possible, I started with your statement: > 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. I think what you mean is that any proof that does no…

Here is a comment where I tried to clarify, but yes you seem to basically understand my point:

https://news.ycombinator.com/item?id=23007600

Re: 0.999...= 1

#197

Post by gspr »

amelius wrote: post #195
gspr wrote: post #176

Earlier quoted context omitted.

> which is indeed false, you can also interpret the expression as: > lim[eps->0] ((1-eps) Assuming you don't mean some special notion of limit, I would guess that by `(1-eps) If so, `f` does indeed have a limit from above that is `false` and a limit from below that is `true`. Where do you wanna go from here? Edit: Corrected stupid wrong assertion about limit from below, d'oh.

Have a look here: https://www.wolframalpha.com/input/?i=Limit%5BSign%5B1-x-1%5... This shows that the limit does exist from both sides (but is different from both sides).

Oops, sorry, that was a stupid blunder. Thanks for the correction.

Re: 0.999...= 1

#198

Post by JoeAltmaier »

ginko wrote: post #147
JoeAltmaier wrote: post #103

Earlier quoted context omitted.

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.

I don't see how it would be significantly different in other bases. In hexadecimal it'd be 0x0.ffff.. * 0x10 = 0xf.fff.. The rules are essentially the same.

Oh right! My mistake. Was stuck in my head with "10 == nine plus one" instead of your insight, multiply by '10' in whatever base.

Anyway it's still an artifact of representation. If rationals were represented as fractions, then it is unrepresentable. 1/1 X 10 == 10/1?

Re: 0.999...= 1

#199

Post by tom-thistime »

Sharlin wrote: post #187
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.

Yes, but at the same time it is common for people to insist that 0.999… only "approaches" unity as if it were a series approaching a limit, rather than an unique number. Intuition is a funny thing.

Saying "a number isn't a limit" is true, but it's only really relevant if you're talking to someone who genuinely has no idea what limits are. In actual math the number 0.99... can be defined as the limit of a series.

Re: 0.999...= 1

#200

Post by tom-thistime »

Okx wrote: post #182
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…

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.

This works because you have defined what equality means.

We all wanna talk infinity because it sounds more exciting, but I think everybody gets "infinitely close to 1" pretty well intuitively. What they don't get is whether "infinitely close to 1" means "equal to 1". That could happen because these people are stupid.[note] But it could also happen if nobody has defined equality.

[note]for example even highly educated people maybe don't listen, which is functionally a lot like being stupid.

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