5 ms·
0.999․․․
- anenefan 4y ago12 hours ago. https://news.ycombinator.com/item?id=34087556 https://news.ycombinator.com/item?id=34087556 Now I've checked the past link, it appears some people struggle with real numbers or haven't touched any areas needing calculus. 2 years ago https://news.ycombinator.com/item?id=23949097 https://news.ycombinator.com/item?id=23949097 3 years ago https://news.ycombinator.com/item?id=23004086 https://news.ycombinator.com/item?id=23004086
- susam 4y agoThanks for sharing the links. Your third link posted 3 years ago has very interesting discussion in the comments! Especially the comment there by lmkg that says the following leads to a very interesting discussion thread. > People seem to have the intuition that the decimal representation of a number is a number.
- anenefan 4y agohttps://hn.algolia.com/?query=0.999%E2%80%A4%E2%80%A4%E2%80%A4&type=story&dateRange=all&sort=byDate&storyText=false&prefix&page=0 https://hn.algolia.com/?query=0.999%E2%80%A4%E2%80%A4%E2%80%... I'm surprised it gets so much attention. I suppose it would be cruel to also point out zero multiplied by infinity = 1 [actually it might be -1, my memory on the maths I used to toy with is at least 30 years distant]
- fknorangesite 4y ago> I'm surprised it gets so much attention. 0.999... = 1 is my favourite internet drama. Unlike picking a fight with other classics like "will an airplane on a treadmill take off?", this one actually has an objectively correct, unambiguous answer. People argue against it so hard and are just so wrong. It's always hilarious.
- anenefan 4y agoIt's the sort of topic an old school Troll might have posted to an usenet group where contributors should know better. As for the everyday person who might have left school as soon as they could, they might not have been paying that much attention in class. Similar occurs when some people who should know better start to confuse integer type numbers with real numbers.
- constantcrying 4y ago>Intuitively, it may feel like the number 0.999... gets closer and closer 1 to without ever reaching 1. The actual proof is almost never discussed, but it is hidden in that statement. By definition of the equality of real numbers two real numbers either are the same, or they differ by a fixed positive ammount. It is easy to except that 0.999... nears 1 arbitrarily close, but that is already proof that they are the same.
- kbrkbr 4y agoRight. Thanks for that thoughtful comment, now I remember epsilon and delta. Just want to add that this way of looking at it is only helpful if reals are the agreed upon number set. If we talk about rational numbers the 1/3 + 1/3 + 1/3 proof is not so bad, is it? Rational numbers can be expressed using repeated decimal notation.
- constantcrying 4y ago>If we talk about rational numbers the 1/3 + 1/3 + 1/3 proof is not so bad, is it? It's not a proof at all, as the author acknowledges. In fact it essentially assumes the consequent. The issue is that writing down 0.333... essentially already assumes that you are working with real numbers. 0.333... should be thought of as an infinite series and you really want to use analysis in that case. You can do analysis on the rationals, but that is a very bad idea. But in the end you would still define equality the same way, which already is the proof.
- sobriquet9 4y ago> these algebraic demonstrations are not rigorous proofs because we have not shown that elementary rules for addition and multiplication extend to repeating decimal numbers too. They do but we have not demonstrated that here. That's why I don't like proofs. There are many things that are not demonstrated here. The same can also be said about the other proof that is presumably complete. Unless you build everything from axioms up, any proof will be incomplete. And building from axioms is too verbose. From the practical perspective it's like proving that 1 is equal to 1.0000...
- r0uv3n 4y agoBuilding from axioms is entirely possible, and in fact you'll often see the construction of first the rational numbers from the natural numbers and then the real numbers from Cauchy sequences or Dirichlet cuts within the first few weeks of studying mathematics in university. The construction of the natural numbers itself is often not discussed deeply, but the actual details aren't all that hard.
- sobriquet9 4y agoEverything you know is easy for you. For others, there's opportunity cost. The time one spent on learning Peano axioms is the time one could have spent on learning something more useful, say, linear algebra.
- xigoi 4y agoYou can take all the well-known properties of numbers as axioms and study linear algebra. Then later, if you're curious, you can prove the presumptions from a smaller set of axioms.
- Someone 4y ago> That's why I don't like proofs. If you don’t like them, I think grudgingly accepting them as a necessary evil is your best option. If you don’t do that, what do you use instead? Intuition? Beliefs? Do you take any statement that, when expressed from the axioms up, is too complex, as unknown?
- russellbeattie 4y agoEven though I know I'm wrong, I like to think of 1 - 0.999... as the reverse of infinity. The latter is the largest number, and the former is the smallest. In my opinion it deserves its own symbol.
- the_shivers 4y agohttps://en.m.wikipedia.org/wiki/Infinitesimal https://en.m.wikipedia.org/wiki/Infinitesimal
- russellbeattie 4y agoExactly! But 1 - 0.999... should be considered the infinitesimalest.
- xigoi 4y agoWhat happens if you divide this “infinitesimalest” number by 2?
- nh23423fefe 4y agobut that symbol already exists. '0'
- 4y ago
- the_shivers 4y agoI detest the algebraic "proofs" used in debates on this. Anybody who thinks that 0.999... does not equal 1 should probably also reject the idea that 1/3 is the same as 0.333... the argument for both is the identical (getting infinitely closer to some x is equivalent to x itself). You're using the thing you set out to prove as an argument in your proof. Online debates on this subject would be a lot better if the (correct) side of 0.999... = 1 was more open about the shortcomings of these simpler proofs.
- edflsafoiewq 4y agoThat's the point of the proof, it's to show you how things you already accept are true entail something you're less certain about. But perhaps the deeper point is this is a lesson for children or students and has a level of rigor appropriate for, and not unusual in, pedagogy, not a ironclad argument for "serious" debates with internet contrarians.
- patrickthebold 4y agoA Professor I TA'd for made a very good point: It's not so important if you think 0.9999... = 1 or not, the important thing is to have a (rigorous) definition for what 0.999... means. So much time is lost to people arguing without first defining what they are talking about. Not to mention the exercised of writing down what a repeating decimal means is a great way to clarify your thinking.
- Koshkin 4y ago> a (rigorous) definition for what 0.999... means But 0.999... is not "defined" to be equal to 1, it is equal to 1 as a matter of fact. (In other words, both are just different spellings of the same number arising, automatically, from the construction of the decimal representation of the reals.)
- patrickthebold 4y agoI'm not familiar with the construction of the reals, but somewhere in there you'd have to say what 0.999... means for it to be equal to 1. I'd probably just define it as an infinite series, where it's straight forward to show it is 1. But on second thought, kids run into decimals before constructing the reals, I'd imagine you could define rational numbers and repeating decimals and end up with a simpler argument.
- deleted 4y ago[deleted]
- pdpi 4y ago0.99… doesn’t equal 1 in, say, base 16. 0.11… can be 1/9 in decimal, or 1 in binary. Those equalities aren’t a “matter of fact”, they’re provable artefacts of positional number systems.
- macrolocal 4y agoSee ~ page 16 here for the (historic) definition: https://personal.math.ubc.ca/~cass/courses/m446-05b/dedekind-book.pdf https://personal.math.ubc.ca/~cass/courses/m446-05b/dedekind... It’s possible to work with both {x : x < r} and {x : x <= r} and maintain the distinction between 0.999… and 1, but then eg. 1 = 0.999… + x has no solution.
- tiffanyh 4y ago1/9 = 0.11111… 2/9 = 0.22222… 3/9 = 0.33333… … 7/9 = 0.77777… 8/9 = 0.88888… 9/9 = 1 Wait what!? it’s not 0.999…? does 0.999… = 1?
- v8xi 4y agoI always liked the second proof they show. You see it and you think something must be wrong with the method... but then try it with other numbers. 10x = 8.888 - x = 0.888 ------- 9x = 8; x = 8/9 = 0.888...
- feoren 4y agoHumans have trouble imagining infinity. This trouble manifests itself in many different misunderstandings and debates in math, and this is one of them. You'll see people say things like "0.999... approaches 1"; no, numbers don't "approach" things. Series can approach things, but they've substituted the number 0.999... with the infinite sum 9x10^-k for k = 1 -> inf (which it also is, and is also equal to 1), and then proceeded to misunderstand that, by imagining that series as being somehow built up over time. No, there's a qualitative, universe-shifting difference between "approaching infinity" and "at infinity", and the above series is defined to be "at infinity". People very often think they're talking about the latter when they're imagining the former. It's true that if you sat down and wrote a sequence of numbers with progressively more 9s at the end, you'd never reach 1. But that's because you never actually write the number we're talking about, which is equal to 1. Another place this shows up is in fractal dimensionality. If you imagine the Sierpinski triangle, it seems like nonsense to say that it is a 1.585-dimensional object. At every single iteration, it has a finite area: it's clearly 2-dimensional, right? But that's the error: you're not really imagining the Sierpinski triangle, because it's only defined at infinity; you're imagining it in steps approaching infinity. At every step approaching infinity, it is a 2-dimensional object with nonzero area. But at infinity, which is the only point it becomes a true fractal, it is a 1.585-dimensional object with no concept of 2-dimensional area.
- constantcrying 4y ago>You'll see people say things like "0.999... approaches 1"; This is somewhat correct. At least it is the correct way to think. A real number is an equivalence class of cauchy series. 0.999... is one representative of 1, as it is shorthand for the Cauchy series (0.9,0.99,0.999,...).
- gauddasa 4y agoThis is not true. The truth is, 0.111111... = 1 . All other bases are fake and they are just imitating this behavior found in binary numbers.