0.999․․․
susam.net
susam.net
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.
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.)
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.
Those equalities aren’t a “matter of fact”, they’re provable artefacts of positional number systems.
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.
And a rigorous definition of what "=" means too, which is another important point.
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.
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,...).
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...
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?
I know I can dig deeper all the way down to axioms if I choose so, but realistically there is not enough time for me to learn all of the underlying math. At most, I can occasionally check if the results are consistent and match reality.
If you're trying to push it all the way to epistemology, then even the most trivial proof doesn't really work. "A, A->B, therefore B" could be wrong because you have only your brain to inspect it. And computers, and other logicians, but they could all be glitchy. Even "Assume A, therefore A" could be wrong.
This is a valid line of reasoning, but if one wants to follow it, it would seem more useful to apply it to trivial proofs rather than get bogged down in a problem people have a weirdly difficult time accepting. I'd kinda like somebody to investigate the properties of proofs that non-mathematicians feel compelled to have an opinion in, though.
Absurd.
>I know I can dig deeper all the way down to axioms if I choose so, but realistically there is not enough time for me to learn all of the underlying math.
So proofs are impossible because you refuse to learn math? What.
Show me the proof of Newton's laws of motion.
If you try to convince a theorem prover like Coq you have to be very explicit of all the annoying details otherwise Coq will complain. It's very tedious work and some of the reasons why it's not more commonly used among programmers.
If you write a math paper you can assume that your audience is quite knowledgeable in the domain and you don't have to be very explicit about every detail (readers can fill in the gaps) but you have to be precise in your notation and reasoning. You provide a formal proof that convinces your peers not a rigid machine.
If you write a blog post then informal proofs are very much sufficient, e.g.
0.3333... = 1/3
0.9999... = 3/3 = 1
Is sufficient to convince your audience. You don't have to write down a fully scientific formal proof or Coq program to elaborate your argument in the context of a blog post.
31 is prime,
331 is prime,
3331 is prime,
33331 is prime,
...
33333331 is prime, therefore
333333331 must be prime (while in fact it's divisible by 17).
I'm not arguing one should never look at proofs and take everything for granted. I'm saying that rigorous proofs going down all the way to Peano axioms are unnecessary. In fact, most math people actually use, including linear algebra, calculus, Fourier transform predates Peano axioms.
In this example the flaw is intentionally obvious, but there are other examples where the flaw is more subtle.
Ridicolous. You use previous results which were built up from axioms. The correct formal proof here is very short, it is just an appeal to the definition of equality for two real numbers.
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.
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.
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.
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?
10x = 8.888
- x = 0.888
-------
9x = 8; x = 8/9 = 0.888...
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.
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.
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 3 years ago https://news.ycombinator.com/item?id=23004086
> People seem to have the intuition that the decimal representation of a number is a number.
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]
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.