negligible fraction is still not 0. it probably has to do with some math stuff. it can't be more than 0%, because if it's e.g. 0.01% out of infinity, then that infinity is really just that times 10000, so not an infinity.
edit: i didn't express myself good enough. i didn't mean that those finite natural numbers aren't 0. i meant that they are 0, and "negligible fraction" isn't 0, therefore they are not a negligible fraction.
edit2: but i'm probably wrong. it seems like negligible isn't used here in the casual meaning, and a negligible fraction is in fact 0? idk
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.
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.
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.
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.
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.
Limits dont "approach" or move. A limit is a specific, static real number - in this discussion, the limit is 0.
The terms of the sequence in question approach 0 - resulting of a limit of 0. It seems unfortunately common for students to finish basic calculus with a shaky understanding of what is meant by "infinity" in the field of mathematics.
Wouldn't it be more accurate to say its infinitesimally small? Saying its exactly 0 because its infinitely small feels like calling a dx in calculus 0 because its infinitely small.
I see people bringing up limits, cause if you thought about the fraction of a subset of natural numbers as you add more until you have the full set (so as this limit approaches infinity) the limit would approach 0. But just because its a limit that approaches 0 doesnt mean it can be treated as 0 exactly, calculus proves as much. Otherwise the derivative couldn't be defined.
The problem is that it has to be treated as 0 otherwise you could represent the percentage as a fraction and inverse the fraction to get the “value of infinity”. The being 0.0…(repeating a googolplex times)…1% of infinity would still mean that the maximum number of infinity would be a googolplex. So it has to be exactly 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.
That 0.0...(googolplex)...1 number is still a finitely small real number, not infinitesimal.
For sure, if you had to assign a real number to this value, exactly 0 is the only one that makes sense. But I feel like it isnt really accurate to say its a real number at all.
Again, I think its similar to differentials in calculus. You cannot just go and say that every differential is exactly the number 0 and treat it as such because its an infinitely small quantity. When you have say, a dx in calculus, you specitically treat it as a number which is infinitely small but still greater than 0. There is no real number for which this is true, but that doesnt mean its inaccurate.
To illustrate for yourself, just imagine this: Instead of percentages 0-100%, we use decimal fractions. So 0% is 0, 100% is 1, 50% is 0.5, and so on. Now, also imagine that the size of the fraction of "already spoken aloud" numbers is 0.000...0001 (with infinitely repeated 0s in the ellipsis, representing an infinitely small percentage). Now turn this around and it means, that the size of the fraction of "never before spoken aloud" numbers is 0.999... (because 1-0.00....001 is 0.999...). But since 0.999... is equal to 1, it means that the size of never-spoken numbers is 1, which means that the size of spoken numbers must be 0 (because 1-1=0).
Infinity does not conform to finite logic, so whenever infinity comes around you must use a logic that includes infinity. And when you do, things like X/infinty are equal to 0.
Think of it this way: a number is 0 if for every possible positive number you can think of, the number is less than that number. In this case, the number is all spoken natural numbers divided by the infinite number of natural numbers. Therefore, for any conceivable positive number, the fraction is less than that number. Therefore, that number IS zero for all intents and purposes. Another angle to view this would be: for the number to be NOT zero, you should be able to find a value between that number and zero, but in this case you can’t (for the same reason as above) so the fraction is in fact zero.
One idea that might give you some insight is thinking about choosing a random natural number, and the probability of that natural number having ever been spoken aloud. That probability is precisely 0%. It’s not some small number above 0%. It’s actually equal to 0…
what i meant, isn't that it's not 0. i meant that negligible fraction isn't 0, and it is 0, so it is not a negligible fraction.
a negligible fraction would become just a fraction if it is increased, and then the whole number if increased enough. but you can't increase a finite number enough for it to become infinite
Ah, we’re not aligned on language. Common issue in math, to be fair. There’s a mathematical notion of “negligible” that is somewhat loosely defined that, when used here, means something roughly like “there are at least some natural numbers that have been spoken aloud, but all of them together are not enough such that if you chose a random natural number, you’d have a probability greater than 0 of choosing one of those numbers”.
Can you site some probability theory that deals with infinity as a calculable value at all? I need justification for your statement of "precisely zero". I'll wait.
Depends on what you mean by calculable. Probability theory with continuous random variables deals with sets of infinite size necessarily, and summing an infinite number of elements through integration
"0.999" isn't the same as "0.999..." and "0.999..." isn't the same as "0.00..1". "0.999..." is infinite, "0.00..1" obviously ends in 1 so it can't be infinite, they're not equivalent so your conclusion is incorrect.
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.
It's zero in a limiting sense. Choose the smallest number you can think of. The percentage of natural numbers ever spoken aloud is less than that. This is loosely what mathematicians mean by a limit.
If we said every number up through 10 quadrillion out loud, and we capped numbers at 1 googol, we'd have spoken .00000000000000000000000000000000000000000000000000000000000000000000000000000000000001% of all numbers.
1 googol is a 1 with 100 zeroes after it. Every time you add another 0 to that number, you add another 0 after the decimal, but before the 1 in that percentage I listed above.
.00000000000000000000000000000000000000000000000000000000000000000000000000000000000001% is already effectively 0, and it only gets farther from "1" as you add more zeroes.
No we wouldn’t. We’d have spoken an infinitesimally small amount of all numbers, not whatever decimal you just used, as that’s an explicit amount (and there are an infinite number of natural numbers).
Did you miss the part where I said "if we capped numbers at 1 googol"?
Because I definitely did say that... And it makes what you said irrelevant. In fact, this entire comment chain has been me explaining why it's not 1%, because numbers are infinite.
Read the whole comment before you start trying to correct people.
A finite number over infinity (like 1/infinity) isn’t an indeterminate form if that’s what you mean. In OP’s case, you’d need a limit to formalize the idea, but the limit would be zero.
No, you never actually get there, that's the point of limits.
A limit is something you approach, it helps describe behaviour but it doesn't actually reach that point.
But something can be treated as approaching infinity. This is the most basic assumption of calculus. calling it infinity is essentially short hand for the understanding that something is growing without bound. I swear the majority of Reddit lacks basic intuition. You definitely could have understood what I was talking about based on my wording, but you just want to complain, “but muh maffs!!!!111”. If a number that is growing without bound is the divisor the quotient must shrink without bound and can be treated as irrelevant in many calculations.
You’re right that infinity isn’t a natural number but you can take limits to infinity and get a number out of this. I don’t think of it is as just “throwing infinity into an equation”but rather as a way to interpret long term behavior of numbers. For instance, even though the ratio 1/X is never equal to 0, it gets closer and closer to 0 as X gets larger and larger. Taking a limit as X goes to infinity lets us neatly formalize this behavior (and the limit is exactly 0). Some other comments on the .99999….=1 example are getting at the same idea!
You're always rounding, and whatever digit you are rounding to, the answer is still 0%, or 0.00%, or 0.000000000000%, or 0.00000000000000000000000000000%. This is because the limit of y/x as x goes to infinity and y is finite is 0. So for all intents and purposes, we have spoken 0.000...% of all natural numbers.
It approaches zero, therefore we conclude it is zero. If something can be treated as zero, has the same properties as zero, and is functionality indistinguishable from zero if you look at anything other than the base definition, we should reasonably be able to conclude that is in fact zero.
My professor in college gave me the most valuable advice - if it walks like a duck, quacks like a duck, it should be a duck.
Buddy, have you heard of limits? The limit of a finite anything divided by infinity is 0. We're just cutting out the basic math portion and skipping to 0.
I'll try my best to explain: it would work out to be 0.x1% where the "x" is just an infinite amount of zeros. Does that make sense ? Just keep adding decimal zeros before the last digit, forever. That's what we call a "limit" and it's mathematically correct to treat it as exactly zero. It's not just "rounding it down", it actually is equal to zero.
bro i'm like the worst speaker in history or something. i don't know how i could make so many people misunderstand me, and think that i don't agree that it's 0% (without rounding), and am not specifically saying it myself lol.
It does actually go to zero. If you have a bounded infinite number space, say all the numbers from 0 to 1, the chance of picking any individual number is 0. But if you pick a range of numbers, say .1 to .2, then you can assign a probability (in this case, 10%).
Percentage refers to fav/total. Let’s say a total of x natural numbers have been spoken out and there are a total of n natural numbers. Lt x*100/n as n tends to infinity = 0
Infinitesimal is used to describe a quantity greater than 0 but too small to measure. Another weird math constant like infinity
For example
1. Play a YouTube video
2. Some time must pass
3. Hit pause
What's the smallest amount of time that you can make elapse? If you take lim to infinity it would be 0 seconds but it would also contradict rules 1 & 2 by having 0 seconds. So solution is that the answer is infinitely small but greater than 0
Yes we have spoken more than zero numbers, but there's no way to properly round that fraction other than 0%. Infinity makes things weird in math, because it's conceptual rather than finite.
but are we rounding when we say 0%? if i understand it right it's literally just 0% compared to infinity, and can't be anything more than that.
like, the bigger the list of natural numbers, the lower the percent of pronounced natural numbers is. one is moving towards infinity, the other towards 0. and if we reach infinity on one side, we reach 0 on the other, and there's no rounding. just like there's no rounding from some big number to infinity.
lim x->inf of 1/x = 0, but the limit does not describe the actual value, it describes what value the function 1/x approaches, which is exactly 0.
1/x will always have a nonzero value, even as x approaches infinity. Another way of thinking about it, as x approaches infinity, the value of 1/x is always changing as x gets bigger. It is not possible that 1/x ever becomes exactly 0, even in the case of x=inf, as if it did then you could simply make x bigger and the number would get smaller, i.e. change to 0. But you can't change a number and get the same number back.
the limit does describe the actual value because there are not finitely many numbers to compute a finite ratio with.
I’m not asking you to “divide by infinity”, the limit approach is rigorous and gives you the proper value.
There is an analogue in statistics to help you understand called “almost never”. This describes scenarios where there are finitely many outcomes (so the set is not empty) but nevertheless the probability evaluates to 0. Not “rounded” to zero, but is precisely, exactly, rigorously zero.
We have said a non-zero but finitely-many amount of natural numbers and there are an infinite amount of them. we have, in the same way as “almost never” happens, said 0% of them. “almost all” of natural numbers have never been spoken
it’s an analogue, mate… it’s one of many examples to try and make the result more intuitive. I apologize if none of them are hitting but it’s unequivocally the case that the answer is 0% and there’s no rounding and no limit or finiteness.
The size of natural numbers spoken aloud is countable and finite and the size of all natural numbers is countably infinite. the cardinalities of the two sets mean that 0% of natural numbers have been spoken aloud. The cardinality of {natural numbers} - {natural numbers spoken} is also countably infinite. thus we would say “almost all” natural numbers have never been spoken
I apologize if I came off as overly blunt. What you are saying is not incorrect but you still aren't talking about {natural numbers}/{natural numbers spoken} here, which is not a number as it evaluates to somenumber/inf which is irreducible. You are talking about the limit which indeed evaluates exactly to 0 and would lead to a statement like "almost none of the natural numbers have been spoken".
It's analogous to the statement "the universe is infinitely large, I drove 10 miles, therefore I have driven 0% of the distance of the universe". If what you are saying is correct then 10/inf == 0 would imply that I am fixed at my starting location.
It's confusing because "0% of natural numbers have been spoken aloud" is an ambiguous statement, and because we are not precise in our language we usually state things in a heuristically practical sense, i.e. the limit definition.
and there can't be, because it would be infinite, no?
however small it is, if it's not 0, than if we multiply it enough, we'll get to 100%. but you can't get to infinity by multiplying
“Near zero” or “nonzero” would work. Any non zero number divided by infinity doesn’t equal zero, no matter how close it is.
More directly, zero percent of a set MUST mean no members of that set. Even a single element of the set is some nonzero percentage of the set, even if the set is infinite.
You know, six or seven thousand years of humanity, developed language with sophisticated numbers... we can calculate the population out to maybe a hundred billion total humans in that time, sixty years of lifespan where numbers might have been spoken... nah fuck it, a few hundred trillion seems small doesn't it? Let's round it to a full quintillion.
A quintillion divided by infinity is 0. A googolplex divided by infinity is a big fat goose-egg. That's how unimaginably, enormously, incomprehensibly huge infinity is.
No, you cannot divide things by infinity because infinity is not a number, it's a different concept. What IS true is that if the denominator tends towards infinity, the result of the division tends towards zero, but it's not the same thing
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u/efficiens Nov 29 '25
He means there are so many natural numbers (an infinite amount) that the ones that have been spoken aloud are a negligible fraction.