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0.999...= 1

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Re: 0.999...= 1

#481

Post by klank »

alanbernstein wrote: post #383
_Donny wrote: post #264

Earlier quoted context omitted.

I think this is spot on, at least for me personally. I am not very good at mathematics, so I never questioned my professors when they said that "You cannot treat infinites as regular numbers". Perhaps due to that statement, I did not really pursue these kinds of equations. For instance, I do not really see how the algebraic argument on the Wiki is any different from: 2 * inf = inf inf + inf = inf (subtract inf from b…

There is this phrase, often used when describing the decimal expansion of pi - "keeps going infinitely". This phrase is not exactly incorrect, but I wonder if it misleads people into thinking that an "infinite decimal" is "a kind of infinity", which it really isn't in any meaningful way.

I think it absolutely gets confused.

Infinity, the number, is routinely confused with creating an onto function mapping digits of pi to a set with a cardinality of the natural numbers. But sadly most people don't have the mathematical maturity to understand the difference when they encounter their first irrational number (normally pi).

Re: 0.999...= 1

#482

Post by momentbruh »

chottocharaii wrote: post #454
twiceaday wrote: post #440

Earlier quoted context omitted.

.999... and 1 exist on a continuous line. If they are different numbers, name a number between them.

? Just because there is nothing between two numbers does not mean the two numbers are equivalent. What nonsense is this

That is true for the reals

Re: 0.999...= 1

#483

Post by brlewis »

gspr wrote: post #477
brlewis wrote: post #436

Earlier quoted context omitted.

They have to know that 0.999... means you never stop writing nines. Put 1.0 on top, 0.9 on the bottom. Start subtracting from left to right, and keep writing nines on the bottom as you go to the right. In no time you'll see that the answer is infinite zeros.

> They have to know that 0.999... means you never stop writing nines. How do they know that that's a real number?

They don't have to know that it's a real number. Knowing that you never stop writing nines is sufficient to perform the calculation.

Re: 0.999...= 1

#484

Post by russellbeattie »

Scarblac wrote: post #144
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.

So 0 and that number are two different numbers. What is the difference between them?

One is positive and the other is negative. I thought that was pretty obvious!

Re: 0.999...= 1

#485

Post by tom-thistime »

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.

Ok. (a) I think you're saying that 0.00 ... 1 is not a real number. (b) Do we agree that lim n->inf 10^-n is a real number? (I.e. zero?) (c) Then you're saying that limit in "(b)" is not a reasonable definition of the string "0.00 ... 1". Is that correct?

Re: 0.999...= 1

#486

Post by ben509 »

twiceaday wrote: post #440
addicted wrote: post #434

Earlier quoted context omitted.

There isn’t a proof because it’s actually kind of arbitrary that 0.999... = 1. Fundamentally, this is true because we chose it to be true. Now, there are good reasons we chose it to be true, and that’s what people usually use as proofs. If it’s not true then a bunch of mathematical expressions become more inconvenient. But there is no reason as such why 0.999... could not have been defined as something that was alway…

.999... and 1 exist on a continuous line. If they are different numbers, name a number between them.

{ 0.999... | 1 } using surreal number notation.

Re: 0.999...= 1

#487

Post by russellbeattie »

mywittyname wrote: post #194
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.

Just to be clear, as I said, I totally accept the correct answer. I was just trying to humorously point out the non-intuitive nature of it.

Re: 0.999...= 1

#488

Post by ubercow13 »

cthalupa wrote: post #452
jefftk wrote: post #151

What if you have 0.9̅4? Can we say 0.9̅5 > 0.9̅4 > 0.9̅3? More on what happens if you allow this: https://mathwithbaddrawings.com/2013/08/13/the-kaufman-decim...

>What if you have 0.9̅4? Well, you fundamentally can't. If the 9s go on for forever then you never reach a point where you can add the 4. The definition of infinity precludes anything after infinity, because it never ends so you can never get there.

That is, until Cantor showed otherwise

Re: 0.999...= 1

#489

Post by RHSeeger »

lukifer wrote: post #375
RHSeeger wrote: post #335

Earlier quoted context omitted.

> it could just as easily be reframed as 0.000...0001 = 0 But it can't be, because there's nothing after "0.000..."; that ... goes on infinitely. It's literally "0s forever, never stopping". It's not a process of "keep adding 0s", it's the end result of never adding 0s. It's not a process, it is a noun.

My argument is that "noun" is a purely human abstraction, and that phenomena that act noun-like in nature are at best snapshots of iterative processes. Within the bounds of the noun abstraction, sure, I'll cede that point. But if one eschews that abstraction and looks at it purely as a process (I want to render 1/3 in decimal notation, then multiply that decimal notation by 3), there is always that niggling 0.000...1…

Regardless of whether or not they are human abstractions, a process (making cheese) and a noun (cheese) are two distinct things.

Re: 0.999...= 1

#490

Post by erodommoc »

laichzeit0 wrote: post #396
LunaSea wrote: post #327

Earlier quoted context omitted.

This is one of my pet peeves in maths. Although I do understand the concepts presented, the notion of "greater" makes no sense when applied to something without boundaries. Yet it's used all the time.

Things quickly fall apart if you rely on your intuition. Let's imagine all the odd numbers: 1, 3, 5, etc. Now imagine all the even numbers: 2, 4, 6, etc. Can we agree that there is an "infinite" amount of numbers in each of those groups? Now imagine all the odd numbers and even numbers together. That's also infinite right? Would you say there are more "all numbers" than just "all odd numbers", or would you say that t…

Well, the answer is equal because of how you define equality of infinite sets (one-to-one and onto). It's a very useful definition, but it's hardly the only possible definition.
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