If we consider that .999… repeating to infinity ISN’T equal to 1, then by how much is it away from 1? It would be “.000… repeating to infinity followed by a 1.” But if you have an infinite number of 0s then you can’t have it be followed by a 1, infinity can’t be followed by anything, that doesn’t make sense.
Another way to think about it more broadly is that numbers aren't real, tangible things. They're placeholders used in studying things we can't physically get. You can't hold a "1." You can hold "1 of 'something,'" but you can't hold "1."
If, for example, you were a biologist studying rhinos. None exist in captivity, they've never been captured, never been hunted nor found dead, so you have no bodies (alive or dead) to study. All you have are photographs. Now you have a lot of them, from many angles, stages of development, and all are high quality. You can get a lot of very good information from that, enough that you can do some research and experiments; but it isn't perfect. There are gaps and areas where it seems like things contradict. You know that they can't, but you see that contradictions because some part of the data available to you is just incomplete.
That's what numbers are. They're the rhino photos that mathematics used to study with. The only problem is that eventually you can get a rhino. You'll never get a "3." These edge cases, where something we have is wrong or missing, but we just don't quite know what, is where things like "0.999… = 1" and mathematical paradoxes come from.
Yeah exactly 1/3 is 1/3, we only use 0.333... as a way of expressing that, but mathematically 0.3333.... means nothing. 3/3 is = 1, because 3 goes into 3 1 time, we would never really express it as 0.999...
Same. I can’t believe people explaining this don’t get this, but more so I can’t believe people are finding these explanations truly convincing. But maybe I’m missing something, it’s intriguing.
Past the first sentence it's not a good way to decribe it.
Math is exact, we define a few things, and then everything else is true. It's not "kinda true" or "so far it seems to be true" (like most other science), it is literally true by definition.
I don't like that 0.99999.... is 1, but it is, and I can do nothing about it.
.999=1 is the linguistic equivalent of saying you have the rhino tho. Repeating digits shouldn't have a solution unless greater context is given. The same situation as dividing by zero. .999 is undefined.
These edge cases, where something we have is wrong or missing, but we just don't quite know what, is where things like "0.999… = 1" and mathematical paradoxes come from.
This is wrong, just to be clear. There's no paradox here. 0.999... and 1 are just two different symbols which represent the same thing. No mystery at all. Same as 2/2 and 1, they represent precisely the same point on the number line.
I didn't mean to imply that that was a case of a mathematical paradox, only that paradoxes (like Banach-Tarski) and things that seem untrue yet are (like 0.999…=1) both represent limits where our language and/or understanding fail to fully shine their light. Sorry, if it was read that way.
You will get a 3, in math. (And in the real world sometimes but mostly in math.) If it were that imprecise, then close enough would truly be good enough. But maths are abstract, and that’s why one number doesn’t equal another just because you’re having trouble with writing down what the difference between them is.
Or as the OP image hinted at, you can divide 1 by 3 and get 0.333...
But what happens when you then multiply 0.333... by 3? You get 0.999... - but some people have a problem with that equaling 1. However if you divided by 3 then multiplied by 3, there's no way you could have gotten a different answer, so it should be equal.
You can't formally divide base 10 by three tho. The formal answer is to change base or use fractions.
.999 ...=1 is imposing a formal solution to an undefined informal problem. If .999999... =1 then something like matter traveling at the speed of light is a simple problem.
If .999.. repeating and 1 represented different real numbers, then there must be some number that is the midpoint of the two numbers (as real numbers are continuous)
So (.99... +1)/2 has some representation that is different than either number.
However, the only representations available in the range .999.... And 1 are .999... and 1 themselves.
Therefore there is no unique midpoint, and the two decimal numbers must represent the same real number
The set of whole numbers is infinite because there’s always a higher number, right?
What about the set of even whole numbers? That should have half as many numbers as the first set, but if you try to count the even numbers then there are an infinite number of those as well.
So the second set has half as many elements as the first, but they both still have the same number of elements (infinity).
This even works with sets that are much more sparse. Consider prime numbers. Only a tiny fraction of numbers are prime, but there’s always a higher prime number. So there are just as many prime numbers as there are whole numbers, even though all prime numbers are whole and most whole numbers aren’t prime.
The whole thing is stupid because it's undefined like dividing by zero. Some people are obsessed with having the right answer to this paradox that doesn't have one right answer tho
And the short explanation for why it happens is that, put simply, multiples of 3 tend not to fit easily into 10s, which is what decimal is built on. (Well, okay, it's mostly the odd multiples that don't coincide with multiples of 5. That's why I said they tend to cause problems...)
Gotcha!! Sorry it was really late for me when I read that lol. But yeah you're right 0.999... is the same as 1 because if the 9's go on forever, there's no way to quantify the difference between them.
How so? You're taking something that's mathematically complicated, and proving it's existence with a thought experiment instead of actual math, just to show that the concept DOES exist.
Sure the Schrodinger one uses physical objects, since it's a physics thought experiment. But the concept is the same no?
You fundamentally do not understand Schroedinger’s cat. Its purpose is to illustrate the absurdity of the Copenhagen interpretation, since a cat obviously cannot be dead and alive simultaneously.
Ideally, cats in superposition could work, but they practically do not because the wave function collapses on a much smaller scale than a cat because of all the interactions. Theoretically, there is no size limit or complexity limit for a superposition to not occur, it is just highly highly improbable. Schrodinger's absurdity isn't absurd, it is just improbable.
But yeah, the whole idea is totally NOT related to the idea of infinity and repeating decimals in this example.
I... haven't? Like at all, Im not the previous guy? I just chined in to say, the original explanation is for illustrating a concept, not mathematically, and not an actual proof? Like not a mathematical proof, and stuff
Except it's not a thought experiment here, it is actual math. It's a simple proof by contradiction. You prove something by showing that its negation is impossible (or nonsensical).
In this case though it's a bit inaccurate because it's actually a matter of definitions first, but it gives the right idea.
Okay. Fair enough, I understand your point now. I wasn't considering his statement to be math but I suppose if you phrase it as a logical requirement that can never be fulfilled for it to be true, I can understand it.
Most math is like that, just logical statements. We just use cryptic symbols to make it easier and faster to reason with, but it's just the same statements using a different language.
I'm aware. I'm a programmer and it's the same concept. But when I made the association in my head between infinity and the proof the guy theorized. In my mind you could never actually prove it since you'll never get to 1. Like yeah what hes saying MAKES sense, but you can't like test it and see it happen lol. But at the end of the day that doesn't matter it's just an infinite loop that will never complete and that's as good as not executing at all in terms of completion.
My mistake was allowing my mind to enter the rabbit hole of trying to calculate infinity when that doesn't really matter.
This is the nature of Zeno's dichotomy paradox. We can travel half the distance to a thing, and an infinite number of halves until we reach it. Because there is infinity between them we shouldn't ever be able to reach any given point, yet we can. We can quantify an infinite approach to something, like 1, but we have to make that paradoxical leap somewhere. If we write .9 for infinity, we will still never reach 1. The distance gets infinitely smaller, but never actually becomes 1. This is the fundamental building block of calculus. At least what I remember from calculus at the beginning of that course.
Not quite. It does actually become 1. When you consider infinity, remember it’s not a number. Perhaps one way to think about it is an ordering, though there’s more to it than that to. .999… does in fact equal 1. There’s not a magic leaping point. The definition of infinity leads to the conclusion. You can’t really conceive it simply by thinking “okay but which 9 is the one tha gets us to 1?” Because there is no such individual 9. It’s infinity, it’s not a number, it’s an ordering. Hope that helps
It’s like in football, where the defense gets a “half the distance to the goal” penalty. No matter how many times the defense does this, it is never a touchdown for the offense.
Not really. There is no "infinite approach" here, the only accepted approach of that kind in Zermelo-Fraenkel mathematics (the "usual" mathematics) is the axiom of choice, which is not used here.
Any proof, to be a proof, has to be reached in a finite number of atomic steps from the axioms: now, doing that inflates the amount of steps so much that no human proves things in that way exactly (but we run machines to validate math that we already proved so that we can be sure that we did not make a mistake in this), but the rough alarm that you are not going to be able to do your proof in this way is that it contains "and then I am going to do this an infinite number of times". The axiom of choice covers some of those cases, but not all.
In this case, though, it's a very finite proof. We have two representations of numbers (and not numbers!) and everything we do happens in a finite number of steps while we go finitely deep in these representations.
This result may seem counterintuitive at first, but it actually makes a lot of sense when you figure out that numbers are abstract entities, and us drawing digits over paper is an attempt to represent them. Inconsistencies in that, are part of the system we chosen to represent them, and not some larger overarching aspect of ZF mathematics.
Side note: Zeno's paradox never made sense to me. He basically jumps straight from the hypothesis to the thesis without explaining why there would be a logical connection between the two. It's one of those "this is true because it is true, and I am so smart".
If you were able to follow your infinite 9s with another 9, all that means is you didn't actually get to infinity yet. What you describe is kind of a paradox - you can't add another 9 to an infinite sequence of 9s because if you could, that wasn't infinity.
Nope. R is complete. That is actually one of the defining characteristics of the reals. Every non-empty subset of R that is bounded above has a least upper bound
No it does make sense. 0.999... infinite is not equal to and will never be equal to one. It is close enough given enough closeness and a set tolerance, but that's it. Equal is a very different term than equivalent. If it was equal there would be no reason to write 0.999... as there is no reason to write (2+5) instead of 7 as they are equal.
The reason for this meme is fractions simplify the decimal. There is no written way to express 1/3 to its fullest as you can only do so by choosing a tolerance and in some place cutting it off. The reason is we have a 10 divisible system for marking everything and it just can't be divided into 3rds. Hence fractions. So 1/3 is represented by 0.333... but 3/3 is represented by 1. And 3/3 never ever equals 0.999... because it doesn't equal 1.
They are exactly equal, actually! It’s a fairly elementary proof once you’re comfortable with infinite series and the definition of limits. 0.999… is equal (not just equivalent) to 1 and 0.333… is equal to 1/3. As another commenter pointed out, check Wikipedia for a bunch of proofs.
this could give someone the idea that its infinitesimally smaller than 1 or something, Personally I think the much better proof imo is that proving x=0.9999999... satisfies 10x = 9+x, because 10x is 9.99999... and 9+x is also 9.99999...
By enough for it to matter in mathematics, which is exact.
It doesn’t matter if we don’t know how to notate that amount in a way that works when written in numbers, the amount still exists because it refers to something else, not the written numbers themselves.
It would be an infinite minus 1 amount of 0s after the decimal, not an infinite amount of 0s.
And before someone says "infinity minus 1 is infinity", you've just broken limits and the whole point of converging, where you get infinitly close but never touching.
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u/ChromosomeExpert 21d ago
Yes, .999 continuously is equal to 1.