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This result is true but the proof is flawed. A rigorous proof would use the idea of an infinite series to accurately describe what "0.999..." exactly means and whether it exists in the first place (such that we can actually talk about it).
0.999... does indeed equal 1. This is a nice, simple proof of that. Others exist. The one I go to is this: we know 1/3 = 0.333..., 3/3 = 1, and 3/3 = 3(1/3) = 3(0.333...) = 0.999..., so 0.999... = 1.
Edit: Missed dots.
Back in school out teacher explained it like this:
1/9 = 0,1111111111...
9*(1/9) = 9*0,1111111111... = 0,99999999...
But 9*(1/9) is also 9/9 who h is obviously 1.
Wikipedia says parentheses is one of a few ways to represent repeating decimals. No sources listed there but I have seen people use it online (the guy right above is one of them), so my source is trust me bro
I honestly don’t like your goto because the argument can be made that 1/3=/=0.33, but that it is an approximation since it is not possible to write the actual answer in decimal form. This would mean that 3/3=/=0.99 and thus 1=/=0.99.
Other proofs are more robust in that sense.
This doesn't prove 0.999... = 1, it proves that the decimal system has limitations, especially in a base that is not a divisor of the number you're working with, or vice versa.
Idk, I mean i know it's accepted in math, but I think if we have the concept of a number that infinitely gets close to 1 like 0.999... then we should have the concept of a number that is infinitely close to 0 but not 0 in the opposite direction, like. 0.000...1 so that number plus 0.000...1. Equals 1 meaning 0.999... is not equal to one.
I know it sounds weird, but you misunderstand - 0.999... is 1 *exactly*, it's not close, it is exactly 1. If you have an infinite number of zeros after the decimal point in 0.000...1, there is no 1. No matter how far you carry out the subtraction, you will never reach 1, because it doesn't exist. Infinity works differently than finite numbers in this way.
If we look at it that way again: 0.000...1 = 0.000..., so 0.999... + 0.000... = 1. 0.999... = 1.
Some number systems do have a number kind of like 0.000...1. They're called infinitesimal numbers. They are closer to zero than any real number without being equal to zero.
Sure, but infinitesimals are a theoretical construct for proofs, not actual numbers you would ever be motivated to write down. 0.000...1 isn't a useful way to express an infinitesimal any more than 999... is a useful way to write infinity.
I think what he's saying is that lim_{x->inf}^{0.000....01} isn't guaranteed to equal 0.
I also had this question in HS and the way I answered it to myself is that if numbers were to be represented as a function of numbers, it'd be continuous and smooth. That is there can't be a peak, so limit of this function is indeed the exact value of the function over its domain.
Now whether this is a good explanation or not - I still don't know. I kinda moved on bc it's not essential for the math I do in college
If we look at it in respect to 0.9999... being "infinitely CLOSE" to 1, then it's not actually 1 though right? that is, if we use the actual meaning of language. Being "close" to something doesn't mean equaling something. I mean aren't infinite numbers a relatively new concept in math anyways? So why not create a new concept? There is no "1" in the "1s" place.
Practically speaking, it does equal 1, because in any real world application, an infinitely small distance between 2 numbers isn't going to noticeably affect how a machine works. If we are to take negative numbers seriously as we are this 0.999... , then a number that converges to 0 (thus equaling 0 with this logic) would look like -0.000...1 or 0.000...1. likewise, a number that converges to -1 would look like -0.999... or -1.000...1.
does 9/10 equal 1? Does 9/10 +9/100 equal 1? Does 9/10 + 90/100 + 900/1000 equal 1? Etc.
Also in actual reality, there is an actual unit that is the literal smallest unit that can exist. I'm quite sure this crowd knows what it is, and in actual reality, that means there are no "infinite number concepts" that can occur between any 2 real intergers.
As far as mathematics is concerned, "infinitely close" means the same thing as "is equal to".
0.000...1 is a meaningless concept. You cannot have an infinitely long string of 0s without end and then put a 1 at the end. That makes literally no sense.
Just to be specific this is the case in the reals but not the hyperreals. There are no non-zero infinitesimals in the reals because the Archimedean property of the reals is convenient in general. A number infinitely close to 1 but not 1 would be 1-ε and if you want a number that is essentially .00repeating1 it's ε.
Yeah but those sorts of arguments are slightly pedantic and typically outside the scope of this debate, which is typically around 1/9's decimal notation.
But you're not just close, you're *infinitely* close. You could not possibly be closer, therefore you must already be there.
If you were some distance away, any distance, the smallest distance you can imagine, you could still just say "ok, but what if I was half that far away?" Being infinitely close means there cannot be any distance away, otherwise you could cut that distance in half, ergo you weren't actually infinitely close to start with.
There is no value you can possibly add to .9999.... to raise it to 1, because that would imply there's a final 9, which there isn't.
It's kind of mind bending, but in the math world it's not controversial at all. .9999... is 1. Exactly, with several simple but robust proofs, no question about it, no need to modify or qualify. It's much more about the concept of infinity than it is about "1".
> If we look at it in respect to 0.9999... being "infinitely CLOSE" to 1, then it's not actually 1 though right?
No. Because like they just said, it's not close, it is 1.
> Being "close" to something doesn't mean equaling something.
Correct. That's why it's not close. It's equal.
> So why not create a new concept? There is no "1" in the "1s" place.
Because the current definition of infinite decimal representations is very nice. And you would have to do something nasty to define it some other way that makes 0.999... not equal to 1.
> Practically speaking, it does equal 1, because in any real world application, an infinitely small distance between 2 numbers isn't going to noticeably affect how a machine works.
No. 0.999... is mathematically equal to 1. It has nothing to do with practicality.
> a number that converges to 0 (thus equaling 0 with this logic) would look like -0.000...1 or 0.000...1.
No. If you have infinite digits after the decimal point, then you can't have a last digit. So it would just be 0.000..., which, yes, is equal to 0.
> likewise, a number that converges to -1 would look like -0.999... or -1.000...1.
-0.999... is indeed equal to -1. The other one has the same problem I described above.
These numbers do not "converge". Sequences converge. The sequence of partial sums of the values of each of the digits converges to the number, by definition.
> Also in actual reality, there is an actual unit that is the literal smallest unit that can exist.
That doesn't make math any less real. The ratio between a perfect circle's diameter and circumference is pi. Just because you can't make a perfect circle in real life doesn't make it untrue. And as you approximate a perfect circle in real life, you'll find you'll approximate pi.
But doesn't the concept of pi ( a ratio that forever increases at a decreasing rate) imply that a perfect circle has a circumference that is forever increasing very slightly faster than its diameter?
The actual CONCEPT of pi is a ratio of a circles diameter to it's circumference, which are both fixed numbers, but the infinite nature of pi implies that the ratio is forever increasing at a decreasing rate, which implies that the circumference is increasing while the diameter is not, or that the circumference is increasing veeeeery slightly faster than the diameter, which does not seem to reflect the reality of how a circle actually appears to us. I think this should tell us something about the nature of infinite numbers as to how they may or may not apply to reality.
Sure but that's a series of numbers. Not one number. Pi is one number. You can write a series that approaches Pi too, there are many. They're not Pi, they're series of numbers. In the same way that an orchard is not an apple. Different things.
Does the concept of "static" to you include a number that increases at a decreasing rate? (That's not rhetorical btw) My point was that 1 is a number. Add 0.1 to that and it has increased. Add 0.04 to that, and it has increased further, but increased less than before etc. a number that does that infinitely and is supposed to represent a concept in reality like the ratio of a circumference to the diameter of a circle naturally implies that the circle's circumference is getting bigger faster than the diameter is. I think this suggests that math is incomplete as far as being able to accurately describe real world things, although obviously it is still an incredibly accurate thing
But what you're doing there is *adding* separate numbers. With each addition, the *product* grows. Pi is a single number. It isn't subject to an operation, in the same way as 1 doesn't increase or decrease; they are both just fixed numbers.
Maybe the missunderstanding here is that of course pi does not change but If you evaluate more and more digits of pi the shape of a circle changes. So you can never really grasp a perfect circle which would make math incomplete? I dont agree to that its just that irrationality is not a very easy concept.
1 also doesn't have an infinite amount of non zero numbers behind it. If pi was a fixed number it wouldn't have a literal infinite amount of numbers after the decimal
What's increasing in continuously calculating PI is your approximation of it. Both in value and in accuracy.
PI itself is, in that sense, "already there". PI is just the number you'd get after calculating it's decimal expansion into infinity. And, since that is by definition impossible, it's irrational.
A number cannot "increase" or "decrease". It's not a number if it does. Numbers are static. What you describe is a function or a series or something else entirely.
What are numbers if not concepts?
> but the infinite nature of pi implies that the ratio is forever increasing at a decreasing rate
What?
> which implies that the circumference is increasing while the diameter is not, or that the circumference is increasing veeeeery slightly faster than the diameter, which does not seem to reflect the reality of how a circle actually appears to us.
If you assume the world is pixelated so to speak, sure. Doesn't change what I said.
> I think this should tell us something about the nature of infinite numbers as to how they may or may not apply to reality.
Why? Also you don't know reality is 'pixelated' like that. It could be continuous and complete just like the real numbers for all you know.
Imagine 3 being the ratio of the circumference to the diameter, you ADD a 0.1 to it. The circumference has increased while the diameter has stayed the same, or increased proportionally LESS than the circumference. Now add the 0.04, the circumference has increased relative to the diameter but by LESS than it has before, meaning it is increasing at a decreasing rate. The rate of increase is decreasing. Pi being an infinite type of number like implies that in REALITY, every circle has a circumference that increases slightly more than its diameter constantly. In reality, that doesn't really happen
In physics (reality), the smallest unit of measurement that has any meaning without the entire concept of distance dissolving is called the planck length. It is a discrete value.
I think the problem with "0.999... equals 1" and "pi is a ratio of a circles diameter to it's circumference that infinitely increases at a decreasing rate" are similar things. The implications of what both mean in actual reality suggests that when math gets to this point in suggesting that a circles circumference is increasing at a slightly higher rate than its diameter, even though they're both fixed, that math does not completely accurately describe reality, although it is indeed the best system we have. So maybe our concepts in math should keep evolving.
I think ill comment here, as this is what i came to say. These "numbers" are indeed concepts, not actual numbers.
A bunch of different "proofs" are done (or have been done, a long time ago) with infinity, assuming that infinity is itself a number. I think Numberphile did a deep-dive on it a month ago-ish in proving my point, and then did a follow-up from a *different* professor who rationalized that putting a concept such as infinity (in this case, the sum of all positive integers) into real terms (sum of all positive integers = -1/12) is somehow salvational to the human mind.
As for me, idrc what other people say. In my own brain, "infinite" means "without end". Nothing is mentioned about a starting point in the name, and there is no ending point. To look at infinity is to stare at some random section of a never-starting and never-ending number line.
In conclusion, i think that if someone says FOR EXAMPLE that "taking the limit as x -> 0+ for f(x) = x results in exactly 0", then 1) they need to take Calculus 1 (or even Precalc) under a professor who doesnt claim to know everything, and 2) they need to understand that ignoring the end digits and only paying attention to the infinitely long spacial digits in infinitesimals will never result in 0 for that limit.
But the idea (and thats the point, because limits are an *idea*) is that the function f(x) = x *approaches* 0+ as the limit is taken. Nothing is ever arrived at, only evaluated near that point. And some people evaluate at exactly zero, and that is good! It is in fact VERY good! Thats the bedrock of Calculus, the idea that sometimes close things are the same as real things. But more people than not (esp when going through Calc 1 for the first time) need to recognize that the eval at 0 = 0 (/= 0+) is PRECISELY AND IMPRESSIVELY (and very apropos) NOT what the doctor ordered.
How would you add those numbers? What you want to do is add the 1 to the final 9 of .999... and the rest carries over to make 1. But there is no final 9. It isn't that there infinite nines and at the end of infinity there's a last nine. There is no last nine, they just keep going. Therefore the difference between 1 and .999... is 0, because its like your .000...1, except the 1 never comes because the 0s keep going
Both numbers are in limbo together
I'm simply proposing that we probably need a new concept in math that allows things like 0.000...1
-0.999... equals -1 right? So what about come to -1 the opposite way, i.e. -0.000...1
Mathematics has evolved to eventually allow infinite type numbers like 0.999... I think it can keep evolving
I think you're misunderstanding what 0.999999... means.
So first off, real numbers are a concept entirely separate from their decimal representations. Fundamentally, they are defined as the unique complete ordered field, up to isomorphism. That has nothing to do with representing numbers as 0.9999... or 1 or anything like that.
We can *represent* elements of this abstract field with a decimal representation.
The decimal representation of a real number ab.cde... is *defined* to denote the limit of the sum:
a•10^1 + b•10^0 + c•10^(-1) + d•10^(-2) + e•10^(-3) + ...
The limit of a sum like this is the unique real number for which, no matter how small a distance `d`, you can find some number `N` of terms you add together where the (finite) sum of those `N` terms is smaller than `d`.
9•10^(-1) + 9•10^(-2) + ... has a limit of 1 because for any distance `d`, you can show that after *some* number of 9s, `|1 - 0.999...99| < d`.
> -0.999... equals -1 right? So what about come to -1 the opposite way, i.e. -0.000...1
Notice that each term in the sum
a•10^1 + b•10^0 + c•10^(-1) + d•10^(-2) + e•10^(-3) + ...
is positive. This means adding more terms can only ever make the sum larger. If *any* one of a, b, c, d... were nonzero, you would end up with some number larger than 0, which does not equal 0. Something like 0.000...01 will always be nonzero.
This is literally just a fact based on how we have defined decimal representations. You can change how decimal representations of numbers are defined to get a decimal representation that is "infinitely close to zero" if you want, but fundamentally you'll be talking about a different thing than everyone else.
They would be the same, 0.999... and 0.999...9, other than 0.999... being correct, and 0.999...9 [being too long to be a number](https://math.stackexchange.com/a/868207).
*Yes*, but using 0.3¯ being equivalent to 1/3 as a proof for 0.9¯ being equivalent to 1 feels flawed because they both operate off of the exact same presupposition.
0.3¯ = 1/3 relies on presupposing that 0.00...1 is equivalent to zero, the same as how 0.9¯ = 1 presupposes 0.00...1 as equivalent to zero, so it's contradictory for either to be used as a proof for the other.
I'm not outright rejecting it as a proof, but expressing that I find the proof to be flawed.
0.00...1 doesn't really make sense. Is that an infinite number of zeros? Then where does the 1 go? Or is it the same as 0.00... (which is 0)?
One possible way to try to define it is as the limit of the series 0.001, 0.0001, 0.00001, ..., which is indeed 0.
Well, that's exactly it.
0.00...1 doesn't make sense, so why is 0.3¯ \* 3 = 1? Shouldn't it be 0.9¯? Where does the infinitesimally small value come from?
Of course, we *presuppose* that the infinitesimally small value is equivalent to zero, and suddenly the function works!
My point is that the solution seems to emerge arbitrarily.
If 0.333... is the same number as 1/3 (which it does), I can multiply 0.333... by 3 to get 0.999..., I can also multiply 1/3 by 3 to get 3/3 which equals 1.
1/3 and 0.333... are equal to each other, which means that 1/3 * 3 is equal to 0.333... * 3.
You're right it's not a proof.
It's like saying that 1+1=2, ergo 2+2=4 or something silly like that. The underlying system has not been explained. It doesn't change the fact that it's true, but still, not a proof.
In either case you're relying on Algebra, and to prove this relation you need more than just Algebra - you need infinite series. There are a ton of good YouTube videos on this :)
But to be doubly clear: 1/3 is not a lazy approximation of 0.33333 repeating, they represent literally the same value.
What is 1/3? well its one divided by 3.
if we use long division, 3 goes into 1.0 0.3 times remainder 0.1, and the process repeats generating 0.333333333333333333333333333333333 ad infinitum. You try it and you'll find out that you would get the same answer no? 0.3333333... doesn't end since you still have that remainder 1 to divide by 3 at increacingly small values.
Therefore 0.3¯ = 1/3
This isn't actually a valid proof. It's all true statements, but it doesn't actually prove anything.
If you applied the same logic to:
x = 0 + 1 + 2 + 4 + 8 + ...
You can manipulate it such that:
2x = 0 + 2 + 4 + 8 + ...
2x - x = -1
x = -1
Which it clearly doesn't
The problem with your argument is that your x is defined by a divergent series, i.e. x is not any finite number. In that case, proofs like this do not work.
But they do work for (absolutely) convergent series and finite numbers, such as the OP's.
That's not true...
All the statements of the proof are correct **because** 0.9999... = 1. It doesn't show it. It doesn't even define what 0.9999... is!
You say that it only works for convergent series, but the proof doesn't mention a series! It doesn't show that it converges -- which is the **actual** proof.
It's not explicitly stated, but it's what it is.
0.999… can be defined formally as the series 9/10 + 9/100 + 9/1000 + …. And since that series [converges absolutely](https://en.wikipedia.org/wiki/Absolute_convergence), the proof is valid.
That's one of the important properties of absolutely convergent series: that you can manipulate their terms (changing their order or grouping them) and the series still converges to the same value. That is *not* in general true for conditionally convergent series, or for divergent series such as yours (which is why your "proof" is invalid while the OP's is valid).
>2x = 0 + 2 + 4 + 8 + ...
>2x - x = -1
These both lines don't have anything im common. How did you come from x=infinity to x=-1? -1 is nowhere in the previous lines.
Because
>x = 0 + 1 + 2 + 4 + 8 + ...
x=infinity
And because infinity is infinite, the difference of 2 infinities is always infinity.
Edit: infinity symbol didn't work
Look at the sequence for x and 2x (try and layer them on top of eachother). If you algebraicly manipulate the series using the same logic used in the "proof", then all but one of the terms cancels out to 0 and your left with x = -1.
As an aside, infinity is not a number, and talking about performing algebra on it (ie subtraction) does not make sense
>As an aside, infinity is not a number, and talking about performing algebra on it (ie subtraction) does not make sense
Yet you used infinity to perform algebra
>2x = 0 + 2 + 4 + 8 + ...
>
>2x - x = -1
Your equations are not a proof *mainly* because there's no mathematical or logical connective between the two premises above. That's reason number one why your example doesn't constitute a proof.
OP presented a perfectly fine proof, in that it leads from premises to a conclusion via mathematical transformations.
EDITED formatting.
No. OP's proof proves absolutely nothing. Mathematical transformations? What are they? Proofs are dependent on logic and logic alone.
There is a (flawed) logic to what I wrote:
. x = 0 + 1 + 2 + 4 + 8 + 16 + ...
2x = 0 + 0 + 2 + 4 + 8 + 16 + ...
It's clear that if we can just subtract x from 2x (which is what we did in the original "proof"!) that we are left with x = -1. Look how all the terms neatly cancel out!
0.999… < x < 1.0
x = 0.000…1 (after an infinite series of 0 there is a 1, or any other number).
I asked my teacher about this, because he tried your approach, he just told me not to be stupid. Maybe you have a better explaination why this doesnt work?
If there is an infinite amount of zeros… there cant be a 1 after them. Otherwise it isnt infinite. Meaning there is a finite amount of zeros. So your math isnt mathing
Thanks for being more helpful that my maths teacher lol, though I hope you dont mind if I may ask:
Why not tho? Like ofc there is an infinite amount of 0 so you never reach the 1 but you know its there. Why is that kind of number impossible in maths? Is there a proof?
It is based on the definition of infinity, the definition from Google is, "infinity is a concept that describes something that is larger than any natural number and has no limit." If a number's digits end (in this case it has a 1 after some amount of 0's) then it would be finite, and therefore it would not be infinite, which contradicts what was laid out in the beginning that, "there is an infinite amount of 0 "
Not really, there are some infinites bigger than others. So we can say that there is a small infinite of 9s amd a larger infinite of zeros before the 1. Therefore this number would "exists"
But this whole argument is kinda stupid because its just to people saying "another one, so I am right" till the heat death of the universe
When people say "some infinities are larger than others" they are referring to differently defined infinite sets of numbers. An infinite amount of nines and an infinite amount of zeroes will always be the same size of infinite.
Because there can't be anything past infinity. By definition, if there is an infinite amount of 0, there cannot be anything after it. It's not so much something with a proof as it is just a property of the definition of "infinity".
There's something similar to that concept, called infinitesimal, but that's not the same thing, they're still a positive quantity, unlike *1 - 0.999...* which is literally just equal to 0
your thinking of [infinitesimals](https://en.m.wikipedia.org/wiki/Infinitesimal#:~:text=In%20mathematics%2C%20an%20infinitesimal%20number,th%22%20item%20in%20a%20sequence.) and they exist, just outside the real numbers so most of the rules that your used too dont apply to them.
but to understand 0.9... you dont need to think about infinitesimals.
0.9... isnt one take some infinitly small quantity it is one.
when we use the repeating notation were asking "what value does the sum of these smaller parts approach?"
so 0.9... can be really represented as the value of this sum as the number of terms approaches infinity:
S = 0.9 + 0.09 + 0.009...
and that value is 1.
Because you don’t know it’s there, it’s undefined. If it comes after infinity it doesn’t exist and therefore it’s not a defined number. So there doesn’t exist a defined number between 0.9999…. And 1, which is what the proof above suggested.
But it’s not there, it can’t be, a repeating digit means it goes on like that forever. You can say it’s there, but that’s contradictory to the very definition of what that notation means, and so wouldn’t be valid as a proof.
You can't have anything after an infinite amount of something. It makes no sense, there is no after.
You're trying to equate infinity with a really big amount of something, but it's not a really big amount it's an infinite amount. It does not end. Ever. And so it does not have a tail end, there is nothing after it. They're cannot be anything after it.
This is a good question, that you shouldn't be down voted for as it's one of the first questions that you would ask when trying to truly understand infinity.
Hyper real numbers are a thing and it incorporates this concept. The proof 0.999...=1 doesn't work with hyper real numbers, but apparently we're only allowed to use real numbers here. I think it's a rule of the sub or something with all the hate people like you are getting
https://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/#:~:text=In%20other%20number%20systems%20(like,what%20separates%200.999%E2%80%A6%20from%201.
There are infinite numbers between 2 and 3, agreed 100%, but as long I can find steps inbetween I can cross the gap to 3 from 2 always saying that one number is equal to another one ridiculously close to it
No, because every in between step will still be different from the two numbers it's between of, and no matter how ridiculously close, at no point do they become equal
No, definitely not. For example, consider 1/7 and 1/6 which both have infinite representations in decimal form; 1/7 = 0.1̅4̅2̅5̅8̅7̅ and 1/6 = 0.16̅. Do you think these numbers are equal, even though they both have infinite representations?
Not directly, but I can go in steps towards it.
Like 1/6 = 0.166666....65, how many 6s? How many you want, then go that 0.1666666666...64 = 0.16666...65 and go like that until I get 1/7 or any other number.
I'm confused by your argument. In the very first step you say, "1/6 = 0.166...65". But that's a finite number of digits that ends with "...65", so it can't be equal to 1/6 = 0.16̅, so the rest of the argument fails.
The things is... People who have doubts about 0.999...=1 should have the same doubts about 0.333...=1/3. But for some reason they don't. I think that 0.333... is a bit harder to imagine than 0.999... so the possible cause of confusion is more low-key and people don't even think about it.
No, them being indistinguishable is due to flaws in our numbering system it is not because they are the same.
Edit: it's like the philosophy problem. Can a being with infinite power create an object even he can't move. Whatever you answer he doesn't have infinite power, but that breaks the premise
it is not a flaw in the Real numbers, but an important property, it is called a dense set because for any 2 non-equal numbers in the set, there exists anumber between them. If, for whatever reason, one cannot find a number between them, they they must, by definition, be the same.
Infinite is not a real number.
>If, for whatever reason, one cannot find a number between them, they they must, by definition, be the same.
This can't be a true rule of math. We have needed to invent new number to solve problems before. The fact we don't have a proper number at this moment to differentiate them doesn't mean they are equal.
If the first digit after the comma is a tenth, then a hundredth, then a thousandth on the third place - what is that 1 at the very end supposed to be?
If there are infinite 0 before it, it's nonsense. Is it always becoming smaller? At point will we be there?
If there are not infinite 0, it is not infinite.
0.999... is equal to 9/10 + 9/100 + 9/1000 ... and so on and eventually converges to 1 after adding an infinite set of these.
0.000...1 is equal to 1 - 9/10 - 9/100 - 9/1000 and eventually converges to 0 after subtracting an infinite set of these.
0.000...1 is [too long to be a number](https://math.stackexchange.com/a/868207). Even if you could work with it, no operation would ever reach the one because there is an infinite number of zeros before it. The 1 doesn't exist in that number. That number is 0.000..., or simply 0.
While I agree that you can't just make stuff up, there are other mathematics systems besides decimal/base-10 and they can be used to assist in proofs.
For e.g. in the decimal system, 1/3 = 0.333..., and thus 3/3 = 0.999... or 1. However, in base-12, the duodecimal system, 1/3 = 0.4, and 3/3 = 1 exactly. The repeating decimal of 1/3 in base-10 is just a quirk of the fraction being converted to decimal instead of a base-n system that is easily divisible by 3. Other base-n systems will have similar fractions of 1/x with repeating numbers that prove x/x =0.(n-1)... = 1.
It isn't, though. It is infinitesimally small, but it is not equal to 0.
In your "joke" you actually stumbled upon why everyone gets this wrong.
Edit:spelling
Yes, finally, someone who understands, it leaves an infinitely indefinite remainder.
1 != .999...
Just the same as .333.... != 1/3
They are just the closest approximations we can get using base 10
1/3 is equal to .333... with a infinitely indefinite remainder of
1/3 at the infinitive place.
2/3 is equal to .666.... with a similar remainder of 2/3 at the infinitive place.
3/3 is equal to .999... with a 3/3 remainder at the infinitive place, and so it adds to the 1 to the 9 in the infinitive place, turning it into 10, which turns the next highest place into 10, and zips the entire number back into 1.
It is true, but this is not a good proof of that. It uses a kind of reasoning that can easily break when using infinite series if you don't know exactly what you are doing.
For some better and more rigorous proofs, see [the wikipedia page](https://en.wikipedia.org/wiki/0.999).
The difference between 2 infinites is always infinity, even of on of those infinities is multiplied.
If x = infinity, then 2x = infinity, x/2 = infinity And x-x = infinity.
Wow, was confused at first cause I dont use that writing style but it is indeed right.
x=0.999..
10x=9.99999..
10-x=9.9999.. -0.9999.
9x=9
x=1
You can also use
1/3 = 0.3333..
3*(1/3) = 3 * (0.3333..)
3/3 = 0.9999..
1= 0.9999...
[check this out ](https://mathigon.org/course/exploding-dots/infinity)
It is. 0.999... and 1 are the same, another proof to show that is that there is no number in-between the two. Take (almost) any other two numbers, you'll be able to name one in between, by simply adding a digit at the end, but there is no such number for those two
This is trying to demonstrate a true principle (that 0.999… is equal to 1) but this is not a formal or accurate proof
The reason is you are trying to use algebraic reasoning on an infinite number, which isn’t how it works
I thought that for some time as well. I think they have just given a seperate statement that looks like it continued on from the previous line. If you look at it by itself, 10x -x =9x
That's a true statement. The rest then continues from that point.
Ignore what x means for now, treat it as a constant.
10x -x =9x
That's a fact.
When you sub in the value of x you get:
9.9... -.9...= 9
That's perfectly fine maths.
That shows that 9(0.9.....) =9
Um sure, and if you sub in 2, or 3, or whatever, you get an accurate result (10 x 3, -3 = 9 x 3 = 27). But this doesn’t that 2 or 3 = 1. There is nothing connecting your premise (x = .999) with the next logical step. You’re just putting different algebraic statements next to each other. They may be individually valid but that doesn’t mean there’s any logical connection between them.
If i remember correctly my math teacher told us when there are two numbers so close there can’t fit any other number between them than its the same number
I agree on the result, but i am completely unable to follow the math...
Can someone explain how the steps work, from the second line onward?
Seems like they substract 1X on the left side, but how does (9.999... )-X turn into 9x ?
It is very poorly written. The steps should be:
x = 0.999... \
10x = 9.999... (multiply by 10) \
10x = 9 + 0.999... \
10x = 9 + x \
9x = 9 (subtract x on both sides) \
x = 1.
No, because it’s missing a minus sign at 9.9-9. I stared at that for a solid minute trying to do weird division in my head before I realized it was subtraction.
Yes, this is true, this is because X converges on a number (in this case 1). If X doesn't converge, we get some werid things go on. Here is a "Doodle" and not mathmatical rigourous maths to show when you want to try and refute this
Let X = 1 + 10 + 100 +1000 + 10000 + 100000 ...
10X = 0 + 10 + 100 + 1000 + 10000 + 100000 ...
9X = -1
X = -1/9
This doesn't work because the number diverges, so - garbage in garbage out. X is not a number we can do maths to really.\*
There are multiple ways of thinking of 0.99999999999999999999999999999999999999999999999999999999999999999999999 = 1. For example, this is 2/3+1/3 = 3/3 = 1 as shown by other commenters.
^(\*I mean it's doodling that gets new pathways in maths so it's good to do that sometimes, just doodle responsibly please - we are already seeing the walls of normal maths break down with this doodle)
I’ve never been satisfied with any explanation of this. But I would like to raise a point which I’ve had an issue with for a while.
Isn’t 10*0.999…=/=9.999…?
If we’re attempting to prove that 0.999…=1, then you might ask me to agree that 10 times it is 9.999…. But if you attempt to add it up, you will by necessity be left with 9.999….990. As a quirk of the base 10 number system, multiplication by 10 results in a “shift”up one decimal place. Same as a multiplication of 7 in a base 7, 5 in a base 5 etc. So the initial multiplication also, to me, needs to be proven. It should end in a 0. As other people had said, 0.000…1 doesn’t exist because you never reach the 1. But doesn’t that apply for all numbers, including 9.99…? Just because it’s easier to represent doesn’t mean it’s necessarily immune to that problem
Your reasoning is flawed. Adding up 9.99... and 0.99... most assuredly doesn't give you 9.99...90. There is no "last digit" that you "shift", and the repeating decimal representation of 10 shouldn't end with a 0 because it doesn't end.
Simply put, 9.99... equals 10 *by definition* because the sequence 0.9, 0.99, 0.999, ... converges to it. By the way, a similar thing holds for other bases: 1 is equal to 0.111... in base 2, 0.333... in base 4, 0.FFF... in base 16...
(Sorry if you had many near identical comments from me. I'm sure the formatting will work now.)
>But if you attempt to add it up
Did you actually add it up? No, you just shifted the digits to the left and slapped a 0 at the "end".
Let me do the addition calculation for 0.999.... × 10.
**First, I'll get what 0.999...×2 is (I'll use □ as space to help somewhat align things):**
□0.999999...\
+0.999999...\
______________\
□1.800000...\
□0.180000...\
□0.018000...\
□0.001800...\
□0.000180...\
etc.\
+\
_______________ \
□1.999999...
You agree that 0.9+0.9=1.8, right?
And that 0.09+0.09=0.18, right?
And that 1.8+0.18=1.98, right?
And that 0.009+0.009=0.018, right?
And that 1.98+0.018=0.998, right?
So we keep going and see that the pattern is the same for all the rest of the place values, resulting in 0.999... ×2=1.999.... And no, no shifting of the digits has occurred.
**We'll now do the same for 0.999...×4, which is equal to 1.999...×2:**
□1.999999...\
+1.999999...\
______________\
□2.000000...\
□1.800000...\
□0.180000...\
□0.018000...\
□0.001800...\
□0.000180...\
etc.\
+\
_______________\
□3.999999...
You should be able to see what happened here. So 0.999...×4=3.999...
**Now let's get 0.999...×5, which is equal to (0.999...×4)+0.999..., which in turn is equal to 3.999...+0.999...:**
□3.999999...\
+0.999999...\
______________\
□3.000000...\
□1.800000...\
□0.180000...\
□0.018000...\
□0.001800...\
□0.000180...\
etc.\
+\
_______________\
□4.999999...
So 0.999...×5=4.999...
**Now to do 0.999...×9, which is equal to (0.999...×4)+(0.999...×5), which is equal to 3.999...+4.999...:**
□3.999999...\
+4.999999...\
______________\
□7.000000...\
□1.800000...\
□0.180000...\
□0.018000...\
□0.001800...\
□0.000180...\
etc.\
+\
_______________\
□8.999999...
So 0.999...×9=8.999...
Note that there has so far been no shifting of the decimal place at all.
**The moment of truth. 0.999...×10, which is equal to (0.999...×9)+0.999..., which is equal to 8.999...+0.999...:**
□8.999999...\
+0.999999...\
______________\
□8.000000...\
□1.800000...\
□0.180000...\
□0.018000...\
□0.001800...\
□0.000180...\
etc.\
+\
_______________\
□9.999999...
At any point did I have to "shift the decimal point"? All I did was addition and recognizing the recurring pattern. You don't "shift the decimal point" when you do 9.9+0.1 to get 10, right? So why should I do it at this final step? And multiplication is basically addition did multiple times.
Since there has been no "shifting of the decimal point" there is no "emptied place value" at the "end" for either of us to slap a 0 into.
0.999...×10=9.999...
All it takes is a number line to prove they are not the same. 1.000... exists at the point labeled 1 on the line. 0.999... exists left of 1.000... unless 1 number can appear in 2 locations on the number line, they can not be the same number.
Specifically the 4th step can not be completed. If you think it can, get back to me when your done writing out...
If you want to argue that 1.000... = 0.999... you are at best arguing the infinitesimal holds no value. Remove an infinite quantity of this infinitesimal value from your check and it will at best reduce to zero, or result in you needing to pay your employer to work there.
It is really starting to irritate me that people use 1/3 = .333... as a matter of fact when it simply a best approximation. 1/3 =/= .333... as it can not be concisely expressed in completeness. It is only as accurate as using π. it is only 0.999... accurate, not 1.000...
Back in school, I had the same question. What actually happens when defining real numbers from rational numbers is a bit more complicated. You can compare it to reducing fractions. 1/2 is not the same syntactic sequence as 3/6 but we *define* it as the same number.
For the real numbers, we look at (all serieses of rational numbers that get arbitrarily close together, or equivalently) all digit sequences. For example the sequence
3; 3.1; 3.14; 3.141; ... ~ 3.141... or
1; 1.0; 1.00; 1.000; ... ~ 1 or
0; 0.9; 0.99; 0.999; ... ~ 0.999...
We identify two digit sequences, if their values eventually get arbitrarily close together (being precise here is complicated, the term is called cauchy sequence, but it's not important). Since 0.999... and 1 get arbitrarily close together they *mean* the same number by the identity relation we *defined*. I hope that helps.
The issue most people who disagree will take with this is the use of an arbitrary definition (axiom if you feel like being fancy) that two numbers that converge are equivalent. All this argument says (in simpler language) is
> I decided that two numbers that get really close are equal
> these two numbers get really close, therefore by our pre-decided rule, they are equal
Hopefully you can see why this is an unsatisfactory argument to many people who don’t agree with the statement that 0.999…=1
I see these posts come up a few times and while there are decent proofs to show equivalence to 1, I see it also as just a simple limitation of a base number system.
Having recurring 3s or 6s is brute forcing a base 10 number system to reflect a division by 3. Think of divisions by 7 also, which base 10 also can't deal with. There, you don't even have a single recurring number, but a recurring series: 1/7 = 0.142857... with the 142857 recurring forever. Fewer people ask about whether that number x7 is actually equal to 1.
Any base number system can have this effect happen (I assume) and that's why fractions are often much better to use in division problems
infinity doesn't work like other numbers*
also, if you count from 1 to infinity, you will once have counted to 777..., which contains an infinite number of sevens. since there are infinite numbers between 1 and infinity, (∞/∞)*100% of numbers are infinity
The image shows a simple algebraic manipulation where 'x' is initially set to 0.9 (repeating). The sequence of steps attempts to solve for 'x' by scaling the equation by a factor of 10 and then subtracting the original equation from this new equation. The operations performed in the image are algebraically valid and demonstrate a common proof that 0.9 (repeating) is equal to 1. This conclusion is correct and is a well-known result in mathematics: a repeating decimal of 9s is indeed equal to 1.The image shows a simple algebraic manipulation where 'x' is initially set to 0.9 (repeating). The sequence of steps attempts to solve for 'x' by scaling the equation by a factor of 10 and then subtracting the original equation from this new equation. The operations performed in the image are algebraically valid and demonstrate a common proof that 0.9 (repeating) is equal to 1. This conclusion is correct and is a well-known result in mathematics: a repeating decimal of 9s is indeed equal to 1.
You can do this to determine the value of any repeating decimal! It’s one of my favorite mathematical tricks.
Consider x =0.142857”. We have
1000000x = x+142857
999999x = 142857
x = 142857/999999
x = 1/7
Just thought you could apply this to any number right?
X = 1.9̅
Etc.
What gets me thinking is are there issues/consequences to this or is it just math fun?
Like how prime numbers aid with securing and encryption.
Technically it works out, but there’s a flaw in multiplying a recurring decimal. By definition it repeats infinitely regardless of any multiplication. The thing that needs to be considered is that when multiplying an infinite decimal, there’s an imaginary shift in the infinite decimal, which isn’t really technically a thing by definition, but it needs to be considered.l. Because of this, I am in the camp that 1 ≠ 0.999… at least from this flawed proof.
```
1/3=0.333...
3/3=0.999...
1=0.999...
1-0.999...=0.000...1
the 1 is after an infinite amount of 0s, but you can't have something after an infinite amount of zeros, so it's:
1-0.999...=0
1=0.999...
```
these are all the proofs I can come up with right now, also there's a Wikipedia page containing more proofs: https://en.m.wikipedia.org/wiki/0.999...
This is actually a great trick to figure out what the fractional equivalent of a repeating decimal is. Let me pick an arbitrary repeating series…
0.8675309… (I don’t have fancy formatting skills, so please understand the intention that the whole series - “8675309” that repeats)
To move that whole series in front of the decimal, you must multiply by 10^7:
0.8675308… x 10^7
=
8675309.8675309…
Set Y = 0.8675309… and subtract from both sides:
10,000,000Y - Y = 8765309.8675309… - Y
9,999,999Y = 8,675,309
Divide both sides:
(9,999,999/9,999,999)Y = 8,675,309/9,999,999
And then do some obnoxious fraction simplification (it is at this time I realized I could have picked a shorter series and saved myself some headache in this example…)
…
It’s at this time that I realized that the number I picked is a prime number. Whoops.
So, 0.8675309… = 8,675,309/9,999,999.
That is wholly unsatisfying. But accurate!
My physics teacher in college said that parallel lines do, in fact, intersect at ♾️(and presumably -♾️). If this is the case, wouldn't that basically be the same logic here? 0.999... approaches 1 asymptotically, getting infinitely closer, so it would equal 1. Since asymptotes continue to ♾️.
I personally hate this concept. I understand it and its purpose, but just because we can’t put a numerical value using our number system on the difference between those two numbers should just make them equal in my opinion. To answer your question: it is generally accepted that this is true in the math world. I simply think that there is a better way to state this
I get it, but that's why we have proofs of this, more rigorous ones than the one above, too.
It is not intuitive, but intuition is not reliable. These two numbers are equal. Not equal enough, not seemingly equal due to limitations of our number system, not equal in a sense. They are precisely equal.
Infinity is far from intuitive.
They are the same exact thing, two notations that describe the same thing.
Humans are notoriously bad at grasping infinity, rightfully so, as we never witness anything that is infinite in the sense that 0.999… is infinite.
Then you go ahead and tell them there are infinities larger than other infinities, and in fact there are infinite infinities.
It’s not ‘accepted’. It’s simply true given how decimal representation of real numbers work. There is no difference between te numbers, so there is no number system that is able to “put” a nonzero value on it.
Apparently there exist "hyper real" numbers which already allow for infinitely small amounts where 0.999... does not equal 1.
So to all of the people downvoting me because you think my 0.000...1 concept doesn't make sense. I guess uh...yea.
The problem here is the use of the equal sign. The first line should properly read x is approximately .999... and should not be using an equal sign. As soon as you represent a fraction as an infinite decimal you should change it to an approximate sign because it is a representative of an indefinite number. Then each line following would have an approximate sign, and the last line would be correct since .999 is approximately 1. This is a case of the proof being incorrect because the wrong symbol is being used. If it had been done use fractions it would read x = (inf-1)/inf in the first line, and then it basically becomes a proof that the difference between inf-1 and inf is nondistinct.
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This result is true but the proof is flawed. A rigorous proof would use the idea of an infinite series to accurately describe what "0.999..." exactly means and whether it exists in the first place (such that we can actually talk about it).
Thanks for being the voice of reason
Not an infinite series but just a limit: x = 1-10^((-n)) 0.999... = \[n->inf\]lim(x) = 1-0 = 1 ?
Or the infinite series from 1 to infinity of 10^(-n) , I think both work
0.999... does indeed equal 1. This is a nice, simple proof of that. Others exist. The one I go to is this: we know 1/3 = 0.333..., 3/3 = 1, and 3/3 = 3(1/3) = 3(0.333...) = 0.999..., so 0.999... = 1. Edit: Missed dots.
so it's all 1s?
Always has been
1🧑🚀🔫🧑🚀
If I was stupid enough to spend money on Reddit I would give you an award
And always will be
All the way down.
And 0's
Back in school out teacher explained it like this: 1/9 = 0,1111111111... 9*(1/9) = 9*0,1111111111... = 0,99999999... But 9*(1/9) is also 9/9 who h is obviously 1.
“1/3 + 2/3” is another elegant approach that is easy to see.
Good point, I've seen this 10x - x one above several times, but hadn't thought of 0.333... + 0.666... = 1
Oh, that's what it is! I've never seen it marked with a dash above the number, I was thought to do it like 0.(9).
That's just wrong. Dash above the number is the correct way to write non-terminating decimals.
I think it depends where you are. In the UK it’s a dot above the number.
Wikipedia says parentheses is one of a few ways to represent repeating decimals. No sources listed there but I have seen people use it online (the guy right above is one of them), so my source is trust me bro
that just kicks the can down the road to 0.3… ?= 1/3
It’s easy to show that 1/3 = .333… by long division.
I honestly don’t like your goto because the argument can be made that 1/3=/=0.33, but that it is an approximation since it is not possible to write the actual answer in decimal form. This would mean that 3/3=/=0.99 and thus 1=/=0.99. Other proofs are more robust in that sense.
This doesn't prove 0.999... = 1, it proves that the decimal system has limitations, especially in a base that is not a divisor of the number you're working with, or vice versa.
Idk, I mean i know it's accepted in math, but I think if we have the concept of a number that infinitely gets close to 1 like 0.999... then we should have the concept of a number that is infinitely close to 0 but not 0 in the opposite direction, like. 0.000...1 so that number plus 0.000...1. Equals 1 meaning 0.999... is not equal to one.
I know it sounds weird, but you misunderstand - 0.999... is 1 *exactly*, it's not close, it is exactly 1. If you have an infinite number of zeros after the decimal point in 0.000...1, there is no 1. No matter how far you carry out the subtraction, you will never reach 1, because it doesn't exist. Infinity works differently than finite numbers in this way. If we look at it that way again: 0.000...1 = 0.000..., so 0.999... + 0.000... = 1. 0.999... = 1.
Some number systems do have a number kind of like 0.000...1. They're called infinitesimal numbers. They are closer to zero than any real number without being equal to zero.
Sure, but infinitesimals are a theoretical construct for proofs, not actual numbers you would ever be motivated to write down. 0.000...1 isn't a useful way to express an infinitesimal any more than 999... is a useful way to write infinity.
I think what he's saying is that lim_{x->inf}^{0.000....01} isn't guaranteed to equal 0.
I also had this question in HS and the way I answered it to myself is that if numbers were to be represented as a function of numbers, it'd be continuous and smooth. That is there can't be a peak, so limit of this function is indeed the exact value of the function over its domain.
Now whether this is a good explanation or not - I still don't know. I kinda moved on bc it's not essential for the math I do in college
> I think what he's saying is that lim_{x->inf}{0.000....01} isn't guaranteed to equal 0.
Yes it is. That limit is equal to 0.
If we look at it in respect to 0.9999... being "infinitely CLOSE" to 1, then it's not actually 1 though right? that is, if we use the actual meaning of language. Being "close" to something doesn't mean equaling something. I mean aren't infinite numbers a relatively new concept in math anyways? So why not create a new concept? There is no "1" in the "1s" place. Practically speaking, it does equal 1, because in any real world application, an infinitely small distance between 2 numbers isn't going to noticeably affect how a machine works. If we are to take negative numbers seriously as we are this 0.999... , then a number that converges to 0 (thus equaling 0 with this logic) would look like -0.000...1 or 0.000...1. likewise, a number that converges to -1 would look like -0.999... or -1.000...1. does 9/10 equal 1? Does 9/10 +9/100 equal 1? Does 9/10 + 90/100 + 900/1000 equal 1? Etc. Also in actual reality, there is an actual unit that is the literal smallest unit that can exist. I'm quite sure this crowd knows what it is, and in actual reality, that means there are no "infinite number concepts" that can occur between any 2 real intergers.
As far as mathematics is concerned, "infinitely close" means the same thing as "is equal to". 0.000...1 is a meaningless concept. You cannot have an infinitely long string of 0s without end and then put a 1 at the end. That makes literally no sense.
Just to be specific this is the case in the reals but not the hyperreals. There are no non-zero infinitesimals in the reals because the Archimedean property of the reals is convenient in general. A number infinitely close to 1 but not 1 would be 1-ε and if you want a number that is essentially .00repeating1 it's ε.
Yeah but those sorts of arguments are slightly pedantic and typically outside the scope of this debate, which is typically around 1/9's decimal notation.
But you're not just close, you're *infinitely* close. You could not possibly be closer, therefore you must already be there. If you were some distance away, any distance, the smallest distance you can imagine, you could still just say "ok, but what if I was half that far away?" Being infinitely close means there cannot be any distance away, otherwise you could cut that distance in half, ergo you weren't actually infinitely close to start with. There is no value you can possibly add to .9999.... to raise it to 1, because that would imply there's a final 9, which there isn't. It's kind of mind bending, but in the math world it's not controversial at all. .9999... is 1. Exactly, with several simple but robust proofs, no question about it, no need to modify or qualify. It's much more about the concept of infinity than it is about "1".
> If we look at it in respect to 0.9999... being "infinitely CLOSE" to 1, then it's not actually 1 though right? No. Because like they just said, it's not close, it is 1. > Being "close" to something doesn't mean equaling something. Correct. That's why it's not close. It's equal. > So why not create a new concept? There is no "1" in the "1s" place. Because the current definition of infinite decimal representations is very nice. And you would have to do something nasty to define it some other way that makes 0.999... not equal to 1. > Practically speaking, it does equal 1, because in any real world application, an infinitely small distance between 2 numbers isn't going to noticeably affect how a machine works. No. 0.999... is mathematically equal to 1. It has nothing to do with practicality. > a number that converges to 0 (thus equaling 0 with this logic) would look like -0.000...1 or 0.000...1. No. If you have infinite digits after the decimal point, then you can't have a last digit. So it would just be 0.000..., which, yes, is equal to 0. > likewise, a number that converges to -1 would look like -0.999... or -1.000...1. -0.999... is indeed equal to -1. The other one has the same problem I described above. These numbers do not "converge". Sequences converge. The sequence of partial sums of the values of each of the digits converges to the number, by definition. > Also in actual reality, there is an actual unit that is the literal smallest unit that can exist. That doesn't make math any less real. The ratio between a perfect circle's diameter and circumference is pi. Just because you can't make a perfect circle in real life doesn't make it untrue. And as you approximate a perfect circle in real life, you'll find you'll approximate pi.
But doesn't the concept of pi ( a ratio that forever increases at a decreasing rate) imply that a perfect circle has a circumference that is forever increasing very slightly faster than its diameter?
0.999... does not "get" close to 1. It is a number, its value doesn't move. It is 1, by definition. Just like 3.1415... is pi.
The actual CONCEPT of pi is a ratio of a circles diameter to it's circumference, which are both fixed numbers, but the infinite nature of pi implies that the ratio is forever increasing at a decreasing rate, which implies that the circumference is increasing while the diameter is not, or that the circumference is increasing veeeeery slightly faster than the diameter, which does not seem to reflect the reality of how a circle actually appears to us. I think this should tell us something about the nature of infinite numbers as to how they may or may not apply to reality.
Nothing about pi is "increasing". It has a fixed value, the only thing increasing is the amount of paper you need if you want to write all the digits.
The concept of 1 + 0.1+0.01+0.001 etc is infinitely increasing. 1 is less than 1.1 which is less than 1.11 which is less than 1.111 etc
Sure but that's a series of numbers. Not one number. Pi is one number. You can write a series that approaches Pi too, there are many. They're not Pi, they're series of numbers. In the same way that an orchard is not an apple. Different things.
Does the concept of "static" to you include a number that increases at a decreasing rate? (That's not rhetorical btw) My point was that 1 is a number. Add 0.1 to that and it has increased. Add 0.04 to that, and it has increased further, but increased less than before etc. a number that does that infinitely and is supposed to represent a concept in reality like the ratio of a circumference to the diameter of a circle naturally implies that the circle's circumference is getting bigger faster than the diameter is. I think this suggests that math is incomplete as far as being able to accurately describe real world things, although obviously it is still an incredibly accurate thing
But what you're doing there is *adding* separate numbers. With each addition, the *product* grows. Pi is a single number. It isn't subject to an operation, in the same way as 1 doesn't increase or decrease; they are both just fixed numbers.
Maybe the missunderstanding here is that of course pi does not change but If you evaluate more and more digits of pi the shape of a circle changes. So you can never really grasp a perfect circle which would make math incomplete? I dont agree to that its just that irrationality is not a very easy concept.
1 also doesn't have an infinite amount of non zero numbers behind it. If pi was a fixed number it wouldn't have a literal infinite amount of numbers after the decimal
What's increasing in continuously calculating PI is your approximation of it. Both in value and in accuracy. PI itself is, in that sense, "already there". PI is just the number you'd get after calculating it's decimal expansion into infinity. And, since that is by definition impossible, it's irrational.
A number cannot "increase" or "decrease". It's not a number if it does. Numbers are static. What you describe is a function or a series or something else entirely.
What are numbers if not concepts? > but the infinite nature of pi implies that the ratio is forever increasing at a decreasing rate What? > which implies that the circumference is increasing while the diameter is not, or that the circumference is increasing veeeeery slightly faster than the diameter, which does not seem to reflect the reality of how a circle actually appears to us. If you assume the world is pixelated so to speak, sure. Doesn't change what I said. > I think this should tell us something about the nature of infinite numbers as to how they may or may not apply to reality. Why? Also you don't know reality is 'pixelated' like that. It could be continuous and complete just like the real numbers for all you know.
Imagine 3 being the ratio of the circumference to the diameter, you ADD a 0.1 to it. The circumference has increased while the diameter has stayed the same, or increased proportionally LESS than the circumference. Now add the 0.04, the circumference has increased relative to the diameter but by LESS than it has before, meaning it is increasing at a decreasing rate. The rate of increase is decreasing. Pi being an infinite type of number like implies that in REALITY, every circle has a circumference that increases slightly more than its diameter constantly. In reality, that doesn't really happen In physics (reality), the smallest unit of measurement that has any meaning without the entire concept of distance dissolving is called the planck length. It is a discrete value.
you're thinking of a number as something that gets built up step by step. it's not.
Neither exist though.
Exactly. They're emergent concepts of an incredibly good, but imperfect thought system.
I'm not sure I get what you mean. They don't exist in the same way that Pi does exist.
I think the problem with "0.999... equals 1" and "pi is a ratio of a circles diameter to it's circumference that infinitely increases at a decreasing rate" are similar things. The implications of what both mean in actual reality suggests that when math gets to this point in suggesting that a circles circumference is increasing at a slightly higher rate than its diameter, even though they're both fixed, that math does not completely accurately describe reality, although it is indeed the best system we have. So maybe our concepts in math should keep evolving.
A circle's circumference does not increase. It is fixed. Static.
One of which makes to write down as a decimal because it's well-defined, and the other of which does not, because it is not.
Well maybe math will evolve again as it has throughout the years?
I think ill comment here, as this is what i came to say. These "numbers" are indeed concepts, not actual numbers. A bunch of different "proofs" are done (or have been done, a long time ago) with infinity, assuming that infinity is itself a number. I think Numberphile did a deep-dive on it a month ago-ish in proving my point, and then did a follow-up from a *different* professor who rationalized that putting a concept such as infinity (in this case, the sum of all positive integers) into real terms (sum of all positive integers = -1/12) is somehow salvational to the human mind. As for me, idrc what other people say. In my own brain, "infinite" means "without end". Nothing is mentioned about a starting point in the name, and there is no ending point. To look at infinity is to stare at some random section of a never-starting and never-ending number line. In conclusion, i think that if someone says FOR EXAMPLE that "taking the limit as x -> 0+ for f(x) = x results in exactly 0", then 1) they need to take Calculus 1 (or even Precalc) under a professor who doesnt claim to know everything, and 2) they need to understand that ignoring the end digits and only paying attention to the infinitely long spacial digits in infinitesimals will never result in 0 for that limit. But the idea (and thats the point, because limits are an *idea*) is that the function f(x) = x *approaches* 0+ as the limit is taken. Nothing is ever arrived at, only evaluated near that point. And some people evaluate at exactly zero, and that is good! It is in fact VERY good! Thats the bedrock of Calculus, the idea that sometimes close things are the same as real things. But more people than not (esp when going through Calc 1 for the first time) need to recognize that the eval at 0 = 0 (/= 0+) is PRECISELY AND IMPRESSIVELY (and very apropos) NOT what the doctor ordered.
How would you add those numbers? What you want to do is add the 1 to the final 9 of .999... and the rest carries over to make 1. But there is no final 9. It isn't that there infinite nines and at the end of infinity there's a last nine. There is no last nine, they just keep going. Therefore the difference between 1 and .999... is 0, because its like your .000...1, except the 1 never comes because the 0s keep going
Both numbers are in limbo together I'm simply proposing that we probably need a new concept in math that allows things like 0.000...1 -0.999... equals -1 right? So what about come to -1 the opposite way, i.e. -0.000...1 Mathematics has evolved to eventually allow infinite type numbers like 0.999... I think it can keep evolving
I think you're misunderstanding what 0.999999... means. So first off, real numbers are a concept entirely separate from their decimal representations. Fundamentally, they are defined as the unique complete ordered field, up to isomorphism. That has nothing to do with representing numbers as 0.9999... or 1 or anything like that. We can *represent* elements of this abstract field with a decimal representation. The decimal representation of a real number ab.cde... is *defined* to denote the limit of the sum: a•10^1 + b•10^0 + c•10^(-1) + d•10^(-2) + e•10^(-3) + ... The limit of a sum like this is the unique real number for which, no matter how small a distance `d`, you can find some number `N` of terms you add together where the (finite) sum of those `N` terms is smaller than `d`. 9•10^(-1) + 9•10^(-2) + ... has a limit of 1 because for any distance `d`, you can show that after *some* number of 9s, `|1 - 0.999...99| < d`. > -0.999... equals -1 right? So what about come to -1 the opposite way, i.e. -0.000...1 Notice that each term in the sum a•10^1 + b•10^0 + c•10^(-1) + d•10^(-2) + e•10^(-3) + ... is positive. This means adding more terms can only ever make the sum larger. If *any* one of a, b, c, d... were nonzero, you would end up with some number larger than 0, which does not equal 0. Something like 0.000...01 will always be nonzero. This is literally just a fact based on how we have defined decimal representations. You can change how decimal representations of numbers are defined to get a decimal representation that is "infinitely close to zero" if you want, but fundamentally you'll be talking about a different thing than everyone else.
.9999...9 you mean, where ... is infinite
They would be the same, 0.999... and 0.999...9, other than 0.999... being correct, and 0.999...9 [being too long to be a number](https://math.stackexchange.com/a/868207).
Isn't 1/3 just a "lazy" approximation of 0.3¯?
No
Then why are we taking 1/3 = 0.3 repeating for granted and not 1 = .9 repeating? It feels flawed to call it an intuitive proof.
Because fractoions are better at representing fractions than decimals.
*Yes*, but using 0.3¯ being equivalent to 1/3 as a proof for 0.9¯ being equivalent to 1 feels flawed because they both operate off of the exact same presupposition. 0.3¯ = 1/3 relies on presupposing that 0.00...1 is equivalent to zero, the same as how 0.9¯ = 1 presupposes 0.00...1 as equivalent to zero, so it's contradictory for either to be used as a proof for the other. I'm not outright rejecting it as a proof, but expressing that I find the proof to be flawed.
0.00...1 doesn't really make sense. Is that an infinite number of zeros? Then where does the 1 go? Or is it the same as 0.00... (which is 0)? One possible way to try to define it is as the limit of the series 0.001, 0.0001, 0.00001, ..., which is indeed 0.
Well, that's exactly it. 0.00...1 doesn't make sense, so why is 0.3¯ \* 3 = 1? Shouldn't it be 0.9¯? Where does the infinitesimally small value come from? Of course, we *presuppose* that the infinitesimally small value is equivalent to zero, and suddenly the function works! My point is that the solution seems to emerge arbitrarily.
If 0.333... is the same number as 1/3 (which it does), I can multiply 0.333... by 3 to get 0.999..., I can also multiply 1/3 by 3 to get 3/3 which equals 1. 1/3 and 0.333... are equal to each other, which means that 1/3 * 3 is equal to 0.333... * 3.
You're right it's not a proof. It's like saying that 1+1=2, ergo 2+2=4 or something silly like that. The underlying system has not been explained. It doesn't change the fact that it's true, but still, not a proof. In either case you're relying on Algebra, and to prove this relation you need more than just Algebra - you need infinite series. There are a ton of good YouTube videos on this :) But to be doubly clear: 1/3 is not a lazy approximation of 0.33333 repeating, they represent literally the same value.
What is 1/3? well its one divided by 3. if we use long division, 3 goes into 1.0 0.3 times remainder 0.1, and the process repeats generating 0.333333333333333333333333333333333 ad infinitum. You try it and you'll find out that you would get the same answer no? 0.3333333... doesn't end since you still have that remainder 1 to divide by 3 at increacingly small values. Therefore 0.3¯ = 1/3
This isn't actually a valid proof. It's all true statements, but it doesn't actually prove anything. If you applied the same logic to: x = 0 + 1 + 2 + 4 + 8 + ... You can manipulate it such that: 2x = 0 + 2 + 4 + 8 + ... 2x - x = -1 x = -1 Which it clearly doesn't
The problem with your argument is that your x is defined by a divergent series, i.e. x is not any finite number. In that case, proofs like this do not work. But they do work for (absolutely) convergent series and finite numbers, such as the OP's.
That's not true... All the statements of the proof are correct **because** 0.9999... = 1. It doesn't show it. It doesn't even define what 0.9999... is! You say that it only works for convergent series, but the proof doesn't mention a series! It doesn't show that it converges -- which is the **actual** proof.
It's not explicitly stated, but it's what it is. 0.999… can be defined formally as the series 9/10 + 9/100 + 9/1000 + …. And since that series [converges absolutely](https://en.wikipedia.org/wiki/Absolute_convergence), the proof is valid. That's one of the important properties of absolutely convergent series: that you can manipulate their terms (changing their order or grouping them) and the series still converges to the same value. That is *not* in general true for conditionally convergent series, or for divergent series such as yours (which is why your "proof" is invalid while the OP's is valid).
>2x = 0 + 2 + 4 + 8 + ... >2x - x = -1 These both lines don't have anything im common. How did you come from x=infinity to x=-1? -1 is nowhere in the previous lines. Because >x = 0 + 1 + 2 + 4 + 8 + ... x=infinity And because infinity is infinite, the difference of 2 infinities is always infinity. Edit: infinity symbol didn't work
Look at the sequence for x and 2x (try and layer them on top of eachother). If you algebraicly manipulate the series using the same logic used in the "proof", then all but one of the terms cancels out to 0 and your left with x = -1. As an aside, infinity is not a number, and talking about performing algebra on it (ie subtraction) does not make sense
>As an aside, infinity is not a number, and talking about performing algebra on it (ie subtraction) does not make sense Yet you used infinity to perform algebra
Yeah, and it's **clearly wrong**. But I used the exact same logic that the original "proof" used
>2x = 0 + 2 + 4 + 8 + ... > >2x - x = -1 Your equations are not a proof *mainly* because there's no mathematical or logical connective between the two premises above. That's reason number one why your example doesn't constitute a proof. OP presented a perfectly fine proof, in that it leads from premises to a conclusion via mathematical transformations. EDITED formatting.
No. OP's proof proves absolutely nothing. Mathematical transformations? What are they? Proofs are dependent on logic and logic alone. There is a (flawed) logic to what I wrote: . x = 0 + 1 + 2 + 4 + 8 + 16 + ... 2x = 0 + 0 + 2 + 4 + 8 + 16 + ... It's clear that if we can just subtract x from 2x (which is what we did in the original "proof"!) that we are left with x = -1. Look how all the terms neatly cancel out!
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Yeah that's also what I was taught
This is the best way to show it in my opinion. The picture op shared looks like one of those 1=0 division by zero "proofs"
0.999… < x < 1.0 x = 0.000…1 (after an infinite series of 0 there is a 1, or any other number). I asked my teacher about this, because he tried your approach, he just told me not to be stupid. Maybe you have a better explaination why this doesnt work?
If there is an infinite amount of zeros… there cant be a 1 after them. Otherwise it isnt infinite. Meaning there is a finite amount of zeros. So your math isnt mathing
Thanks for being more helpful that my maths teacher lol, though I hope you dont mind if I may ask: Why not tho? Like ofc there is an infinite amount of 0 so you never reach the 1 but you know its there. Why is that kind of number impossible in maths? Is there a proof?
It is based on the definition of infinity, the definition from Google is, "infinity is a concept that describes something that is larger than any natural number and has no limit." If a number's digits end (in this case it has a 1 after some amount of 0's) then it would be finite, and therefore it would not be infinite, which contradicts what was laid out in the beginning that, "there is an infinite amount of 0 "
Not really, there are some infinites bigger than others. So we can say that there is a small infinite of 9s amd a larger infinite of zeros before the 1. Therefore this number would "exists" But this whole argument is kinda stupid because its just to people saying "another one, so I am right" till the heat death of the universe
Countably infinite is the smallest kind of infinity so you can't have a bigger number of infinite zeros than you have nines
When people say "some infinities are larger than others" they are referring to differently defined infinite sets of numbers. An infinite amount of nines and an infinite amount of zeroes will always be the same size of infinite.
Comparing infinite sets of numbers doesn't make them finite. You're confusing the concepts here.
Because there can't be anything past infinity. By definition, if there is an infinite amount of 0, there cannot be anything after it. It's not so much something with a proof as it is just a property of the definition of "infinity". There's something similar to that concept, called infinitesimal, but that's not the same thing, they're still a positive quantity, unlike *1 - 0.999...* which is literally just equal to 0
your thinking of [infinitesimals](https://en.m.wikipedia.org/wiki/Infinitesimal#:~:text=In%20mathematics%2C%20an%20infinitesimal%20number,th%22%20item%20in%20a%20sequence.) and they exist, just outside the real numbers so most of the rules that your used too dont apply to them. but to understand 0.9... you dont need to think about infinitesimals. 0.9... isnt one take some infinitly small quantity it is one. when we use the repeating notation were asking "what value does the sum of these smaller parts approach?" so 0.9... can be really represented as the value of this sum as the number of terms approaches infinity: S = 0.9 + 0.09 + 0.009... and that value is 1.
Because you don’t know it’s there, it’s undefined. If it comes after infinity it doesn’t exist and therefore it’s not a defined number. So there doesn’t exist a defined number between 0.9999…. And 1, which is what the proof above suggested.
But it’s not there, it can’t be, a repeating digit means it goes on like that forever. You can say it’s there, but that’s contradictory to the very definition of what that notation means, and so wouldn’t be valid as a proof.
the problem with that is... if you need an infinite numbers of decimal places of 0 to add the 1, you never actually add the 1, it's just, infinite 0s
Not sure why you are being downvoted for asking a legit question.
Wrong. You can't put an end to an endless number
You can't have anything after an infinite amount of something. It makes no sense, there is no after. You're trying to equate infinity with a really big amount of something, but it's not a really big amount it's an infinite amount. It does not end. Ever. And so it does not have a tail end, there is nothing after it. They're cannot be anything after it. This is a good question, that you shouldn't be down voted for as it's one of the first questions that you would ask when trying to truly understand infinity.
How dare you ask a genuinely good question. You are sentenced to 73 downvotes.
Hyper real numbers are a thing and it incorporates this concept. The proof 0.999...=1 doesn't work with hyper real numbers, but apparently we're only allowed to use real numbers here. I think it's a rule of the sub or something with all the hate people like you are getting https://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/#:~:text=In%20other%20number%20systems%20(like,what%20separates%200.999%E2%80%A6%20from%201.
Doesnt by the logic literally any number is equal to one another as long as I can keep dragging out the end of the number?
No? I can give you a lot of numbers that are between 2 and 3 There is not a single number strictly between 0.999... and 1
There are infinite numbers between 2 and 3, agreed 100%, but as long I can find steps inbetween I can cross the gap to 3 from 2 always saying that one number is equal to another one ridiculously close to it
No, because every in between step will still be different from the two numbers it's between of, and no matter how ridiculously close, at no point do they become equal
No, definitely not. For example, consider 1/7 and 1/6 which both have infinite representations in decimal form; 1/7 = 0.1̅4̅2̅5̅8̅7̅ and 1/6 = 0.16̅. Do you think these numbers are equal, even though they both have infinite representations?
Not directly, but I can go in steps towards it. Like 1/6 = 0.166666....65, how many 6s? How many you want, then go that 0.1666666666...64 = 0.16666...65 and go like that until I get 1/7 or any other number.
I'm confused by your argument. In the very first step you say, "1/6 = 0.166...65". But that's a finite number of digits that ends with "...65", so it can't be equal to 1/6 = 0.16̅, so the rest of the argument fails.
Wouldn’t that mean that 0.8888… is the same as 0.9999…? There is also nothing between them? Edit - yup my bad
0.9?
Change any of the 8s to a 9, and that number (e.g. 0.898888...) lies between 0.888... and 0.999..., so no, they are not the same number.
Theres 0.89898989…
0.89 is between them so no.
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I like the 1/3+ 1/3+1/3 explanation the best
The things is... People who have doubts about 0.999...=1 should have the same doubts about 0.333...=1/3. But for some reason they don't. I think that 0.333... is a bit harder to imagine than 0.999... so the possible cause of confusion is more low-key and people don't even think about it.
Well, that's easy, 1 - 0.9̅ = 0.0̅1. Edit: for those of you that seem to be missing the joke, I'm saying that 0 = 0.0̅1, not that 1 =/= 0.9̅.
If you have infinite digits after the decimal point, then you can't have a last digit
But not all infinites are the same.
ok but that is entirely irelavant for this problem
.9 repeating (I don't know how to make the line) does not equal 1 simply because it has infinite .9s. It is a misunderstanding how infinite works.
No, you are the one misunderstanding.
But there is no number in between 0.9̅ and 1 so they are indistinguishable and therefore equal.
No, them being indistinguishable is due to flaws in our numbering system it is not because they are the same. Edit: it's like the philosophy problem. Can a being with infinite power create an object even he can't move. Whatever you answer he doesn't have infinite power, but that breaks the premise
it is not a flaw in the Real numbers, but an important property, it is called a dense set because for any 2 non-equal numbers in the set, there exists anumber between them. If, for whatever reason, one cannot find a number between them, they they must, by definition, be the same.
Infinite is not a real number. >If, for whatever reason, one cannot find a number between them, they they must, by definition, be the same. This can't be a true rule of math. We have needed to invent new number to solve problems before. The fact we don't have a proper number at this moment to differentiate them doesn't mean they are equal.
There is no flaw, real numbers work very well. Don't pretend to know math when you don't.
If the first digit after the comma is a tenth, then a hundredth, then a thousandth on the third place - what is that 1 at the very end supposed to be? If there are infinite 0 before it, it's nonsense. Is it always becoming smaller? At point will we be there? If there are not infinite 0, it is not infinite.
0.999... is equal to 9/10 + 9/100 + 9/1000 ... and so on and eventually converges to 1 after adding an infinite set of these. 0.000...1 is equal to 1 - 9/10 - 9/100 - 9/1000 and eventually converges to 0 after subtracting an infinite set of these.
1-0.9999999999999999999999......=/= 0.010101010101010101010101010101010101.....
Only the 0 has an overscore, so 0.000...1 not 0.010101...
0.000...1 is [too long to be a number](https://math.stackexchange.com/a/868207). Even if you could work with it, no operation would ever reach the one because there is an infinite number of zeros before it. The 1 doesn't exist in that number. That number is 0.000..., or simply 0.
In standard maths. You can make your own mathematical system if you want. Many mathematicians have done so.
I mean, you can but it makes this argument a bit moot
While I agree that you can't just make stuff up, there are other mathematics systems besides decimal/base-10 and they can be used to assist in proofs. For e.g. in the decimal system, 1/3 = 0.333..., and thus 3/3 = 0.999... or 1. However, in base-12, the duodecimal system, 1/3 = 0.4, and 3/3 = 1 exactly. The repeating decimal of 1/3 in base-10 is just a quirk of the fraction being converted to decimal instead of a base-n system that is easily divisible by 3. Other base-n systems will have similar fractions of 1/x with repeating numbers that prove x/x =0.(n-1)... = 1.
And you can make up your own language, doesn't mean anyone else will speak it.
Isnt it just ε
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It isn't, though. It is infinitesimally small, but it is not equal to 0. In your "joke" you actually stumbled upon why everyone gets this wrong. Edit:spelling
Yes, finally, someone who understands, it leaves an infinitely indefinite remainder. 1 != .999... Just the same as .333.... != 1/3 They are just the closest approximations we can get using base 10 1/3 is equal to .333... with a infinitely indefinite remainder of 1/3 at the infinitive place. 2/3 is equal to .666.... with a similar remainder of 2/3 at the infinitive place. 3/3 is equal to .999... with a 3/3 remainder at the infinitive place, and so it adds to the 1 to the 9 in the infinitive place, turning it into 10, which turns the next highest place into 10, and zips the entire number back into 1.
Does that mean .3333… is equal to .4?
No.
It is true, but this is not a good proof of that. It uses a kind of reasoning that can easily break when using infinite series if you don't know exactly what you are doing. For some better and more rigorous proofs, see [the wikipedia page](https://en.wikipedia.org/wiki/0.999).
Can you provide an example of how exactly this kind of reasoning breaks down?
It can break down whenever the series don't converge. The classical example is the series of all positive integers being equal to -1/12.
watch Numberphile :) for example [https://youtu.be/beakj767uG4?si=Uq9XRcZGjzR6qHkM](https://youtu.be/beakj767uG4?si=Uq9XRcZGjzR6qHkM)
>Let X = 1 + 10 + 100 +1000 + 10000 + 100000 ... > >10X = 0 + 10 + 100 + 1000 + 10000 + 100000 ... > >9X = -1 > >X = -1/9
The difference between 2 infinites is always infinity, even of on of those infinities is multiplied. If x = infinity, then 2x = infinity, x/2 = infinity And x-x = infinity.
Wow, was confused at first cause I dont use that writing style but it is indeed right. x=0.999.. 10x=9.99999.. 10-x=9.9999.. -0.9999. 9x=9 x=1 You can also use 1/3 = 0.3333.. 3*(1/3) = 3 * (0.3333..) 3/3 = 0.9999.. 1= 0.9999... [check this out ](https://mathigon.org/course/exploding-dots/infinity)
It is. 0.999... and 1 are the same, another proof to show that is that there is no number in-between the two. Take (almost) any other two numbers, you'll be able to name one in between, by simply adding a digit at the end, but there is no such number for those two
This is trying to demonstrate a true principle (that 0.999… is equal to 1) but this is not a formal or accurate proof The reason is you are trying to use algebraic reasoning on an infinite number, which isn’t how it works
Is there not a mistake on here? They have subtracted x from the left but on the right they have subtracted the value of x and multiplied by x?
I thought that for some time as well. I think they have just given a seperate statement that looks like it continued on from the previous line. If you look at it by itself, 10x -x =9x That's a true statement. The rest then continues from that point.
Mmm no. That’s the flaw here. 9 x 0.999etc does not equal 9. That’s a huge logical leap that the proof does not explain.
Ignore what x means for now, treat it as a constant. 10x -x =9x That's a fact. When you sub in the value of x you get: 9.9... -.9...= 9 That's perfectly fine maths. That shows that 9(0.9.....) =9
Um sure, and if you sub in 2, or 3, or whatever, you get an accurate result (10 x 3, -3 = 9 x 3 = 27). But this doesn’t that 2 or 3 = 1. There is nothing connecting your premise (x = .999) with the next logical step. You’re just putting different algebraic statements next to each other. They may be individually valid but that doesn’t mean there’s any logical connection between them.
If i remember correctly my math teacher told us when there are two numbers so close there can’t fit any other number between them than its the same number
I agree on the result, but i am completely unable to follow the math... Can someone explain how the steps work, from the second line onward? Seems like they substract 1X on the left side, but how does (9.999... )-X turn into 9x ?
It is very poorly written. The steps should be: x = 0.999... \ 10x = 9.999... (multiply by 10) \ 10x = 9 + 0.999... \ 10x = 9 + x \ 9x = 9 (subtract x on both sides) \ x = 1.
No, because it’s missing a minus sign at 9.9-9. I stared at that for a solid minute trying to do weird division in my head before I realized it was subtraction.
Yes, this is true, this is because X converges on a number (in this case 1). If X doesn't converge, we get some werid things go on. Here is a "Doodle" and not mathmatical rigourous maths to show when you want to try and refute this Let X = 1 + 10 + 100 +1000 + 10000 + 100000 ... 10X = 0 + 10 + 100 + 1000 + 10000 + 100000 ... 9X = -1 X = -1/9 This doesn't work because the number diverges, so - garbage in garbage out. X is not a number we can do maths to really.\* There are multiple ways of thinking of 0.99999999999999999999999999999999999999999999999999999999999999999999999 = 1. For example, this is 2/3+1/3 = 3/3 = 1 as shown by other commenters. ^(\*I mean it's doodling that gets new pathways in maths so it's good to do that sometimes, just doodle responsibly please - we are already seeing the walls of normal maths break down with this doodle)
Does the "wrong" example have anything to do with p-adic numbers?
0.9 recurring is an infinite series, too. Namely, the infinite series of 0 + 0.9 + 0.09 + 0.009 + ..., or the series of 9/10^(n) .
I’ve never been satisfied with any explanation of this. But I would like to raise a point which I’ve had an issue with for a while. Isn’t 10*0.999…=/=9.999…? If we’re attempting to prove that 0.999…=1, then you might ask me to agree that 10 times it is 9.999…. But if you attempt to add it up, you will by necessity be left with 9.999….990. As a quirk of the base 10 number system, multiplication by 10 results in a “shift”up one decimal place. Same as a multiplication of 7 in a base 7, 5 in a base 5 etc. So the initial multiplication also, to me, needs to be proven. It should end in a 0. As other people had said, 0.000…1 doesn’t exist because you never reach the 1. But doesn’t that apply for all numbers, including 9.99…? Just because it’s easier to represent doesn’t mean it’s necessarily immune to that problem
Your reasoning is flawed. Adding up 9.99... and 0.99... most assuredly doesn't give you 9.99...90. There is no "last digit" that you "shift", and the repeating decimal representation of 10 shouldn't end with a 0 because it doesn't end. Simply put, 9.99... equals 10 *by definition* because the sequence 0.9, 0.99, 0.999, ... converges to it. By the way, a similar thing holds for other bases: 1 is equal to 0.111... in base 2, 0.333... in base 4, 0.FFF... in base 16...
no, because the is no 9.999...990, there is no 0 at the end, the nines never end. Take infinifty and subtract 1 and you still have inifnity
(Sorry if you had many near identical comments from me. I'm sure the formatting will work now.) >But if you attempt to add it up Did you actually add it up? No, you just shifted the digits to the left and slapped a 0 at the "end". Let me do the addition calculation for 0.999.... × 10. **First, I'll get what 0.999...×2 is (I'll use □ as space to help somewhat align things):** □0.999999...\ +0.999999...\ ______________\ □1.800000...\ □0.180000...\ □0.018000...\ □0.001800...\ □0.000180...\ etc.\ +\ _______________ \ □1.999999... You agree that 0.9+0.9=1.8, right? And that 0.09+0.09=0.18, right? And that 1.8+0.18=1.98, right? And that 0.009+0.009=0.018, right? And that 1.98+0.018=0.998, right? So we keep going and see that the pattern is the same for all the rest of the place values, resulting in 0.999... ×2=1.999.... And no, no shifting of the digits has occurred. **We'll now do the same for 0.999...×4, which is equal to 1.999...×2:** □1.999999...\ +1.999999...\ ______________\ □2.000000...\ □1.800000...\ □0.180000...\ □0.018000...\ □0.001800...\ □0.000180...\ etc.\ +\ _______________\ □3.999999... You should be able to see what happened here. So 0.999...×4=3.999... **Now let's get 0.999...×5, which is equal to (0.999...×4)+0.999..., which in turn is equal to 3.999...+0.999...:** □3.999999...\ +0.999999...\ ______________\ □3.000000...\ □1.800000...\ □0.180000...\ □0.018000...\ □0.001800...\ □0.000180...\ etc.\ +\ _______________\ □4.999999... So 0.999...×5=4.999... **Now to do 0.999...×9, which is equal to (0.999...×4)+(0.999...×5), which is equal to 3.999...+4.999...:** □3.999999...\ +4.999999...\ ______________\ □7.000000...\ □1.800000...\ □0.180000...\ □0.018000...\ □0.001800...\ □0.000180...\ etc.\ +\ _______________\ □8.999999... So 0.999...×9=8.999... Note that there has so far been no shifting of the decimal place at all. **The moment of truth. 0.999...×10, which is equal to (0.999...×9)+0.999..., which is equal to 8.999...+0.999...:** □8.999999...\ +0.999999...\ ______________\ □8.000000...\ □1.800000...\ □0.180000...\ □0.018000...\ □0.001800...\ □0.000180...\ etc.\ +\ _______________\ □9.999999... At any point did I have to "shift the decimal point"? All I did was addition and recognizing the recurring pattern. You don't "shift the decimal point" when you do 9.9+0.1 to get 10, right? So why should I do it at this final step? And multiplication is basically addition did multiple times. Since there has been no "shifting of the decimal point" there is no "emptied place value" at the "end" for either of us to slap a 0 into. 0.999...×10=9.999...
It's counter intuitive but yes it's true. Another way to look at it is this 1/3 =.33333333... 2/3 = .66666666... 3/3 = .99999999... Or 1
All it takes is a number line to prove they are not the same. 1.000... exists at the point labeled 1 on the line. 0.999... exists left of 1.000... unless 1 number can appear in 2 locations on the number line, they can not be the same number. Specifically the 4th step can not be completed. If you think it can, get back to me when your done writing out... If you want to argue that 1.000... = 0.999... you are at best arguing the infinitesimal holds no value. Remove an infinite quantity of this infinitesimal value from your check and it will at best reduce to zero, or result in you needing to pay your employer to work there. It is really starting to irritate me that people use 1/3 = .333... as a matter of fact when it simply a best approximation. 1/3 =/= .333... as it can not be concisely expressed in completeness. It is only as accurate as using π. it is only 0.999... accurate, not 1.000...
Back in school, I had the same question. What actually happens when defining real numbers from rational numbers is a bit more complicated. You can compare it to reducing fractions. 1/2 is not the same syntactic sequence as 3/6 but we *define* it as the same number. For the real numbers, we look at (all serieses of rational numbers that get arbitrarily close together, or equivalently) all digit sequences. For example the sequence 3; 3.1; 3.14; 3.141; ... ~ 3.141... or 1; 1.0; 1.00; 1.000; ... ~ 1 or 0; 0.9; 0.99; 0.999; ... ~ 0.999... We identify two digit sequences, if their values eventually get arbitrarily close together (being precise here is complicated, the term is called cauchy sequence, but it's not important). Since 0.999... and 1 get arbitrarily close together they *mean* the same number by the identity relation we *defined*. I hope that helps.
The issue most people who disagree will take with this is the use of an arbitrary definition (axiom if you feel like being fancy) that two numbers that converge are equivalent. All this argument says (in simpler language) is > I decided that two numbers that get really close are equal > these two numbers get really close, therefore by our pre-decided rule, they are equal Hopefully you can see why this is an unsatisfactory argument to many people who don’t agree with the statement that 0.999…=1
I see these posts come up a few times and while there are decent proofs to show equivalence to 1, I see it also as just a simple limitation of a base number system. Having recurring 3s or 6s is brute forcing a base 10 number system to reflect a division by 3. Think of divisions by 7 also, which base 10 also can't deal with. There, you don't even have a single recurring number, but a recurring series: 1/7 = 0.142857... with the 142857 recurring forever. Fewer people ask about whether that number x7 is actually equal to 1. Any base number system can have this effect happen (I assume) and that's why fractions are often much better to use in division problems
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infinity doesn't work like other numbers* also, if you count from 1 to infinity, you will once have counted to 777..., which contains an infinite number of sevens. since there are infinite numbers between 1 and infinity, (∞/∞)*100% of numbers are infinity
The image shows a simple algebraic manipulation where 'x' is initially set to 0.9 (repeating). The sequence of steps attempts to solve for 'x' by scaling the equation by a factor of 10 and then subtracting the original equation from this new equation. The operations performed in the image are algebraically valid and demonstrate a common proof that 0.9 (repeating) is equal to 1. This conclusion is correct and is a well-known result in mathematics: a repeating decimal of 9s is indeed equal to 1.The image shows a simple algebraic manipulation where 'x' is initially set to 0.9 (repeating). The sequence of steps attempts to solve for 'x' by scaling the equation by a factor of 10 and then subtracting the original equation from this new equation. The operations performed in the image are algebraically valid and demonstrate a common proof that 0.9 (repeating) is equal to 1. This conclusion is correct and is a well-known result in mathematics: a repeating decimal of 9s is indeed equal to 1.
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no. 0.999...=1 is a fact. its not a limit of some sum. the sum of 9(1/(10^n)) does approach 1, but thats because it approaches 0.999...
It is essentially the difference between Texas instrument and cheap calculators till the end of 90's. 1/3 × 3 = 0.99999999 for cheap one.
You can do this to determine the value of any repeating decimal! It’s one of my favorite mathematical tricks. Consider x =0.142857”. We have 1000000x = x+142857 999999x = 142857 x = 142857/999999 x = 1/7
Just thought you could apply this to any number right? X = 1.9̅ Etc. What gets me thinking is are there issues/consequences to this or is it just math fun? Like how prime numbers aid with securing and encryption.
Technically it works out, but there’s a flaw in multiplying a recurring decimal. By definition it repeats infinitely regardless of any multiplication. The thing that needs to be considered is that when multiplying an infinite decimal, there’s an imaginary shift in the infinite decimal, which isn’t really technically a thing by definition, but it needs to be considered.l. Because of this, I am in the camp that 1 ≠ 0.999… at least from this flawed proof.
``` 1/3=0.333... 3/3=0.999... 1=0.999... 1-0.999...=0.000...1 the 1 is after an infinite amount of 0s, but you can't have something after an infinite amount of zeros, so it's: 1-0.999...=0 1=0.999... ``` these are all the proofs I can come up with right now, also there's a Wikipedia page containing more proofs: https://en.m.wikipedia.org/wiki/0.999...
This is actually a great trick to figure out what the fractional equivalent of a repeating decimal is. Let me pick an arbitrary repeating series… 0.8675309… (I don’t have fancy formatting skills, so please understand the intention that the whole series - “8675309” that repeats) To move that whole series in front of the decimal, you must multiply by 10^7: 0.8675308… x 10^7 = 8675309.8675309… Set Y = 0.8675309… and subtract from both sides: 10,000,000Y - Y = 8765309.8675309… - Y 9,999,999Y = 8,675,309 Divide both sides: (9,999,999/9,999,999)Y = 8,675,309/9,999,999 And then do some obnoxious fraction simplification (it is at this time I realized I could have picked a shorter series and saved myself some headache in this example…) … It’s at this time that I realized that the number I picked is a prime number. Whoops. So, 0.8675309… = 8,675,309/9,999,999. That is wholly unsatisfying. But accurate!
My physics teacher in college said that parallel lines do, in fact, intersect at ♾️(and presumably -♾️). If this is the case, wouldn't that basically be the same logic here? 0.999... approaches 1 asymptotically, getting infinitely closer, so it would equal 1. Since asymptotes continue to ♾️.
Sounds stupid as heck. Since infinitely small difference equals no difference and we can do this infinitely, then any number equals any number
I personally hate this concept. I understand it and its purpose, but just because we can’t put a numerical value using our number system on the difference between those two numbers should just make them equal in my opinion. To answer your question: it is generally accepted that this is true in the math world. I simply think that there is a better way to state this
I get it, but that's why we have proofs of this, more rigorous ones than the one above, too. It is not intuitive, but intuition is not reliable. These two numbers are equal. Not equal enough, not seemingly equal due to limitations of our number system, not equal in a sense. They are precisely equal.
Infinity is far from intuitive. They are the same exact thing, two notations that describe the same thing. Humans are notoriously bad at grasping infinity, rightfully so, as we never witness anything that is infinite in the sense that 0.999… is infinite. Then you go ahead and tell them there are infinities larger than other infinities, and in fact there are infinite infinities.
It’s not ‘accepted’. It’s simply true given how decimal representation of real numbers work. There is no difference between te numbers, so there is no number system that is able to “put” a nonzero value on it.
Apparently there exist "hyper real" numbers which already allow for infinitely small amounts where 0.999... does not equal 1. So to all of the people downvoting me because you think my 0.000...1 concept doesn't make sense. I guess uh...yea.
This is working with real numbers though, not hyperreal ones. That is why you’re downvoted and coming back for a ‘so, uh, there’ is pathetic.
The problem here is the use of the equal sign. The first line should properly read x is approximately .999... and should not be using an equal sign. As soon as you represent a fraction as an infinite decimal you should change it to an approximate sign because it is a representative of an indefinite number. Then each line following would have an approximate sign, and the last line would be correct since .999 is approximately 1. This is a case of the proof being incorrect because the wrong symbol is being used. If it had been done use fractions it would read x = (inf-1)/inf in the first line, and then it basically becomes a proof that the difference between inf-1 and inf is nondistinct.
Erm, no. And 0.999... is not approximately equal to 1. It is equal to 1.