Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

0.999...= 1

en.wikipedia.org

Post reply on HN
Print view View original

581–590 of 647 posts

Re: 0.999...= 1

#581

Post by nialv7 »

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…

Infinity will forever be an abstraction, as there is no infinite physical actions you can carry out.

Which means what infinite actions actually constitue is purely by definition, as you cannot experimentally verify it. And that's why under some definition it makes sense to say 1+2+3+4+5+... = -1/12

Re: 0.999...= 1

#582

Post by vfinn »

kosievdmerwe wrote: post #501
vfinn wrote: post #433

Earlier quoted context omitted.

"If 0.999...!= 1 then there must exist a number A, such that 0.999... Thanks for the reply, but I don't know how you can make the above claim, because to me the question of what we mean by 0.999... is intertwined with the claim itself. I mean to me it seems to be a matter of how you interpret the approach to infinity. I don't see why there has to be A in between, if you interpret 0.999... as the "biggest possible num…

That statement is based on: > First hopefully we can agree that if we have two real numbers x and y that are not equal then we have a number z, such that x It's one of the properties of the reals and rationals that if two numbers aren't equal, then there are infinitely many numbers between them. It doesn't work with the integers, 2 and 3 aren't equal, but there's no integer x, such that 2 So if we say 0.999... != 1 t…

Thanks :). I think the resolution to my problem is moving away from infinitesimals. I did learn some calculus, linear algebra, number theory etc. in university, but I did it quite superficially, since it didn't feel that relevant to me at that time. Now I feel different.

Re: 0.999...= 1

#584

Post by perl4ever »

gowld wrote: post #548
naikrovek wrote: post #541

Earlier quoted context omitted.

I don't understand how they're the same number. I will never accept that they are the same. The difference between 0.9 repeating infinitely and 1 is infinitely small, but it isn't zero.

What is an "infinitely small" number? Is 9999..... the same as infinity? What is 1.0 - 0.99999.... = ? 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?

I don't have a strong opinion or much mathematical knowledge, but an "infinitesimal" number is a thing that most people have heard of even if they're fuzzy on what it is. If there is such a thing, what is the difference between 0.999... and 1 - 1/∞?

Re: 0.999...= 1

#585

Post by mninm »

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 too was unable to convince my family but based on your comment I just thought of a new (to me) example I might have tried. It leverages the grasp of fractions that you mentioned people already have.

Everyone knows that 1/3 = .333... and it can be pretty easily shown that 1/3 + 1/3 = 2/3 = .666...

So I would ask them that since .333... + .333... = .666... does it make sense that .333... + .333... + .333... = .999...? And since .333... = 1/3 isn't .333... + .333... + .333... = .999... the same as saying 1/3 + 1/3 + 1/3 = 3/3? And since 3/3 = 1 and 3/3 = .999... it makes sense that 1 = 3/3 = .999...

This might work on your kids but in my experience recalcitrant people will either act bored as if they don't care or will try to claim that somehow they understood it all along.

Re: 0.999...= 1

#586

Post by MrManatee »

gigatexal wrote: post #498

I hate to say it but I still don't believe this, it just goes against all intuition that I have, but people much smarter than I have proven it so I take it on faith for doing things like calculus etc just my lizard brain won't let me accept something that looks like less than 1 being 1 the same way that the limit of 1/x as x goes to infinity is zero but it doesn't seem like ti should be. The number gets infinitesimal…

Don't feel too bad.

The notation "0.999..." looks non-threatening, which tricks people into believing that they understand what it means. We could make "0.999... = 1" look scarier by writing it as [n ↦ 1 - 10^(-n)] = [n ↦ 1], where [n ↦ a_n] denotes the equivalence class of a Cauchy sequence of rational numbers. These statements mean the same thing, but with the scarier notation much fewer people would mistakenly believe that they understand what it says.

I would expect mathematics majors to learn what 0.999... means during their undergraduate university courses. But then there's still the question of why mathematicians chose to define it that way. To really understand that, you need to be able to come up with alternative definitions and to investigate the consequences of those definitions. And for most undergraduates, it might still take a few years to build that level of mathematical maturity.

For anyone who is not a math major, I certainly don't want to discourage any curiosity about this subject. Just don't be discouraged if you feel you can't fully understand what's going on. Understanding what 0.999... means and why mathematicians chose to define it that way is quite subtle.

Re: 0.999...= 1

#587

Post by Gehinnn »

empath75 wrote: post #574
Gehinnn wrote: post #467

Earlier quoted context omitted.

This thought can be made much more concrete when talking about sets. Clearly we can understand a collection of things as a set. Clearly numbers are things, so we can talk about the set of all numbers. But how many elements are in this set? A clever answer could be: It has as many elements as there are natural numbers! As subsets are also a thing, we could ask next how many subsets the set of all numbers has. A clever…

That’s actually not true, the power set of the naturals is uncountable.

As I said, it would be a very problematic answer. But only if you properly define when two non-finite sets have the same size, you can lead this to a contradiction. Infinity can be understood intuitively, extending it to the cardinal numbers not.

Re: 0.999...= 1

#588

Post by gigatexal »

MrManatee wrote: post #586
gigatexal wrote: post #498

I hate to say it but I still don't believe this, it just goes against all intuition that I have, but people much smarter than I have proven it so I take it on faith for doing things like calculus etc just my lizard brain won't let me accept something that looks like less than 1 being 1 the same way that the limit of 1/x as x goes to infinity is zero but it doesn't seem like ti should be. The number gets infinitesimal…

Don't feel too bad. The notation "0.999..." looks non-threatening, which tricks people into believing that they understand what it means. We could make "0.999... = 1" look scarier by writing it as [n ↦ 1 - 10^(-n)] = [n ↦ 1], where [n ↦ a_n] denotes the equivalence class of a Cauchy sequence of rational numbers. These statements mean the same thing, but with the scarier notation much fewer people would mistakenly bel…

Thank you. I had totally expected to get dunked on and find your empathy refreshing.

I’ll keep trying to understand it.

Re: 0.999...= 1

#589

Post by gigatexal »

gigatexal wrote: post #588
MrManatee wrote: post #586

Earlier quoted context omitted.

Don't feel too bad. The notation "0.999..." looks non-threatening, which tricks people into believing that they understand what it means. We could make "0.999... = 1" look scarier by writing it as [n ↦ 1 - 10^(-n)] = [n ↦ 1], where [n ↦ a_n] denotes the equivalence class of a Cauchy sequence of rational numbers. These statements mean the same thing, but with the scarier notation much fewer people would mistakenly bel…

Thank you. I had totally expected to get dunked on and find your empathy refreshing. I’ll keep trying to understand it.

It has to do with infinities (infinite sums) which might explain why it’s so interesting

Re: 0.999...= 1

#590

Post by thinkharderdev »

SkyBelow wrote: post #506
thinkharderdev wrote: post #288

Earlier quoted context omitted.

That doesn't exist. An open interval doesn't have a smallest number.

It does exist. The other poster just clearly showed that it exists by referring to it. The problem is that if we include such a number in our formal system of math, we quickly find contradictions and the whole system falls apart. So such a number is incompatible with any formal system of math (though I guess you could start building one which does include such a number and see what properties it has). Herein lies the…

You are certainly correct that people arguing the opposite side probably don't have a formal system in mind, but I think the intuition that an open interval in the Reals doesn't have a smallest number is easy to grasp even without any formal training. So you can force them to see the consequences of it through fairly straightforward logical contradictions.

Assume x is the smallest real number greater than 0. Then x/2 is also a real number and is greater than 0 but less than x. Therefore, x can't be the smallest real number greater than 0.

Post reply on HN