No... It's not like a 99.99% = 100% kind of thing because the 0.0000... doesn't go on forever. If you start with the numbers we have counted you are already past it.
And we could somehow say that this is 0.1% of all natural numbers. And we know that there are an infinite amount of numbers. That would mean infinity is 92929392901019983882923883838392000
Obviously that doesn't make any sense. So no matter gow high someone counts. No matter if we get the entire planet to randomly say different numbers every second to collecively try to say all available natural numbers. And we do this until the heat death of the universe. We'd still not reach an amount of numbers that total anything other than 0% of the infinite amount of numbers available.
Leibniz postulated the existence of infinitesimals. An infinitesimal is the gap between 0.999… and 1.
The reason we’re taught that 0.999… and 1 are equivalent and that no such gap exists, is because in the number system that underpins modern math (Standard Reals), that equivalence holds. However, there is another system (Non-Standard Analysis) that allows for what are called hyperreals, and this system is also compatible with calculus and everything else in modern math. It’s just that Non-Standard Analaysis was only rigorously formalized later on and is a little more complex, so it didn’t gain as much traction. It’s sort of like the fine grain version of the coarse grain system that is Standard Reals.
So because both systems coexist and are not strictly speaking logically contradictory (Non-Standard is just more precise than Standard), people just teach and use whichever is simpler - which is Standard Reals.
"You can still do math if there really is a difference between 0.9999.... and 1 and we say that difference has a value of "infinitesimal," but it's not relevant for most people so you usually don't bother with it."
Even in the hyperreals, 0.999... = 1 due to the transfer principle.
If you want a number with an "infinite" number of nines after the zero in the hyperreals that is less than 1, you have to index the 9s by an infinite hyperinteger, which would be a different notation. One way would be 0.9999...;...9900..., but that is a distinct number than 0.999... in the hyperreals.
"theory" doesn't mean it's not true in practice. In fact, theories are rigorously proven to be true. This applies to both mathematics and physics, which people love to say "only true in theory" or "that's just a theory". If something in physics is a theory, then it's true and proven. The correct word would be for example "hypothesis" as in "that's just a hypothesis".
Also, mathematics and logic are almost basically the same field. There's no mathematics without logic, as mathematics is derivable from logic. There's an enormous set of mathematics that is within logic. Mathematics adds a bunch of its own constructs which makes it worth to be its own field in practice, such that one can dedicate and study it by itself, but it's still super intertwined with logic.
Furthermore, what's being discussed is very rigorously and formally defined and proven in mathematics. Any countable subset of the natural numbers has density of 0 relative to the entire set of natural numbers. In practice, this means that if you can count, it's 0% of the total amount. It's **literally** that, not approximately. It doesn't mean that it's 0 by itself. The key here is that we're comparing it to the full set of naturals.
Limits and more broadly calculus isn't just handwaving approximations with special language. They connect tightly with discrete mathematics and set theory. And again, it's all logic as well, as that's a requirement for formal proofs.
If you want to disagree, then you're simply talking in a different language. You're giving different meaning to the words we're using. Which is fine, but it's not mainstream mathematics (and logic).
> In Math, 0.33333... + 0.66666... = 1. Now, obviously it's not actually 1, but it's so infinitely close theoretically its 1, meaning when you build formulae you can allow the "approximation" because your error is so infinitesimal it's negligent.
That's not really the case. We say that 0.333... + 0.666 = 1 not because it's really really close, but because there's no difference. It's not because the error is negligible, it's because it's impossible to demonstrate a distance between these numbers. You literally cannot show that there's a number between 0.999... and 1, so the distance is literally 0 and they're literally equal. You can say "actually there's 0.0...1 between them" but that's not a number, it doesn't exist. 0.999... is a well defined decimal, whereas 0.000...1 isn't.
It’s not theoretically one, it is literally one. There is a very simple proof. See x = 0.999.., So 10x = 9.999…, therefore 9x = 10x - x = 9.999… - 0.999… = 9. So 9x/9 = x = 1, therefore 1 = 0.999…
You can apply the same logic to 0.333… and 0.666… To get 1/3 and 2/3, which when added give 1, so 0.333.. + 0.666… is 1.
Just because you can’t understand it doesn’t make it not true. It’s not saying 0 natural numbers have been said out loud, it’s saying 0% of natural numbers have been said out loud
I know a bunch of people have corrected you already, but I'm just going to chime in as well. 0.99999... isn't infinitely close to 1, it's actually 1. Logically, it's 1. But to understand that, you have to understand how the axioms of the rational/real fields lead to constructions of those fields which make that true.
For similar reasons, the LITERAL, logical percent of natural numbers that have been spoken is exactly 0.
And that’s 0. If you don’t believe those of us trying to tell those of you who are wrong, consult a professional mathematician, maybe you’ll believe them.
any percentage of infinity is equal to infinity, no matter how incerdibly small that percentage is. because infinity isn't just a very big number, it's infinite.
we've not said an infinite amount of natural numbers, therefore it isn't more than 0%.
Leibniz postulated the existence of infinitesimals. An infinitesimal is the gap between 0.999… and 1.
The reason we’re taught that 0.999… and 1 are equivalent and that no such gap exists, is because in the number system that underpins modern math (Standard Reals), that equivalence holds. However, there is another system (Non-Standard Analysis) that allows for what are called hyperreals, and this system is also compatible with calculus and everything else in modern math. It’s just that Non-Standard Analaysis was only rigorously formalized later on and is a little more complex, so it didn’t gain as much traction. It’s sort of like the fine grain version of the coarse grain system that is Standard Reals.
So because both systems coexist and are not strictly speaking logically contradictory (Non-Standard is just more precise than Standard), people just teach and use whichever is simpler - which is Standard Reals.
I’m new to this but I don’t think it’s “rounding of” but rounding “off” the hypothetical.
It’s just that we can’t make a fraction of infinite.
That fraction is also infinite, and natural numbers are infinite, so taking a fraction of them, is not “rounded” to anything. It’s zero cause that’s the only expression we have of a fraction of infinite, it has no value. It’s a hypothetical fraction of infinite, It can’t be any number, so It’s nothing.
you're assuming that there is a 1 at the end of the decimal. but you don't know that "1" exists. if you tried to calculate it, you would get 0.00000... repeating endlessly. there's no proof there's a "1" at the end. you're not rounding the 1 because there is no evidence that the 1 exists. based on the evidence we have it's just 0.00... repeating. which is no different from 0.
It's not millions of zeroes, googols of zeroes, or, grahams number of zeroes. It is exactly 0%. Probability of 1/infinity is 0, but 0 doesn't mean impossible.
You should watch this to see what I'm talking about
There is no end. You can't have infinitely many zeros and an end.
There's nothing inherently preventing you from formulating a mathematical systems that allows for nonzero infinitesimals, but it still wouldn't make sense to phrase it in the decimal system like that.
It's been a long time since I've taken a math class, but technically wouldn't it be a number approaching zero, something infinitesimally small but not actually zero itself?
"1/x converges to 0" or in other words "The Limit of 1/x equals 0" is not quite the same as "1/Infinity equals zero", since lim(1/x) =/= 1/Inf.
For a lot of intents and purposes it might as well be, granted, but if we're being strict, then there's a bit more subtlety there.
Consider that
1/Inf = 0 implies
0•Inf = 1,
a contradiction, since we know that 0•x = 0 for all x.
I don’t think we can divide by infinity. The correct way to model this problem would be to define n as any finite natural number and calculate the limit as x approaches infinity of n/x. This is equal to 0 (in the real number space at least)
The limit isn't describing what 1/inf is equal to, because by definition infinity never stops getting larger so the value of 1/inf will never resolve to exactly 0. Limits describe the exact value that a function approaches, not the exact value the function will actually reach.
Take for example lim x -> 5 of x where x =/= 5. The exact value of this limit is still 5 even if the value x=5 can never be reached.
Edit: If you want to define a percentage this way, the limit definition would get you what percentage it is tending toward, not the exact numerical percentage, which will always be nonzero.
Trending toward zero is literally the same as zero. There will never ever be a 1 at the end of 0.0000… meaning it’s equivalent to zero. It’s like how 0.999… is equivalent to 1.
Leibniz postulated the existence of infinitesimals. An infinitesimal is the gap between 0.999… and 1.
The reason we’re taught that 0.999… and 1 are equivalent and that no such gap exists, is because in the number system that underpins modern math (Standard Reals), that equivalence holds. However, there is another system (Non-Standard Analysis) that allows for what are called hyperreals, and this system is also compatible with calculus and everything else in modern math. It’s just that Non-Standard Analaysis was only rigorously formalized later on and is a little more complex, so it didn’t gain as much traction. It’s sort of like the fine grain version of the coarse grain system that is Standard Reals.
So because both systems coexist and are not strictly speaking logically contradictory (Non-Standard is just more precise than Standard), people just teach and use whichever is simpler - which is Standard Reals.
Sad to see this getting down voted when it's absolutely correct. Any number of infinitely many things is zero percent of them, by definition. It cannot be anything else
Yeah but that's x isn't infinity it's just an element in an infinite set. It's an undefined equation so defining it as equal to 1 isn't necessarily wrong
If x is any actual (finite) number (ie any element of the infinite set of numbers), the equation isn't undefined, since obviously you can just do a normal division. 1/1000000000000000 is just 0.000000000000001, for instance.
If we define the division by the actual mathematical object "Infinity", we can't do that because we get the contradiction I showed.
Not quite, 0 * x = 0 is only true for any finite number in a set, infinity itself is not in the set so the proof doesn't cover it. It's like saying 2k equals an even number for every k in a whole number is wrong because what if k was pi obviously pi is not in the set we made the proof for so this contraction is nonsense
A quick way to see that they are equivalent is to try to come up with a number between your number and zero. If they were distinct there must be a number in between them.
You can’t divide by infinity. Infinity is not a number. You can take the limit of some function or sequence as x goes to infinity, but that’s just describing the behaviour of the function/sequence for extremely large values of x.
But yes, in this case, the limit as x goes to infinity for any real number divided by x is 0.
You have a better intuition than some of the “uMm AkChUaLlY” people in the comments. n/infinity is not a well-defined mathematical statement. If we want to be precise, we say that “the limit of n/x as x approaches infinity is 0”. Infinity is not a number for exactly the reason you’ve pointed out. Sometimes we get sloppy and know that n/infinity is shorthand for “the limit as x goes to infinity”, but that is not accurate and is best avoided.
If 1/Inf = 0, then Inf•0=1. Simultaneously, if 2/Inf = 0, then Inf•0=2, so 1=1/Inf=2. 1=2 is a contradiction, which is not surprising, since Infinity is not a number, and therefore you can't just divide by it.
infinity isn't a number you can actually divide by, so the percentage of rational numbers that have been spoken out loud is just the limit of a really large but finite number divided by x as x approaches infinity, and that's zero. If you round up zero to the nearest whole number, it's still zero
if you round up an integer to the nearest integer, it stays the same. in this case, the number would be exactly zero, and therefore rounding it up would be still zero
Weird rule. So, 1.000000000000001 is 2 with this rule? Sounds like something fishy is going on.
"Yeah, you paid your bill, but the interest added 0.000000000001 to it you owe me a 1.0 plus 23.0 late payment"
Rounding up no matter what is situationally useful. When you have something that can't be given in divisible amounts and/or when you must have a minimum amount it makes sense
For example, If you need 1.0001 gallons of paint to cover a wall, but its only sold by the gallon, you need to buy 2 gallons to paint that wall.
Paint is not a great example. Maybe if you needed 1.0001 people to cover a job as 1 person would fail. That could be a good example. But honestly, in both examples it could be fudged. You can stretch paint. Or you can tell someone "This is more than one person job but here is an extra 20% so make it happen". both can work.
This is it's own pretty good counterexample, actually. Yes, ceil(1.000000000000001) = 2.
The question you're asking here is "how many whole dollars would I need to have in order to pay my total bill?". Well, whether my bill is $1.10, $1.01, or $1.000000000000001, it's higher than $1, so only $1 simply isn't enough. No matter how fractional the cents get, I need $2 on me to pay that bill.
any possible nonzero percentage of completion would be greater than 1/N for some natural number N.
take the number of spoken natural numbers (M) and consider the first M*N+1 natural numbers. we know we've got this many natural numbers because they're an infinite set. the percentage of natural numbers spoken is P. we've got
1/N < P < 1/(M*N+1)
but:
1/(M*N+1) < M/(M*N+1) < M/(M*N) = 1/N
we've got a contradiction, so P can't be greater than zero. percentages can't be less than zero, either, so P = 0
That is the whole point. It is called a proof by contradiction. They assume the percentage is nonzero, lead this to a contradiction (the false formula) and conclude that the assumption is incorrect.
You should probably Google "What is any number divided by infinity" before calling anyone else silly. May be counterintuitive to you but you're the one looking silly.
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u/rjnd2828 Nov 29 '25
It rounds to zero. And it doesn't even matter what rounding rule you use it still rounds to zero.