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0.999...= 1

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Re: 0.999...= 1

#611

Post by TheRealPomax »

gspr wrote: post #598
TheRealPomax wrote: post #540

Earlier quoted context omitted.

This reasoning is understandable, but also incorrect, and we can point to where it breaks down: the idea that the 7 is "at a specific decimal" doesn't hold true, due to that pesky ellipsis. The crazy bit about the infinite repetition is that the 7 in 0.666...7 is not at a specific decimal. It's not even at "the last decimal" because there is no last. So, let's show this via a proof by contradiction: --- 1. we assert…

> 1. we assert that 0.666...7 is a sequence of digits. A sequence in the mathematical sense is a function whose domain is the natural numbers. Please define that function for the creature you're working with here. Otherwise you're trying to prove things about an object with no definition. You will end up in trouble.

Why? https://news.ycombinator.com/item?id=23009160 posited that we were in a situation where (1) holds, so that's where we're starting. If we assume (1) holds, then (2) cannot hold. We can abandon (1) but then we're no longer replying to that specific comment, now we're trying to prove something else.

Re: 0.999...= 1

#612

Post by nybble41 »

drran wrote: post #534
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.

0.000...1 can be written as 1/inf, which has sense in Surreal Numbers math.

You could just as easily say that 0.000…54234 can be written as 1/∞. Surreal Numbers is a bit of a detour in this case. The premise of the idea that 1 - 0.999… could be written 0.000…1 is the mistaken concept that there is some point "after an infinite number of steps" where the expansion of 0.999… stops and you can leave the remaining 1. The expansion never stops and there is no final remainder. The result is 0.000…, which is more typically written as 0. Plain zero, not an infinitesimal.

Personally I find the following algebraic proof to be the most approachable:

          x = 0.999…
          x = 0.9 + 0.0999…
          x = 0.9 + (0.999… ÷ 10)
          x = 0.9 + (x ÷ 10)
    x - 0.9 = x ÷ 10
    10x - 9 = x
     9x - 9 = 0
         9x = 9
          x = 1
     0.999… = 1

Re: 0.999...= 1

#613

Post by naikrovek »

gspr wrote: post #594
naikrovek wrote: post #577

Earlier quoted context omitted.

> What is an "infinitely small" number? What is an infinitely large number? > What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer? By that logic, 0.99 repeating isn't a number at all, and therefore can't be equivalent to 1, because you can't subtract it from 1. So my understanding that they are different is correct.

> > What is an "infinitely small" number? > What is an infinitely large number? Neither is a well-defined concept within the standard reals, and completely unnecessary for understanding that 0.999…=1. > > What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer? > By that logic, 0.99 repeating isn't a number at all, and therefore can't be equivalent to 1, beca…

    0.9      is not equal to 1,
    0.99     is not equal to 1,
    0.999    is not equal to 1,
    0.9999   is not equal to 1,
    0.99999  is not equal to 1,
    0.999999 is not equal to 1,
and so on, ad infinitum.

Saying that if you add enough "9"s it suddenly equals 1.0 makes absolutely no sense to me, and I seriously doubt that anyone will be able to convince me that it does make sense. I've read every single post in this thread and none of you have gotten me any closer at all to believing or understanding that 0.9 repeating equals 1.

Maybe I'm too old to understand this "new math" where all numbers are equal to each other.

Re: 0.999...= 1

#615

Post by sv_h1b »

0.999...=1 is true in the mathematical sense, period.

However as a representation of physical world, there is a caveat. What we understand is physical world appears and behaves discretely, because at planck scale (approx. 10^-35) the distances seem to behave discretely.

Although common people don't know/ understand planck scale, they do grasp this concept intuitively. What they are really saying is that in physical world there's some small interval (more precisely, about[1 - 10^-35, 1]) which can't be subdivided further, based on our current knowledge.

Same thing applies to planck time (approx. 5 * 10^-43) too.

So people are arguing two different things - the pure maths concept, or the real world interpretation.

Re: 0.999...= 1

#616

Post by rs23296008n1 »

kps wrote: post #142
rs23296008n1 wrote: post #93

Earlier quoted context omitted.

I have a pizza. I divide it into three parts. You'd be asserting that if I eat the three parts I have not eaten the whole pizza. I'm unconvinced.

You lost (1 − 0.999…)th of a pizza to crumbs when you divided it.

Get a sharper knife.

Re: 0.999...= 1

#617

Post by dahart »

CogitoCogito wrote: post #600
dahart wrote: post #401

Earlier quoted context omitted.

I don't necessarily understand your use of the word "invalid", when what it seems you mean is incomplete and/or too informal for your taste. > does it matter if the proof of a fact is incorrect if the fact itself is correct? Your language isn't allowing for a notion of precision, or for multiple forms of correctness, and it's not considering audience, communication or level of expertise either. I don't think it's a q…

I'm only going to respond to a few chosen points here because I find your post kind of meandering and hard to follow. I'm not cherry-picking (I don't care about "winning" any argument), I'm just trying to focus my response a bit so it increases the likelihood that you'll understand what I'm trying to say. > I don't necessarily understand your use of the word "invalid", when what it seems you mean is incomplete and/or…

Sorry, I misread the blog post’s statement about infinitesimals, I thought it said surreals.

How did Euler actually write his proof, do you know, or have a link? I’ve poked around online but can’t find it.

Re: 0.999...= 1

#618

Post by anewvillager »

tom-thistime wrote: post #121
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. T…

Even in surreal numbers, where there indeed exists numbers "infinitely close" to 1 but smaller than it, 0.9999.. would still be exactly 1. That's a quirk/feature of the decimal representation, not the underlying theory of numbers.

In surreal numbers there's a number called ε a number infinitely close to 0 (but larger than it), so what you would think that 0.9999.. represent is actually written 1-ε maybe?

But there's another number ε/2 that is between 0 and ε; 1-ε/2 is even closer to 1 than 1-ε is. Indeed, there are infinite numbers infinitely close to 1! (and none is really represented by 0.999...)

Re: 0.999...= 1

#619

Post by andrewprock »

tom-thistime wrote: post #606
andrewprock wrote: post #601

Earlier quoted context omitted.

This is usually described mathematically by saying that the representation of a decimal has to be countable, whereas the number 0.000...1 is not a countable representation. I will say though that if your explanation of 0.999... = 1.0 requires that you explain the distinction between countable and uncountable infinities, that's a big ask for most lay people.

Could you please explain a little more what "not a countable representation" means?

A countable set is one where you can reach any element in a finite amount of steps. See:

https://en.wikipedia.org/wiki/Countable_set

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