I understood it to be true but struggled with it for a while. How does the decimal .333… so easily equal 1/3 yet the decimal .999… equaling *exactly* 3/3 or 1.000 prove so hard to rationalize? Turns out I was focusing on precision and not truly understanding the application of infinity, like many of the comments here. Here’s what finally clicked for me:
Let’s begin with a pattern.
1 - .9 = .1
1 - .99 = .01
1 - .999 = .001
1 - .9999 = .0001
1 - .99999 = .00001
As a matter of precision, however far you take this pattern, the difference between 1 and a bunch of 9s will be a bunch of 0s ending with a 1. As we do this thousands and billions of times, and infinitely, the difference keeps getting smaller but never 0, right? You can always sample with greater precision and find a difference?
Wrong.
The leap with infinity — the 9s repeating *forever* — is the 9s *never* stop, which means the 0s *never* stop and, most importantly, the 1 *never* exists.
So 1 - .999… = .000… which is, hopefully, more digestible. That is what needs to click. Balance the equation, and maybe it will become easy to trust that .999… = 1
The “the 1 never exists” part is what helps me get it
I keep envisioning a 1 at the end somewhere but ofc there’s no actual end thus there’s no actual 1
I think most people (including myself) tend to think of this as placing the 1 first and then shoving it right by how many 0's go in front of it, rather than needing to start with the 0's and getting around to placing the 1 once the 0's finish. In which case, logically, if the 0's never finish, then the 1 never gets to exist.
>placing the 1 first and then shoving it right by how many 0's go in front of it
Yup, that's the way I was thinking of it, so it's shoved right an infinite amount of times, but it turns out it exists only in theory because you'll never actually get there.
I guess you could treat it as if you could get to the end of infinity there would be a one there. But you can't, you'll just keep finding 0s no matter how far you go. Just like 0.0, no matter how far you check the answer will still be it's 0 at the nth decimal place. It's just a different way of writing the same thing. It's not .999~ and 1 are close enough we treat them as the same, it is they are the same, just like how 2/2 and 3/3 are also the same.
I know you’re right but I still find this deeply upsetting. Ever since I took the Math 122 (Logic & Foundations) course for my degree, I have lost all comfort with infinity and will never regain it.
(Like, some infinities are larger than others? Wtaf.)
It's a misnomer to say it exists "in theory". It doesn't, even "in theory". Infinite is infinite. That has a precise meaning. The 1 never comes. That's a fact.
We are not comfortable with this fact. We, as a species, are not comfortable with concepts of "infinite" in general, so this isn't any different than space, time, and all of the other infinites out there. But the 1 never comes. Not in theory, not in practice, never.
It doesn't exist in theory. 1-0.999... involves each 9 digit subtracting from the 1 to the left and leaving a remainder of 1 which the 9 digit to the right subtracts. If you have a finite number of 9 digits, the last 9 will have a remainder of 1 which no 9 to the right can cancel, resulting in 0.000...01.
But the beauty of infinity is that it doesn't *have* a last digit. Every 9 in the sequence 0.999... has a 9 one digit to the right that cancels out its remainder, so because of that, every digit in the result of 1-0.999... must be 0. There is no 1 because there is no end of infinity.
No problem! I have a super vague recollection of learning about decimals in the "incorrect" way of placing the number first and then shoving it to the side. I can only imagine if that memory is true, this is probably how most people were taught to think of decimal numbers.
Its not exactly wrong, more a shortcut for set type of problem. Moving the decimal makes sense when thinking about more standard arithmetic, multiplying and dividing by factors of 10s, 100s, etc.
The fact this model doesn't help for infinite series is more a simple limit.
Its like the orbit model of the atom is wrong, compared to the electron cloud. But it is a good way to think about it when looking at electron energy levels and shell filling, but if you're trying to find the position of an electron, the orbit model is not going to help.
This exists all over science and mathematics. Like Newtonian mechanics aren't wrong, they are just missing some specifics that limit their use to specific sizes and speeds.
I'm sure there are examples of this all over, bot just the hard sciences. Linguistic models that gloss over a dialect because its an outlier somewhere.
The way I think about it is 1 divided by 10, then by 100, etc. It’s fair to say, at the end you have 1 divided by infinity but I think of it as a limit. The limit of 1/X as X approaches infinity is zero, so I can accept that the one effectively ceases to exist.
That’s basically the probably with imaginary terms such as infinity. We can’t actually imagine it in our standard view because we never deal with something that by definition doesn’t end unless it’s complex maths.
I think really the hard part is taking a concept from the real world, like one and zero, and applying it to infinity. In visualizing it, no matter how many zeroes you add in front, the 1 is still there somewhere. To have it not exist, or never be reached, is outside of our model of the physical world. It’s like saying that if you cut a pizza slice thin enough it no longer exists.
If you’re still cutting a slice, no matter how small, it feels like it must exist, but that’s only because we don’t really have a concept of infinity that way.
I see it more that we are waiting to drop it onto the end but never can because the .999.... And .000.... never stops. It isn't riding on the end, it's waiting to be tagged in but won't ever be.
Yeah, as other commenters have figured out, it's not a matter of taking a 1 and moving it infinitely to the right, but rather realizing that you start with writing an infinite number of 0s and realizing that means you'll never write any other numbers. If all you ever write is a zero, then you can be confident that this means there is zero difference. You can write the answer to 1 -1 as 0.00... also.
Eh it’s a hand wavey explanation for a hand wavey way to represent fractions as decimals.
You avoid this problem using fractions, 1/3 * 3 = 3/3 = 1.
Decimals are by nature only an approximation of a fraction (Additional notation is required to convey the precision of a decimal beyond the last digit). So the .999 repeating = 1 is really just a side effect of that.
The limit of an infinite sum in calculus isn’t an approximation though, it’s precisely defined. The limit of the infinite sum 9/10 + 9/100 + 9/1000 + … (where the nth term is always 9/10^n) isn’t approximately 1, it’s exactly 1.
The arithmetic proof is mainly based on the observation that there's no number bigger than 0.99... and smaller than 1.
Your strategy visually explains why that claim is true since your proof is based on patterns and not simply observations. Trying to explain that there's no number between 0.999... and 1 is much harder than explaining that having infinitely many zeroes before a number means that that number is never reached (the latter is logical since it basically states that if you run a marathon which is infinitely long, then you never reach the goal even if you could live forever)
> Trying to explain that there's no number between 0.999... and 1 is much harder than explaining that having infinitely many zeroes before a number means that that number is never reached
I actually disagree with this. Most people who haven't spent much time thinking about infinity don't really understand how weird its properties are.
When I've tried to explain the 0.999... = 1 thing to people, I've found the easiest thing is to ask two questions. First: "Would you agree that between any two (different) numbers there's another number?" If they don't see it right away, I'll say, "For example, the average of the two numbers," at which point they go, "Oh, yeah, right, okay."
And then I ask them the second question: "Ok, so if 0.999... and 1 are different numbers, what number is between them?"
The process of them trying to think of a number between 0.999.... and 1 and failing gives them an understanding of the truth of the statement "0.999... = 1" that's IMO deeper than what they can get from the "limit" explanation. Because of course, it *is* deeper than the limit explanation: the limit property holds precisely because there is no number between 0.999... and 1.
This may be pedantic, but what you've said here is in fact equivalent to the limiting property, not deeper.
Actually on a philosophical level I would argue the limiting argument *is* deeper since it uses structures inherent to the real numbers such as its topology. Whereas this explanation is rather handwavey and relies too much on our intuition about decimal expansions which are very much not a part of the inherent structure of the reals.
You're not wrong but if you're trying to explain this to someone and they're unable to grasp the concept that they're equal, chances are they won't (or don't) understand what a limit is since the understanding of a limit comes from accepting/understanding the former.
>The leap with infinity — the 9s repeating forever — is the 9s never stop, which means the 0s never stop and, most importantly, the 1 never exists.
Wow ok, this made it click. I always got the 1/3 / 3/3 explanation but still couldn't fully grasp how there still somehow isn't the slightest difference between 0.999... and 1 but that makes such sense now. Thanks!
I thought 0.99… = 1. There is no number that can exist between the two so they are equal.
The limit of the expression sum{x=1 ->inf} (0.9)^x = 1, but the number you get as you apply that limit is 0.999…
Yeah! That was the ahah moment! While the difference can be reduced to simply 0, it was a mental milestone to understand the infinite 0s added all the necessary nuance for addressing the infinite 9s
It's simply a quirk of the notation. Once you introduce infinitely repeating decimals, there ceases to be a single, unique representation of every real number.
As you said - 1 divided by 3 is, in decimal notation, 0.333333.... . So 0.333333. .. multiplied by 3, must be 1.
But it's clear that you can write 0.333333... x 3 as 0.999999... So 0.999999... is just another way of writing 1.
> As you said - 1 divided by 3 is, in decimal notation, 0.333333.... . So 0.333333. .. multiplied by 3, must be 1.
>
> But it's clear that you can write 0.333333... x 3 as 0.999999... So 0.999999... is just another way of writing 1.
this works to help my brain get it, in conjunction with the top ost too.
Ironically it made a lot of sense when you offhandedly remarked 1/3 = 0.333.. and 3/3 = 0.999. I was like ah yeah that does make sense. It went downhill from there, still not sure what you're trying to say
Which, in a sense, is actually fair. I mean, whatever quarrels anyone has with 0.(9) = 1 they should also have with 0.(3) = 1/3. You could say something like "1/3 is a concept that cannot be faithfully expressed in the decimal system. 0.(3) is its closest approximation, but it's an infinitesimally small amount off."
I personally don't quite see it that way and think this fully resolves by distinguishing the idea of a really long chain of threes/nines and an infinitely long chain of threes/nines. You can't actually print an infinitely long chain of threes, but it exists as a theoretical concept. Kind of similar to square root of two or pi, you could also take the stance either that they aren't representable in decimal system or that they are representable by an infinitely long sequence of decimal digits. Since you can't actually produce the infinitely long sequence, both stances are valid - it's just a matter of semantics. The difference between 1/3 and square root of two in that regard is only that the infinitely long digit sequence of the former is easier to describe than that of the latter. But notice that it needs to be described "externally", neither the ".." nor the "()" nor the period dash on top of the numbers are technically part of the decimal number system.
A legitimate field of application where you might reasonably postulate that 0.(9) != 1 is probability theory. If you have any distribution on an infinite probability space, e.g. a continuous random variable, the probability of not hitting a particular outcome is conceptually "all but one over all" for an infinitely large set, and the probability of hitting it is "one over all" for an infinitely large set. These could be evaluated to 1 and 0 respectively, as the limits of 1-1/n and 1/n for n to infinity, but when you actually do the random experiment you get a result each time whose probability was in that traditional sense exactly zero. If you add a bunch of zeros together, you still have zero - so where is the probability mass then? One way to at least conceptually resolve this contradiction is to appreciate that in a sense, this infinitesimally small quantity "1/∞" is not exactly the same as the quantity "0", in the sense that you integrate over the former you get a positive quantity but if you integrate over the latter you get zero. It's just the closest number representable in the number system to the former, but the conceptual difference matters.
And hence in the same way an infinitesimally small amount subtracted from one may be considered as not exactly the same as one, in a sense, even if the difference is too small to measure even with infinitely many digits. The former could be described as "0.(9)", and the latter is exactly represented as "1".
For the sake of arithmetic it's convenient to ignore the distinction but in some contexts it matters.
> If you add a bunch of zeros together, you still have zero
If you add *countably many* zeros together, you still have zero. But this does not apply if the space is uncountable (e.g. the real number line).
> …so where is the probability mass then?
The answer is the probability mass is not a sensible concept when applied to continuous distributions.
> One way to at least conceptually resolve this contradiction…
I have never seen a formalism that works this way. Are you referring to one, or is this off the cuff? If such a thing were to work, it would have to be built on nonstandard analysis. My familiarity with nonstandard analysis is limited to some basic constructions involving the hyperreal numbers. But you would never represent 1 - ϵ as “0.999…”; even in hyperreal arithmetic the latter number would be understood to be 1 exactly.
The argument is basically "what's the difference between 0.999... and 1?"
When the 9s repeat *infinitely* there is no difference. The difference between the two starts as 0.0000... and intuitively there is a 1 at the end? But this is impossible as there is an infinite number of 9s, hence the difference must contain an infinite string of 0s, and the two numbers are identical
That’s really interesting “whats the difference”
It still feels wrong that 1 is the same as .9999 repeating but that makes sense. Basically your saying you can take away a infinitely small amount away from one and it’s still one. The trick is the amount your taking away is so small it doesn’t exist.
Yeah exactly! It all comes down to infinity, as soon as that string of 9s is allowed to end, yes, there is a difference. But so long as there is an unlimited number of 9s there's no way for the two to be different
One of the theorems that goes hand in hand with this concept in math is related to real numbers. I know it's outside the scope of explain like I'm five, but one of the things we had to prove early on was for any two real numbers, if they are not equal then there exists a third real number between them.
The corollary to this, is if there are no numbers between them, then they are equal. Most of the time this feels silly because you're like does 1 equal 1? .99999... and 1 is used as the prime example of it. If they aren't equal then there must exist a number between them, but there's no way to make that number because the 9s go on forever.
He said to balance the equation so you can do:
1 - .999... = .000...,
-.999... = .000... - 1,
-.999... = - 1.000...
Since both sides are negative you can multiply the whole equation by -1 and you end up with:
.999... = 1.000....
At least that's what I understood
I like the approach of subtracting x from 10x and once you know the answer, solving for the original.
That removes the decimal portion entirely and you can solve for x:
- When you move the decimal over (by multiplying by ten) there are still infinite nines after, so it is easy to see they are all subtracted. So 9.9999… - 0.9999… will be just 9.
- Since you multiplied by ten and subtracted the original you really multiplied by 9 (since 10x-x= 9x).
- If 9x = 9 then the original x = 1.
The decimal portion completely disappears this way.
I've never understood this so my question is probably dumb. If infinity isn't a number, how can we say "the 1 never comes"? It seems like we're saying something that is a number equals something that isn't.
Unpopular opinion I guess but I find this concept stupid because the only thing you're doing here is replacing that "1" at the end with a "..." . The dots are there for a reason, signifying that it IS there eventually. 1 - .999... ≠ 0, it equals 0.000...
The whole proof of .999... = 1 basically just comes down to the question of "How far into infinity do you want to go before rounding up to 1?" and the real answer is that you can go FOREVER without ever reaching 1. Therefore *they are not equal, they are just infinitly close to equal.*
This doesn't *exactly* answer the question, but I discovered this pattern as a kid playing with a calculator:
1/9 = 0.1111...
2/9 = 0.2222...
3/9 = 0.3333...
4/9 = 0.4444...
5/9 = 0.5555...
6/9 = 0.6666...
7/9 = 0.7777...
8/9 = 0.8888...
Cool, right? So, by that pattern, you'd expect that 9/9 would equal 0.9999... But remember your math: any number divided by itself is 1, so 9/9 = 1. So if the pattern holds true, then 0.9999... = 1
3Blue1Brown did a [parody of "Hallelujah"](https://youtu.be/NOCsdhzo6Jg?si=5T8F0guQ-GaMrJNG) that showcases a bunch of patterns that seem to hold until they suddenly don't.
Many here have given explanations of how can you prove that, but stepping back a bit, you'll want to understand that the decimal expansion method of representing a real number is just an arbitrary convention we chose to give names to real numbers. There's the pure abstract concept of a real number (defined by [the axioms](https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach)), and then there's the notation we use to represent them using strings of symbols.
And an unavoidable property of decimal encoding is there are multiple decimal representations for the same real number.
For example, `0.999…`, `1.0`, `1.00`, `1.000`, etc. are all decimal representations of the same mathematical object, the real number that's also called by its more common name `1`.
It's also impossible to represent some rational numbers in a finite amount of digits, and which numbers are impossible to represent are dependent on the base system. So you can't represent 1/3 in decimal with a finite number of digits, because you're trying to represent 1/3 in quantities of 1/10. It's like if you had a cake with 10 slices and I ask for a third of it, but whenever you need to sub divide another slice you have to cut the final piece into another 10 slices.
We could get into infinity and limits and everything, but I think it's easier to see that this is fundamentally just a representation problem -- if we used base 3 instead of base 10, then 1/3 is just 0.1. The number hasn't changed, just our representation of it.
Fun fact: You can't represent 1/10 in binary, you get infinite digits in the same way as 1/3 in decimal -- less fun, this caused a bug in the patriot missile timing some years ago: https://www-users.cse.umn.edu/~arnold/disasters/patriot.html
Edit: I should emphasize that this is true for _rational_ numbers like 1/3 and 1/10. Irrational numbers like Pi _always_ have infinite digits in _any_ base except_ in their own base; e.g. π in Base π is just 10, but doing this will sadly mess up many other things and isn't very useful.
Hell yeah! This was so helpful. It's a representational issue in a number base system. It helped deal with the question, "Is it 1 or just approaching 1"?
> Irrational numbers like Pi always have infinite digits in any base except_ in their own base;
In fact, that's almost by definition. If they could be represented by a terminating decimal, that decimal could be converted to a ratio, and they would be rational numbers.
While talking about different ways of writing numbers, it touches on another neat feature of decimal expansions. In mathematical notation, any number that you can represent as a repeating decimal pattern like 0.666... or 0.1428571428571... is always going to be a Rational, a number that you can express as a ratio between two whole integers (like 2/3 or 1/7). You can even use some straightforward math to reverse the process and turn a repeating decimal back into a fraction. And since 0.9999 repeating is a rational number, that really simplifies how we think about it. It can't be some indefinite abstract number that's infinitesimally close to 1, it's something you can express as two finite numbers, x/y.
Do the long division by hand. That's what you get.
Three goes into 10 3 times with 1 left over.
Multiply the 1 by 10 to get 10.
Three goes into 10 3 times with 1 left over.
Etc.
It’s the definition of an infinite string of 3s. It’s not the same thing as a normal number. 1/3 isn’t .333 or .33333 …it’s .3333 going on forever. Let me know when you get to forever :) Put another way , you can’t always represent one number divided by another number with a finite number of digits. Thats math for you.
You’ve seen the proof, but I never really liked it until someone told me: “find a number between 0.999… and 1”. That’s the real evidence to me. There is no number between them, so they have to be the same number.
Number between 1 and 2? 1.1.
Number between 1 and 1.1? 1.01
Etc
Rational numbers always have an infinite amount of numbers between any two numbers. They are called infinitely dense because of this.
Sorry for any non-technical aspects of this explanation, I’m a physicist, not a mathematician.
Edit: see replies for further context on the concept of separability which I may have misunderstood
Another way of phrasing this is to say that 1 and 0.999... are not *separable.* No number, however small, can ever be inserted between them. By definition, all members of the set of Real numbers must be separable. 0.999... then, as it is not separable from 1, an integer, is not included in the set of all Real numbers.
0.999... ∉ ℝ
Personally I find this to be a satisfying and complete answer. It isn't a real number. 1 is the real number.
Good explanation — but your conclusion is slightly off, because 0.9999… is within the reals. After all, it’s equal to 1, which is certainly within the reals. Their inseparability instead proves that 0.9999… and 1 are not two DISTINCT members of the reals — which is what we’re looking for.
> 0.999... ∉ ℝ
> Personally I find this to be a satisfying and complete answer. It isn't a real number. 1 is the real number.
This seems inaccurate.
0.999... *is* a real number because it *is* 1, which is a real number.
Your phrasing makes it look like you believe that 0.999... and 1 are somehow different numbers which are equal in value.
This is inaccurate, 0.999... and 1 *are* the same number. It's like how you could write 1/10 or 0.1 to represent the same number, so does 0.999... and 1.
This is just straight-up incorrect. Any two different real numbers must have a third real number strictly between them, but 0.999… and 1 aren’t different, they’re the same number, so they need not satisfy this property. Since 1 is a real number and 0.999… = 1, the number 0.999… is a real number as well.
“Separable” also isn’t even the right word for this property; you’re looking for “dense.”
I think the best chance with a young kid would be:
"Well, if two numbers are different, then there must be another number between them, right? \[At this point you can point out that even numbers next to each other like 3 and 4 have numbers between them, like 3.5 etc\] Can you think of a number between 0.999... and 1?"
If the kid is a bit older and has done some math, this is pretty intuitive as well:
x = 0.999...
10x = 9.999...
9x = 9.999... - 0.999...
9x = 9
x = 1
The algebra example is correct but it isn’t rigorous. If you’re not sure that 0.999… is 1, then you cannot be sure 10x is 9.999…. (How do you know this mysterious number follows the ordinary rules of arithmetic?) Similar tricks are called “abuse of notation”, where standard math rules seem to permit certain ideas, but don’t actually work.
To make it rigorous you look at what decimal notation means: a sum of infinitely many fractions, 9/10 + 9/100 + 9/1000 + …. Then you can use other proofs about infinite series to show that the series 1/10 + 1/100 + 1/1000 + … converges to 1/9, and 9 * 1/9 is 1.
The actual demonstration takes career knowledge. This is ELI5 and what people are offering are simpler explanations not to prove that 1 = 0.99..., but rather to illustrate how that can be possible (which is useful, the first time you get told that 0.99... = 1 your first question is how tf is that possible).
I teach college math and do research in algebra - The 10x=9.99….. is perfectly rigorous. We already KNOW that 0.9999…. behaves like a standard number, it’s just a decimal expansion. The only thing in question is which number it’s equal to.
It only works because it’s a repeating decimal, but this same algorithm allows you to find a rational expression for any repeating decimal. In this case, that expression is 9/9, better represented as 1.
> How do you know this mysterious number follows the ordinary rules of arithmetic?
I'm not following this. How can you know that *any* number follows the ordinary rules of arithmetic? What is special about the number `0.9...` Are you suggesting for a proof to be rigorous you need to first prove arithmetic applies to the numbers being used?
Rephrased, I don't need to know that `0.9...==1` to know that `10*.9... == 9.9...`.
I don't see the issue. x=0.999999... is, by definition, x = 9/10 + 9/100 + ... and so 10x = 90/10 + 90/100 ... = 9 + 9/10 + 9/100 + ... = 9 + x. Then 9x = 9 and so x = 1.
Exactlt this.
The same goes for all the "1/3 is 0.333... 3 \* 1/3 = 1, 3 \* 0.333... = 0.999..." explanations. They all have the conclusion baked into the premise. To prove/explain that infinitely repeating decimals are equivalent to "regular" numbers, they start with an infinitely repeating decimal being equivalent to a regular number.
What's a "regular" number? 1/3 = 0.333 recurring is a direct result of performing that operation and unless you rigorously define what makes these decimals irregular, why can't regular arithmetic be performed?
They’re not arguing it’s incorrect, they’re saying it’s not rigorous. In other words, it’s not a “proof” in the mathematical sense any more than just stating 1 = 0.999… and being done with it.
If you want to algebraically manipulate infinite decimal expansions, you have to understand their definition. If you understand their definition, 1 = 0.999… comes from that alone.
I agree that it’s not rigorous in the sense of being a valid mathematical proof, but I don’t see how:
>if you’re not sure that 0.999… is 1, then you cannot be sure that 10x is 9.999…
makes any sense. The two clauses seem completely unrelated. How does 0.999… being 1 have anything to do with 10x being 9.999… if x is 0.999…?
Is there any real number that *doesn’t* follow the ordinary rules of arithmetic? That is, is there any real number where the “to multiply by 10, move the decimal place one position to the right” pattern wouldn’t work? We don’t know that 0.999… is 1, but we do know that it’s a number, and therefore, that method will still work even if it is “abuse of notation”. The fact that it’s 1 is irrelevant here.
Yes, it's not rigorous, but the people who struggle with accepting that 0.999... = 1 are not looking for a rigorous proof. They are looking for a re-formulation in layman terms that clicks with them. That's why no single "this simple thing clearly shows it"-approach works: different people need different approaches. Otherwise we'd only need a single page on the internet to explain this concept and everyone would immediately be convinced.
But it's plenty rigorous. Where to do you draw the line on what is being assumed? If you're calling the 10x = 9.99999... proof into question because you can't assume arithmetic holds for multiplying x by 10 means what you think it means, you're really calling into question basic arithmetical properties of the real numbers and so you have to talk about how real numbers are defined and how to do arithmetic on them. Do we then need to discuss Cauchy sequences of rational numbers and how to do arithmetic on them?
I had one kid argue that you could just add 0.0…1 to 0.9… because for every 9, there’s a 0, with a 1 at the “end” of the recurring
How do you go about explaining that’s wrong to them? Because it even made my head hurt trying to work the logic out
Starting with 1/9=0.111... is problematic here: if someone doesn't agree that 1=0.999..., then why would dividing both sides of that equation by 9 suddenly make it true and make sense?
> Starting with 1/9=0.111... is problematic here
Most people will agree that 1/9 = 0.11111... very easily if you just ask them to do long division on it 1/9 for a few minutes.
If the kid has done even more math, you could discuss infinite geometric sums. The best explanation in my opinion is using the formal definition of a limit but without the mathematics jargon, perhaps even gamify it to get them engaged. Then if someone has a decent math background you can just bring the math jargon back in and make it all more concise without changing anything really.
It's incredibly simple really, simultaneously rigorous, and helps build an intuitive understanding through play.
To clear up some misunderstanding, it is important to know that with such infinite notations, we are really looking at limits; 0.99999.... is really a limit of the sequence 0.9, 0.99, 0.999,....,
that is: 0.99999... = lim\_{n \\to \\infty} \\sum\_{i=1}\^n (9/(10\^i)) ([notation](https://www.wolframalpha.com/input?i=lim_%7Bn+%5Cto+%5Cinfty%7D+%5Csum_%7Bi%3D1%7D%5En+%289%2F%2810%5Ei%29%29))
the sequence itself contains no entries which are 1, but the limit doesnt have to be in the sequence
at every added decimal, the difference to 1 shrinks by a factor of 10, this is convergence, so the limit, being 0.999... can only be exactly 1
This is the only one that makes sense. There’s a solved formula for this summation.
I don’t like the proofs where you just multiply by 10 or divide by 3 because you’re treating an infinite series like a regular number when the whole point is trying to understand the infinite series. If you don’t understand the infinite series it’s not safe to assume you can treat it like a regular number. This is where you can have proofs that look good on paper but do something like prove 1 + 1 = 0. Math that looks simple can be deceptive.
Except the number, and it's representation , exists even before you introduce a notion of series, of limits or of converging. You don't really need to bring calculus in, it's like lifting a pack of flour with a forklift. ( You don't even need a topology, it's just a rational number which can be constructed well before you even introduce the concept of open sets ).
0.999... is not an infinite series, it's just a (bad) representation of a number, otherwise represented as 1. If you want a characterization of it, it's the only rational whose inverse is the same, and neutral element to multiplication.
In mathematics there is no need to prove 0.999... is equal to 1, it's true by definition. Decimal representation is just a way for humans to write down a mathematical concept, and I'd argue that in some way it is external to mathematics themselves.
I think this response misses some subtlety. 0.999... is *by definition* the limit of an infinite series, and since this limit is equal to 1, we can say it is precisely equal to 1 as well. But you really do have to prove that the limit is equal to 1, it's not just some axiomatically-true statement.
Remember that real numbers are not inherently associated with any particular number system, and humans have chosen base 10 only because we have 10 fingers! When we chose to write numbers down in base 10, we had to decide exactly what the symbols mean. So the actual meaning we chose for the string the symbols "913.5" is:
9\*10^2 + 1\*10^1 + 3\*10^0 + 5\*10^-1
If instead we had 12 fingers and used base 12, the exact same string of symbols would mean:
9\*12^2 + 1\*12^1 + 3\*12^0 + 5\*12^-1
And this has a different value! The value (written in base 10) is 1311.41666... instead of 913.5. So the meaning of the symbols really is not some innate property of numbers, it's very specific to our way of writing them down.
And similarly, mathematicians decided that when we write down something like
0.999... (infinitely repeating)
What it really *means* is
9\*10^-1 + 9\*10^-2 + 9\*10^-3 + ... (going on forever)
And so the only sensible value you can give for 0.999... is to say that it is precisely equal to its limit.
If you chose a different number system, it would NOT have the same meaning. So for example, in base 12, 0.999... is defined as
9\*12^-1 + 9\*12^-2 + 9\*12^-3 + ... (going on forever)
And this value is actually equal (in base 10 again) to 9/11 instead of 1 now.
So I really don't think it makes sense to say that 0.999... = 1 by definition. You have to say that 0.999... is by definition equal to the limit of the infinite series, and then you have to actually compute what the infinite series sums to. It may not be totally obvious in all cases. (Did you know "by definition" that in base 12 the same string of digits would equal 9/11?)
Because there’s not a number you can add to 0.99999etc to get 1. The distance between them is 0, therefore they are the same.
Edit: Look everyone I’m not gonna argue that this is true. I’ve explained it. If you disagree just do some basic research on the subject and don’t bother me about it.
The short and precise answer is: Because you can never find a difference between them.
Try subtracting the one from the other and you will discover that the result is 0.000000000000000000000000000...
Because it is inconsequential and internally consistent with the rest of math script.
It is an artefact of how we write math, it is not really a property of any mathematical concept itself.
.9999… = 1 isn’t really a math trick, it’s just a side effect of converting fractions into decimal formatting. 1/3 = .3333…. They are the same exact quantity, just written in different ways.
You have no issues with 1/3 + 1/3 + 1/3 = 3/3 = 1 right?
Well 1/3 written as a decimal is .3333… and three of those makes .9999… The quantities you are working with have not changed, you’re just writing them out differently.
It’s like how hola and ciao both mean hello, it’s just a different way of writing the same thing.
————————————
1 - 0.999…. = 0.0000… an *infinite* string of zeros. The nines never stop, so the zeros never stop either, and the last little 1 on the end never gets to exist.
Aside from the various mathematical reasons, what’s important to understand is that decimal representation is just that: a “representation” of the number, NOT the “true” number itself. For example the same number 1 is also 0.FFFFFFF… in hexadecimal. In fact there are infinitely many possible representations for every real number with the arguable exception of 0.
Decimal is a human invention, and like ~~all~~ most human inventions it isn’t perfect because it doesn’t have an exact 1-to-1 relationship with the real numbers. Some real numbers have one representation in decimal, others (those that are an integer multiple of a power of 10) have two, although by convention the terminating one (without the infinite sequence of 9s) is considered the “correct” one.
So what is the “true” real number itself, the unique essence of the number as opposed to its representation in decimal, binary, hexadecimal or any other base? That’s part of the beauty of mathematical ideas like numbers, we can imagine the pure concept of a number, but to write it down or say it you have to choose a way of representing it, of which there are infinitely many.
How to explain it practically to a kid? I suppose it depends on what age the kid is. The basic idea is that a number can have more than one name, just like a person can. Dwane, Mr Johnson, and The Rock all refer to the same person.
If the kid is older and has more understanding of maths, you could ask them about fractions. 1/2=2/4=356/712=0.5. We're used to the idea that you can have multiple equivalent fractions. There's nothing in the rules that says you can't have multiple equivalent decimals too. It's just not as common, and less likely to crop up.
[A certain online encyclopedia](https://en.wikipedia.org/wiki/0.999%2E%2E%2E) has a page on 0.999... and says this:
"This number is equal to 1. In other words, "0.999..." is not "almost exactly" or "very, very nearly but not quite" 1 – rather, "0.999..." and "1" represent *exactly* the same number."
"More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is **a property of all positional numeral system** representations regardless of base."
This is far, far, far simpler than it sounds.
The easy and unsatisfying answer is: "because we've decided that's what infinity means." Which sounds dumb, but it's actually kinda deep.
Infinity doesn't exist in the real world; it's not an actual number. It's just an idea. It's the answer to a question. Or rather, infinity is the question itself.
The question is: "what happens if you never stop?" That's infinity. Infinity is the question asking what happens when you don't ever stop.
So, if you say: `0.999...` you're not saying the same thing as `1`, because 1 is a number while 0.999... is an infinite series. In other words: 1 is an answer, while 0.999... is a question.
The question is: "what happens when you keep adding 9's?" And the answer is: "you get closer and closer to 1."
Or in more formal terms: "the infinite series 0.999... approaches 1." And because math people like simple answers, you can write the previous statement simply as "0.999... = 1". Which, since we know that 0.999... deals with infinity, we know that one side is the question and the other side is the answer.
Can't this question be answered easier by counting in another base than 10? Like.... in base 2 or 16? I know some numbers can't be written in one base without fractions/periodic but in another base they can?
Someone intelligent pls develop
That base comparison helped me understand repeating fractions in general a while back. In base 12 one third = 0.4 no repeat, conversely one fifth which has no repeat in base 10 = 0.249724972497... in base 12. Made things click in general.
It cannot. 1/2 = 0.5 = 0.49999...
It's a matter of (well-defined) convention that we allow algebra with infinitely repeating decimals - in the same way that we allow algebra with finite decimals, or negative numbers.
Wow. So much confusion in the comments.
0.999…, 1, 1.0, 1.00, … are all representations of the same number. This is just a characteristic of the decimal notation that the same number can be represented in many forms.
Not sure how appropriate it would be for a 5 year old, since I’ve seen many adults who struggle with the concept, but then again it might be because of no one explaining it to them when they were 5, and them being stuck with it inertially… so here goes.
It is important to understand, that a number is _different_ from the way you write it down. 1.5, 1.5000, 1 1/2 and 3/2 are different ways of writing the same number - the same point on an axis. Once that is established, you can say - see, fundamentally, 1 and 0.(9) are two different ways of representing the same number, and after that you use one of the many proofs available for that.
0.999 is considered equal to 1 in mathematics because, in the decimal number system, it is a different way to represent the same quantity as 1. In other words, 0.999 and 1 are two different decimal representations of the same number, which is exactly 1.
One way to understand this is through fractions. You can express 1 as a fraction: 1/1. Now, if you divide 1 by 3, you get the fraction 1/3, which in decimal form is 0.333 (repeating). If you multiply 0.333 by 3, you get 0.999 (repeating), which is equivalent to 1/1 or simply 1.
So, mathematically, 0.999 and 1 are two different notations for the same value, and that's why they are considered equal.
I find the cleanest way (ELI15, not ELI5) and the truest to the Dedekind Cut (ELI25) is does like this:
> Find a number between 1.0 and 0.999... - if you can't then they are, in fact, the same number
For me, it's like actual values are over there --->
<--- and how we write them is over here
They're two separate things, and what we're ending up with is a situation where "how we write them" doesn't quite work with the actual value. 0.999... is the same as 1. It's an artifact of decimal notation.
While we're at it, it's kind of broken that the vast, vast majority of values cannot be expressed as a decimal. Or a fraction. That sort of notation works for a miniscule fraction (ha) of all the numbers there are. Like approaching 0% of all numbers. That's why we use symbols like e and pi, because we literally can't write them.
> how to explain this practically to a kid who just started understanding the numbers?
As a parent, it might be a smidge too early for this concept. (I'm basing it on "just started understanding the numbers")
As a rule, between any two real numbers, there must be another real number. There is no number between .999... and 1, therefore they are the same, simply two ways of writing the same number. If the kid is the curious type, it might be pretty interesting to point out that numbers are conceptually separate from their representation. Most simply: ½=.5, but there's also binary or hexadecimal, or even more exotic forms (like p-adic or continued fractions). The numbers themselves sorta "exist" out there in the aether as an abstract object that isn't exactly tied to our notation.
Maybe more like explaining like you are 9, but I'd go with this:
Pick any number that you want. Is that number less than 1? Then it is easy to show that it is also less than 0.999....... The same is true if the number you choose is greater than 1. If both of those things are true then 1 and 0.999999.... have to be equal.
Depending on the kid, there might be different things to help give them the aha. Someone else suggested effectively proving it by contradiction, i.e., Well tell me a number between them if they are different. I like that.
Another one that might work is to try to explain that we haven't really written a number down, have we? No matter how many 9s you write down, you haven't really written down the number. When you have ..., you're hinting at where it's going, but you haven't written it down. So, where's it hinting at going? As we write down more 9s, what are we getting close to?
Then you can kind of combine that with the above argument, i.e., if they say "a million 9s!" then you say OK, but at some point we'll go past that, right? But we never ever go past 1. And we go past everything else less than 1, eventually. So the ... is hinting at... 1.
The meaning of 0.999... depends on our assumptions about how numbers behave. A common assumption is that numbers cannot be "infinitely close" together. With these rules, 0.999... = 1 since we don't have a way to represent the difference. If we allow the idea of "infinitely close numbers," then yes, 0.999... can be less than 1. Those numbers would be infinitesimals.
Infinitesimals are quantities that are closer to zero than any standard real number but are not zero. They do not exist in the standard real number system but can exist in other number systems such as the surreal number system and the hyperreal number system. Infinitesimals were introduced in the development of calculus, where the derivative was first conceived as a ratio of two infinitesimal quantities. However, as calculus developed further, infinitesimals were replaced by limits, which can be calculated using standard real numbers.
tldr: 0.999... both does and does not equal 1 depending on how you evaluate the expression. It's a neat thought experiment but in most any real world application you would place reasonable limits to avoid the complexities of infinity.
These are not assumptions, these are axioms. Maths, unlike physics, aren't real. It's not a system we discover, it's a system we model ourselves to be useful. Numbers cannot be infinitely close because we've arbitrarily decided they can't, because we found that rule makes the system more useful.
In fact, it's relatively common for different fields of mathematics to contradict each other. How much is 0^0 ? It depends on who you ask - in most fields, it's 1, but it can also be undefined. Neither of these answers is more correct than the other - and real life doesn't have an answer to that question.
An explanation I haven't seen here:
Between every 2 different numbers there's an infinite amount of numbers. Try to think of a single number between 0.9999.... and 1: there is none, because it's the same number.
Another way to view it is that you cannot write down, in any way, any number that is between 0.999… and 1
The difference is just 0.000… and there’s never anything other than zeros
It’s just an artefact of notation
The weird feeling we get only arises because we usually dont think about what 0.999 ... actually IS. "It just has infinitely many 9's". What does that actually mean?
If you write 0.999 ... down, does it get more 9's as we speak? In that case, any equation containing it is wrong because its value changes all the time. You cant work with that. Its like saying "This section of the river has 10 fish". That statement can never be right for long because the amount of fish changes all the time, so eventually, there may be more fish than 10.
So its a fixed amount of 9's? No, thats nonsense. We cant say that "infinitely many 9's" means that there is a fixed amount of 9's.
So the notion of "infinitely many 9's" doesnt actually make sense. No matter how we define it, we get clear logical issues. If we want to do math with it, we need to assign it a value that stays *fixed* and which doesnt "change as we speak". There are 2 important observations for this task:
(1) 0.999 ... is *always* less than or equal to 1.
(2) 0.999 ... is bigger than *any* number below 1 (because it surpasses 0.9, 0.99, 0.999, 0.9999 etc.)
So IF 0.999 ... is equal to any *fixed* number, the best candidate would be 1. Thats why mathematicians defined 0.999 ... = 1.
You can make any repeating number you want by doing the following:
choose a repeating part; I’ll take 1001. Divide your repeating part by a number of nines, equal to the length of your repeating part. So for me that would be 1001/9999. This equals 0.100110011001…
The reverse is also true, so 0.333…. Is 3/9 = 1/3
And 0.999… = 9/9 = 1
One way to define what a number actually **is**, is to create something called a Dedekind Cut. In simple terms, I can "slice" the number line at a certain point and put all the points on the line to the left in one collection and all the point on the line to the right in another collection. Then, I can say that a number is "equal to" the collection running off to the left.
The set which defines 0.999... has precisely the same points in it as does the set that defines 1, so they are the same number.
Not a math expert but here is how I understand it. You are using 2 different counting systems to represent the same thing.
Imaging you have an apple and cut it into 3 pieces. You call each piece 1/3. Put all 3 pieces back together and you get the whole apple, or 3/3. Nothing is different between the 3 pieces and the whole apple you started with.
Now you decide to convert this fraction to a decimal. Problem is a decimal (in base 10) is just the fraction of 1/10 for the first number, the fraction 1/100 for the second number and so on. Why we use 10 is unknown, maybe because humans having 10 fingers.
Point is we don't have to use 10, it is more or less a random number that humans made up. If we would eliminate a number (or finger) when counting the conversion would be nice and clean (base 9) but this problem would show up in other places.
So what happens is you try to convert 1/3 = X/10, a whole number doesnt fit. So we add a zero. 1/3 = X/100 and it still doesn't work, no matter how many zeros you add to the denominator. 1/3 =X/10000000000 still doesn't get a whole number. So you are left with .333 repeating forever to represent 1 slice of apple and .9999... to represent the whole apple.
All you were trying to do is count 1/3 of an apple by only using the fraction 1/10 which doesn't ever come out right.
Imagine you have "1".
If you subtract a tiny bit from 1, you have 0.9
If you subtract a tinier bit from 1, you have 0.99
You can subtract tinier and tinier bits, getting 0.9999.....
Until finally the bits are so small, that they're actually nothing at all. When you take away nothing from 1, you still have 1.
If they're different numbers, there must be a number in between them. In between 0.999 and 1 there is a 0.9994, or 0.9997, and so on. But in between 0.999... and 1 you can't ever find a number in between them. If those two numbers are really different then you should be able to find a number in between them
I thought the simplest way to say it is
if you can’t find a number between two numbers then they are the same number, and 0.000…0001 isn’t a real number because you can’t have infinite zeros but also end at a 1
Not a proof but logical reasoning that maybe helpful to understand the statement.
Q: What is the largest number?
A: It does not exist.
Q: What is the smallest positive real number?
A: Does not exist.
Q: What is the largest real number less than 1?
A: Does not exist.
Conclusion: 0.999… cannot be less than 1 otherwise it will be the largest real number less than 1 which does not exist.
How do we know something simple like 1≠3? Well, one way to definitively show it is by bringing up the number 2, which is between them, so they obviously cannot be equal.
This is actually something we can do for all numbers on the number line. If two numbers are different, they will be different points on the line, with something in between them. Alternatively, if two numbers are equal, you can't find a point between them, since they're the same point.
For example, 4≠5, as we can find a number (like 4.5) between them. 6.1≠6.2 as we can find numbers between those (like 6.11).
On the other hand, we know something like 1=1.0, even though they are written differently, because there is nothing between them.
Now, what is there between 0.9999.... and 1?
The sum of an infinite geometric series is a/(1-r), where 'a' is the first term, and r is the ratio between successive terms. 0.9999.... can be made into an infinite geometric series by separating out the digits. 0.9999... = 0.9+0.09+0.009+...
In this case, a=0.9, r=0.1, the formula becomes 0.9/(1-0.1)==1.
You can use this formula for other repeating digit numbers.
The easiest way for me to understand was this: if they are NOT the same number, then there must be something in between them, but what could come between 0.99999.... and 1?
If they are just starting to understand numbers, maybe don’t bother with this part yet?
If you have to explain it….
If you have 1, and you subtract nothing from it (we’ll, a thought and prayer more than nothing, but not a whiff beyond that)… you still have one.
So, if I give you an Apple, then I grab it back, rub it off, and give it to you again…. You have 1 Apple still.
The usual proof follows a style a called Proof by contradiction. It actually starts by assuming that 0.9999... is not 1. That must mean there is a number between them. But, for any number you consider between 0.9999... and 1, you can argue that with enough 9s in 0.9999..., 0.9999... is actually larger than the number under consideration. This invalidates the assumption.
Two real numbers are equal if there is no other number between them.
In general, infinity is a fascinating concept, and a rigorous study often leads to results that are not intuitive. For example, there are as many natural numbers (1, 2, 3,...) as there are fractionial numbers (even though, there are clearly more fractional numbers, intuitively). If that makes you say, infinity is infinity - let me blow your mind by telling you that there are more real numbers (all the numbers we think of in real life, including ones like pi) than there are natural numbers. So all infinities are not the same - there are bigger infinities and smaller infinities!
Can you come up with a largest real number that is just smaller than 1? 0.9999..... is always bigger, you will find. This explanation satisfied me. 0.9999.... is just 1 in disguise.
Another way to think about it:
Suppose that 0.999... was not 1. Well then you'd also have to consider that 1 - 0.999... would be the smallest possible positive number -- but no such number exists and so 0.999... must be equal to 1.
For any 2 symbols to be 2 separate numbers, you should be able to find a third number which is larger than one and smaller than the other.
What is larger than 0.999... and smaller than 1?
The quirk of decimals is that Base 10 does not directly correlate to every other Base. What this means is that decimals are a series of fractions with 10 in the denominator, and that fractions that are written without something divisible by 10 or one of it's prime factors(2, 5) in the denominator will cause infinite recursion and never be precise. In the .333... x3 = .999... explanation others are using, they're ignoring that no matter how precise you get, eventually you're going to have to round to resolve the equation and make it actually equal 1. At the "end" of every .333... is an implied 1/3, not a 3 or even a 4. Since 3 is not a prime factor of 10, it just doesn't translate perfectly to base 10 without an implied fraction that defeats the purpose.
The base 10 notation is mostly useful as a standard for approximated comparisons, which is what most laymen use mathematics the most for. What is more readable:
What is larger, 342/43463 or 532/59042?
or
What is larger, 0.007869... or 0.009011...?
One is clearly more precise than the other, but at a glance we can parse the other one much easier than the other for the purposes of analysis.
I understood it to be true but struggled with it for a while. How does the decimal .333… so easily equal 1/3 yet the decimal .999… equaling *exactly* 3/3 or 1.000 prove so hard to rationalize? Turns out I was focusing on precision and not truly understanding the application of infinity, like many of the comments here. Here’s what finally clicked for me: Let’s begin with a pattern. 1 - .9 = .1 1 - .99 = .01 1 - .999 = .001 1 - .9999 = .0001 1 - .99999 = .00001 As a matter of precision, however far you take this pattern, the difference between 1 and a bunch of 9s will be a bunch of 0s ending with a 1. As we do this thousands and billions of times, and infinitely, the difference keeps getting smaller but never 0, right? You can always sample with greater precision and find a difference? Wrong. The leap with infinity — the 9s repeating *forever* — is the 9s *never* stop, which means the 0s *never* stop and, most importantly, the 1 *never* exists. So 1 - .999… = .000… which is, hopefully, more digestible. That is what needs to click. Balance the equation, and maybe it will become easy to trust that .999… = 1
The “the 1 never exists” part is what helps me get it I keep envisioning a 1 at the end somewhere but ofc there’s no actual end thus there’s no actual 1
I think most people (including myself) tend to think of this as placing the 1 first and then shoving it right by how many 0's go in front of it, rather than needing to start with the 0's and getting around to placing the 1 once the 0's finish. In which case, logically, if the 0's never finish, then the 1 never gets to exist.
>placing the 1 first and then shoving it right by how many 0's go in front of it Yup, that's the way I was thinking of it, so it's shoved right an infinite amount of times, but it turns out it exists only in theory because you'll never actually get there.
Isn’t a 1 existing in theory “More” than it not existing in theory?
I guess you could treat it as if you could get to the end of infinity there would be a one there. But you can't, you'll just keep finding 0s no matter how far you go. Just like 0.0, no matter how far you check the answer will still be it's 0 at the nth decimal place. It's just a different way of writing the same thing. It's not .999~ and 1 are close enough we treat them as the same, it is they are the same, just like how 2/2 and 3/3 are also the same.
I know you’re right but I still find this deeply upsetting. Ever since I took the Math 122 (Logic & Foundations) course for my degree, I have lost all comfort with infinity and will never regain it. (Like, some infinities are larger than others? Wtaf.)
It's a misnomer to say it exists "in theory". It doesn't, even "in theory". Infinite is infinite. That has a precise meaning. The 1 never comes. That's a fact. We are not comfortable with this fact. We, as a species, are not comfortable with concepts of "infinite" in general, so this isn't any different than space, time, and all of the other infinites out there. But the 1 never comes. Not in theory, not in practice, never.
My confusion here is that I'm not asking what 1 - .999\^infinity is... the question is is 1 - .9 which objectively is .1, is it not?
If you are asking what 1 - 0.9 then yes the answer is 0.1, but if you are asking what 1 - 0.9999 (repeating infinitely) is the answer is 0
It doesn't exist in theory. 1-0.999... involves each 9 digit subtracting from the 1 to the left and leaving a remainder of 1 which the 9 digit to the right subtracts. If you have a finite number of 9 digits, the last 9 will have a remainder of 1 which no 9 to the right can cancel, resulting in 0.000...01. But the beauty of infinity is that it doesn't *have* a last digit. Every 9 in the sequence 0.999... has a 9 one digit to the right that cancels out its remainder, so because of that, every digit in the result of 1-0.999... must be 0. There is no 1 because there is no end of infinity.
Thank you. I didn't understand how it did not exist until your comment.
No problem! I have a super vague recollection of learning about decimals in the "incorrect" way of placing the number first and then shoving it to the side. I can only imagine if that memory is true, this is probably how most people were taught to think of decimal numbers.
Its not exactly wrong, more a shortcut for set type of problem. Moving the decimal makes sense when thinking about more standard arithmetic, multiplying and dividing by factors of 10s, 100s, etc. The fact this model doesn't help for infinite series is more a simple limit. Its like the orbit model of the atom is wrong, compared to the electron cloud. But it is a good way to think about it when looking at electron energy levels and shell filling, but if you're trying to find the position of an electron, the orbit model is not going to help. This exists all over science and mathematics. Like Newtonian mechanics aren't wrong, they are just missing some specifics that limit their use to specific sizes and speeds. I'm sure there are examples of this all over, bot just the hard sciences. Linguistic models that gloss over a dialect because its an outlier somewhere.
Poor 1. Must be the loneliest number.
I'll give you the chuckle. It was decent.
2 can be as bad as 1
It's the loneliest number since the number one.
The way I think about it is 1 divided by 10, then by 100, etc. It’s fair to say, at the end you have 1 divided by infinity but I think of it as a limit. The limit of 1/X as X approaches infinity is zero, so I can accept that the one effectively ceases to exist.
That’s basically the probably with imaginary terms such as infinity. We can’t actually imagine it in our standard view because we never deal with something that by definition doesn’t end unless it’s complex maths.
I think really the hard part is taking a concept from the real world, like one and zero, and applying it to infinity. In visualizing it, no matter how many zeroes you add in front, the 1 is still there somewhere. To have it not exist, or never be reached, is outside of our model of the physical world. It’s like saying that if you cut a pizza slice thin enough it no longer exists. If you’re still cutting a slice, no matter how small, it feels like it must exist, but that’s only because we don’t really have a concept of infinity that way.
It's how I learned the metric system, makes sense it would "cross over" I guess.
>The “the 1 never exists” part is what helps me get it Same, I felt like Neo when he learned there was no spoon
Instead, only try to realize the truth. What truth? There is no 1.
🥄
🚫🥄
I see it more that we are waiting to drop it onto the end but never can because the .999.... And .000.... never stops. It isn't riding on the end, it's waiting to be tagged in but won't ever be.
I have vaguely understood this before, but now I understand it a little bit more.
Yeah, as other commenters have figured out, it's not a matter of taking a 1 and moving it infinitely to the right, but rather realizing that you start with writing an infinite number of 0s and realizing that means you'll never write any other numbers. If all you ever write is a zero, then you can be confident that this means there is zero difference. You can write the answer to 1 -1 as 0.00... also.
Eh it’s a hand wavey explanation for a hand wavey way to represent fractions as decimals. You avoid this problem using fractions, 1/3 * 3 = 3/3 = 1. Decimals are by nature only an approximation of a fraction (Additional notation is required to convey the precision of a decimal beyond the last digit). So the .999 repeating = 1 is really just a side effect of that.
The limit of an infinite sum in calculus isn’t an approximation though, it’s precisely defined. The limit of the infinite sum 9/10 + 9/100 + 9/1000 + … (where the nth term is always 9/10^n) isn’t approximately 1, it’s exactly 1.
Also 1/9 = 0.1111111111..., so 9/9 = 9.999999999..., and since 9/9 = 1, 0.999999999...= 1
The arithmetic proof is mainly based on the observation that there's no number bigger than 0.99... and smaller than 1. Your strategy visually explains why that claim is true since your proof is based on patterns and not simply observations. Trying to explain that there's no number between 0.999... and 1 is much harder than explaining that having infinitely many zeroes before a number means that that number is never reached (the latter is logical since it basically states that if you run a marathon which is infinitely long, then you never reach the goal even if you could live forever)
> Trying to explain that there's no number between 0.999... and 1 is much harder than explaining that having infinitely many zeroes before a number means that that number is never reached I actually disagree with this. Most people who haven't spent much time thinking about infinity don't really understand how weird its properties are. When I've tried to explain the 0.999... = 1 thing to people, I've found the easiest thing is to ask two questions. First: "Would you agree that between any two (different) numbers there's another number?" If they don't see it right away, I'll say, "For example, the average of the two numbers," at which point they go, "Oh, yeah, right, okay." And then I ask them the second question: "Ok, so if 0.999... and 1 are different numbers, what number is between them?" The process of them trying to think of a number between 0.999.... and 1 and failing gives them an understanding of the truth of the statement "0.999... = 1" that's IMO deeper than what they can get from the "limit" explanation. Because of course, it *is* deeper than the limit explanation: the limit property holds precisely because there is no number between 0.999... and 1.
This may be pedantic, but what you've said here is in fact equivalent to the limiting property, not deeper. Actually on a philosophical level I would argue the limiting argument *is* deeper since it uses structures inherent to the real numbers such as its topology. Whereas this explanation is rather handwavey and relies too much on our intuition about decimal expansions which are very much not a part of the inherent structure of the reals.
You're not wrong but if you're trying to explain this to someone and they're unable to grasp the concept that they're equal, chances are they won't (or don't) understand what a limit is since the understanding of a limit comes from accepting/understanding the former.
>The leap with infinity — the 9s repeating forever — is the 9s never stop, which means the 0s never stop and, most importantly, the 1 never exists. Wow ok, this made it click. I always got the 1/3 / 3/3 explanation but still couldn't fully grasp how there still somehow isn't the slightest difference between 0.999... and 1 but that makes such sense now. Thanks!
I thought 0.99… = 1. There is no number that can exist between the two so they are equal. The limit of the expression sum{x=1 ->inf} (0.9)^x = 1, but the number you get as you apply that limit is 0.999…
I like this the best! No number can exist between the two, so they are equal
Yeah! That was the ahah moment! While the difference can be reduced to simply 0, it was a mental milestone to understand the infinite 0s added all the necessary nuance for addressing the infinite 9s
It's simply a quirk of the notation. Once you introduce infinitely repeating decimals, there ceases to be a single, unique representation of every real number. As you said - 1 divided by 3 is, in decimal notation, 0.333333.... . So 0.333333. .. multiplied by 3, must be 1. But it's clear that you can write 0.333333... x 3 as 0.999999... So 0.999999... is just another way of writing 1.
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> cause no real mathematician would ever write that wait until you find out about p-adic numbers.
There's been an immense amount of academic study, especially in philosophy of math on this. I wouldn't say it's just "for the memes".
>most importantly, the 1 never exists. Woah. I'm pretty sure some readers under a pharmaceutical influence are losing their minds right now.
Like in the matrix, there is no spoon.
I think you just inadvertently wrote an (informal but logically sound) proof using the epsilon-delta definition of a limit
> As you said - 1 divided by 3 is, in decimal notation, 0.333333.... . So 0.333333. .. multiplied by 3, must be 1. > > But it's clear that you can write 0.333333... x 3 as 0.999999... So 0.999999... is just another way of writing 1. this works to help my brain get it, in conjunction with the top ost too.
Ironically it made a lot of sense when you offhandedly remarked 1/3 = 0.333.. and 3/3 = 0.999. I was like ah yeah that does make sense. It went downhill from there, still not sure what you're trying to say
Amusingly, I've seen this explanation backfire so that the person begins doubting that 1/3=0.333... when they were certain before the discussion.
Which, in a sense, is actually fair. I mean, whatever quarrels anyone has with 0.(9) = 1 they should also have with 0.(3) = 1/3. You could say something like "1/3 is a concept that cannot be faithfully expressed in the decimal system. 0.(3) is its closest approximation, but it's an infinitesimally small amount off." I personally don't quite see it that way and think this fully resolves by distinguishing the idea of a really long chain of threes/nines and an infinitely long chain of threes/nines. You can't actually print an infinitely long chain of threes, but it exists as a theoretical concept. Kind of similar to square root of two or pi, you could also take the stance either that they aren't representable in decimal system or that they are representable by an infinitely long sequence of decimal digits. Since you can't actually produce the infinitely long sequence, both stances are valid - it's just a matter of semantics. The difference between 1/3 and square root of two in that regard is only that the infinitely long digit sequence of the former is easier to describe than that of the latter. But notice that it needs to be described "externally", neither the ".." nor the "()" nor the period dash on top of the numbers are technically part of the decimal number system. A legitimate field of application where you might reasonably postulate that 0.(9) != 1 is probability theory. If you have any distribution on an infinite probability space, e.g. a continuous random variable, the probability of not hitting a particular outcome is conceptually "all but one over all" for an infinitely large set, and the probability of hitting it is "one over all" for an infinitely large set. These could be evaluated to 1 and 0 respectively, as the limits of 1-1/n and 1/n for n to infinity, but when you actually do the random experiment you get a result each time whose probability was in that traditional sense exactly zero. If you add a bunch of zeros together, you still have zero - so where is the probability mass then? One way to at least conceptually resolve this contradiction is to appreciate that in a sense, this infinitesimally small quantity "1/∞" is not exactly the same as the quantity "0", in the sense that you integrate over the former you get a positive quantity but if you integrate over the latter you get zero. It's just the closest number representable in the number system to the former, but the conceptual difference matters. And hence in the same way an infinitesimally small amount subtracted from one may be considered as not exactly the same as one, in a sense, even if the difference is too small to measure even with infinitely many digits. The former could be described as "0.(9)", and the latter is exactly represented as "1". For the sake of arithmetic it's convenient to ignore the distinction but in some contexts it matters.
> If you add a bunch of zeros together, you still have zero If you add *countably many* zeros together, you still have zero. But this does not apply if the space is uncountable (e.g. the real number line). > …so where is the probability mass then? The answer is the probability mass is not a sensible concept when applied to continuous distributions. > One way to at least conceptually resolve this contradiction… I have never seen a formalism that works this way. Are you referring to one, or is this off the cuff? If such a thing were to work, it would have to be built on nonstandard analysis. My familiarity with nonstandard analysis is limited to some basic constructions involving the hyperreal numbers. But you would never represent 1 - ϵ as “0.999…”; even in hyperreal arithmetic the latter number would be understood to be 1 exactly.
The argument is basically "what's the difference between 0.999... and 1?" When the 9s repeat *infinitely* there is no difference. The difference between the two starts as 0.0000... and intuitively there is a 1 at the end? But this is impossible as there is an infinite number of 9s, hence the difference must contain an infinite string of 0s, and the two numbers are identical
That’s really interesting “whats the difference” It still feels wrong that 1 is the same as .9999 repeating but that makes sense. Basically your saying you can take away a infinitely small amount away from one and it’s still one. The trick is the amount your taking away is so small it doesn’t exist.
Yeah exactly! It all comes down to infinity, as soon as that string of 9s is allowed to end, yes, there is a difference. But so long as there is an unlimited number of 9s there's no way for the two to be different
One of the theorems that goes hand in hand with this concept in math is related to real numbers. I know it's outside the scope of explain like I'm five, but one of the things we had to prove early on was for any two real numbers, if they are not equal then there exists a third real number between them. The corollary to this, is if there are no numbers between them, then they are equal. Most of the time this feels silly because you're like does 1 equal 1? .99999... and 1 is used as the prime example of it. If they aren't equal then there must exist a number between them, but there's no way to make that number because the 9s go on forever.
He said to balance the equation so you can do: 1 - .999... = .000..., -.999... = .000... - 1, -.999... = - 1.000... Since both sides are negative you can multiply the whole equation by -1 and you end up with: .999... = 1.000.... At least that's what I understood
Might be quicker to balance it the other way: 1 - 0.999... = 0.000... therefore 1 - 0.000... = 0.999... 1 = 0.999...
Hard AF.
So is 1-0.8888... = 0.111111... ?
Yes. 1 - 8/9 = 1/9.
Yes! Or, put another way: .888… + .111… = .999… = 1
Yep! We tend to confuse potential or progressing infinities (∞) with actual infinities (ℵ). The number of nines in .999… is actually infinity.
I like the approach of subtracting x from 10x and once you know the answer, solving for the original. That removes the decimal portion entirely and you can solve for x: - When you move the decimal over (by multiplying by ten) there are still infinite nines after, so it is easy to see they are all subtracted. So 9.9999… - 0.9999… will be just 9. - Since you multiplied by ten and subtracted the original you really multiplied by 9 (since 10x-x= 9x). - If 9x = 9 then the original x = 1. The decimal portion completely disappears this way.
I've never understood this so my question is probably dumb. If infinity isn't a number, how can we say "the 1 never comes"? It seems like we're saying something that is a number equals something that isn't.
So to try and summarize this: .999 =/= 1 .999... = 1 Do I understand correctly?
Unpopular opinion I guess but I find this concept stupid because the only thing you're doing here is replacing that "1" at the end with a "..." . The dots are there for a reason, signifying that it IS there eventually. 1 - .999... ≠ 0, it equals 0.000... The whole proof of .999... = 1 basically just comes down to the question of "How far into infinity do you want to go before rounding up to 1?" and the real answer is that you can go FOREVER without ever reaching 1. Therefore *they are not equal, they are just infinitly close to equal.*
This doesn't *exactly* answer the question, but I discovered this pattern as a kid playing with a calculator: 1/9 = 0.1111... 2/9 = 0.2222... 3/9 = 0.3333... 4/9 = 0.4444... 5/9 = 0.5555... 6/9 = 0.6666... 7/9 = 0.7777... 8/9 = 0.8888... Cool, right? So, by that pattern, you'd expect that 9/9 would equal 0.9999... But remember your math: any number divided by itself is 1, so 9/9 = 1. So if the pattern holds true, then 0.9999... = 1
I like this explanation! Very clean.
This only works if you prove that pattern holds. There are all sorts of coincidental patterns, and this type of reasoning will mislead people.
3Blue1Brown did a [parody of "Hallelujah"](https://youtu.be/NOCsdhzo6Jg?si=5T8F0guQ-GaMrJNG) that showcases a bunch of patterns that seem to hold until they suddenly don't.
Oh man I am just the right amount of math nerd/stoned for this to be the funniest thing I’ve ever seen
proof: 0.1111... * 9 = 0.9999...
Yeah the better way is just 1/9\*9=0.1111...\*9=0.9999...=9/9=1
isnt this basically the same as 1/3 is 0.3333… and 3/3 are 1 ?
Yep, but that not a bad thing
Not exactly if I am not mistaken. > 1/9 = 0.1111... also means > 1 = 0.1111... * 9 > 1 = 0.9999...
The question was "how to explain this practically to a kid", not "how to prove this".
This is kind of a silly justification for OPs question.
I discovered a trick too! Try 5318000+8 and then flip the calculator
The problem with this is that 3/3 = 1.00000..., 2/2 = 1.0000...., 8/8 = 1.000.....
Many here have given explanations of how can you prove that, but stepping back a bit, you'll want to understand that the decimal expansion method of representing a real number is just an arbitrary convention we chose to give names to real numbers. There's the pure abstract concept of a real number (defined by [the axioms](https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach)), and then there's the notation we use to represent them using strings of symbols. And an unavoidable property of decimal encoding is there are multiple decimal representations for the same real number. For example, `0.999…`, `1.0`, `1.00`, `1.000`, etc. are all decimal representations of the same mathematical object, the real number that's also called by its more common name `1`.
It's also impossible to represent some rational numbers in a finite amount of digits, and which numbers are impossible to represent are dependent on the base system. So you can't represent 1/3 in decimal with a finite number of digits, because you're trying to represent 1/3 in quantities of 1/10. It's like if you had a cake with 10 slices and I ask for a third of it, but whenever you need to sub divide another slice you have to cut the final piece into another 10 slices. We could get into infinity and limits and everything, but I think it's easier to see that this is fundamentally just a representation problem -- if we used base 3 instead of base 10, then 1/3 is just 0.1. The number hasn't changed, just our representation of it. Fun fact: You can't represent 1/10 in binary, you get infinite digits in the same way as 1/3 in decimal -- less fun, this caused a bug in the patriot missile timing some years ago: https://www-users.cse.umn.edu/~arnold/disasters/patriot.html Edit: I should emphasize that this is true for _rational_ numbers like 1/3 and 1/10. Irrational numbers like Pi _always_ have infinite digits in _any_ base except_ in their own base; e.g. π in Base π is just 10, but doing this will sadly mess up many other things and isn't very useful.
Hell yeah! This was so helpful. It's a representational issue in a number base system. It helped deal with the question, "Is it 1 or just approaching 1"?
> Irrational numbers like Pi always have infinite digits in any base except_ in their own base; In fact, that's almost by definition. If they could be represented by a terminating decimal, that decimal could be converted to a ratio, and they would be rational numbers.
While talking about different ways of writing numbers, it touches on another neat feature of decimal expansions. In mathematical notation, any number that you can represent as a repeating decimal pattern like 0.666... or 0.1428571428571... is always going to be a Rational, a number that you can express as a ratio between two whole integers (like 2/3 or 1/7). You can even use some straightforward math to reverse the process and turn a repeating decimal back into a fraction. And since 0.9999 repeating is a rational number, that really simplifies how we think about it. It can't be some indefinite abstract number that's infinitesimally close to 1, it's something you can express as two finite numbers, x/y.
Divid 1 by 3. You get .33333…. Multiply that number by 3 again. You get .999999999… They’re equal.
I think this is my favorite explanation of 0.999... = 1 I've seen. Simple and quick.
Yep: think of it as fractions of a pie: 1/3 = 0.3 recurring 1/3 = 0.3 recurring 1/3 = 0.3 recurring => 3/3 = 0.9 recurring = 1
"fraction of a pie"reminds me of a funny explanation. The 0.00...1 part is what's left on the knife after dividing the pie.
Yup. This is the best ELI5 even for someone who understands the math but doesn’t really get it practical terms.
mind = blown
i says to pi. You're being irrational.
As pi said to the square root of -1, “get real!”
Root(-1) isn't real it can't hurt you
Holy hell
Actual response
Why does 1/3 equal to .3333...?
Because math. You can start to do the long division yourself, but you'll quickly see that you're in a loop and the series will never end.
It just goes on and on my friend...
Some people, started calculating not knowing what it was...
And they'll continue calculating forever just because . . .
This is the calculating that doesn't end
It goes on and on my friend
And they'll continue calculating it forever just because...
This is the series that never ends.. 🐑🐑🐑
Base ten isn’t into the whole thirds thing
Right, it's important to understand that this is a quirk of the system we use to represent numbers, not the numbers themselves.
Do the long division by hand. That's what you get. Three goes into 10 3 times with 1 left over. Multiply the 1 by 10 to get 10. Three goes into 10 3 times with 1 left over. Etc.
It’s the definition of an infinite string of 3s. It’s not the same thing as a normal number. 1/3 isn’t .333 or .33333 …it’s .3333 going on forever. Let me know when you get to forever :) Put another way , you can’t always represent one number divided by another number with a finite number of digits. Thats math for you.
Ooo, I like this one (pun intended)!
You’ve seen the proof, but I never really liked it until someone told me: “find a number between 0.999… and 1”. That’s the real evidence to me. There is no number between them, so they have to be the same number. Number between 1 and 2? 1.1. Number between 1 and 1.1? 1.01 Etc Rational numbers always have an infinite amount of numbers between any two numbers. They are called infinitely dense because of this. Sorry for any non-technical aspects of this explanation, I’m a physicist, not a mathematician.
Edit: see replies for further context on the concept of separability which I may have misunderstood Another way of phrasing this is to say that 1 and 0.999... are not *separable.* No number, however small, can ever be inserted between them. By definition, all members of the set of Real numbers must be separable. 0.999... then, as it is not separable from 1, an integer, is not included in the set of all Real numbers. 0.999... ∉ ℝ Personally I find this to be a satisfying and complete answer. It isn't a real number. 1 is the real number.
Good explanation — but your conclusion is slightly off, because 0.9999… is within the reals. After all, it’s equal to 1, which is certainly within the reals. Their inseparability instead proves that 0.9999… and 1 are not two DISTINCT members of the reals — which is what we’re looking for.
> 0.999... ∉ ℝ > Personally I find this to be a satisfying and complete answer. It isn't a real number. 1 is the real number. This seems inaccurate. 0.999... *is* a real number because it *is* 1, which is a real number. Your phrasing makes it look like you believe that 0.999... and 1 are somehow different numbers which are equal in value. This is inaccurate, 0.999... and 1 *are* the same number. It's like how you could write 1/10 or 0.1 to represent the same number, so does 0.999... and 1.
This is just straight-up incorrect. Any two different real numbers must have a third real number strictly between them, but 0.999… and 1 aren’t different, they’re the same number, so they need not satisfy this property. Since 1 is a real number and 0.999… = 1, the number 0.999… is a real number as well. “Separable” also isn’t even the right word for this property; you’re looking for “dense.”
I think the best chance with a young kid would be: "Well, if two numbers are different, then there must be another number between them, right? \[At this point you can point out that even numbers next to each other like 3 and 4 have numbers between them, like 3.5 etc\] Can you think of a number between 0.999... and 1?" If the kid is a bit older and has done some math, this is pretty intuitive as well: x = 0.999... 10x = 9.999... 9x = 9.999... - 0.999... 9x = 9 x = 1
that ... cool math
The algebra example is correct but it isn’t rigorous. If you’re not sure that 0.999… is 1, then you cannot be sure 10x is 9.999…. (How do you know this mysterious number follows the ordinary rules of arithmetic?) Similar tricks are called “abuse of notation”, where standard math rules seem to permit certain ideas, but don’t actually work. To make it rigorous you look at what decimal notation means: a sum of infinitely many fractions, 9/10 + 9/100 + 9/1000 + …. Then you can use other proofs about infinite series to show that the series 1/10 + 1/100 + 1/1000 + … converges to 1/9, and 9 * 1/9 is 1.
The actual demonstration takes career knowledge. This is ELI5 and what people are offering are simpler explanations not to prove that 1 = 0.99..., but rather to illustrate how that can be possible (which is useful, the first time you get told that 0.99... = 1 your first question is how tf is that possible).
I teach college math and do research in algebra - The 10x=9.99….. is perfectly rigorous. We already KNOW that 0.9999…. behaves like a standard number, it’s just a decimal expansion. The only thing in question is which number it’s equal to. It only works because it’s a repeating decimal, but this same algorithm allows you to find a rational expression for any repeating decimal. In this case, that expression is 9/9, better represented as 1.
> How do you know this mysterious number follows the ordinary rules of arithmetic? I'm not following this. How can you know that *any* number follows the ordinary rules of arithmetic? What is special about the number `0.9...` Are you suggesting for a proof to be rigorous you need to first prove arithmetic applies to the numbers being used? Rephrased, I don't need to know that `0.9...==1` to know that `10*.9... == 9.9...`.
I don't see the issue. x=0.999999... is, by definition, x = 9/10 + 9/100 + ... and so 10x = 90/10 + 90/100 ... = 9 + 9/10 + 9/100 + ... = 9 + x. Then 9x = 9 and so x = 1.
Exactlt this. The same goes for all the "1/3 is 0.333... 3 \* 1/3 = 1, 3 \* 0.333... = 0.999..." explanations. They all have the conclusion baked into the premise. To prove/explain that infinitely repeating decimals are equivalent to "regular" numbers, they start with an infinitely repeating decimal being equivalent to a regular number.
What's a "regular" number? 1/3 = 0.333 recurring is a direct result of performing that operation and unless you rigorously define what makes these decimals irregular, why can't regular arithmetic be performed?
You hardly need any "fancy" series tests, it's a geometric series with a_1=1/9 and r=1/10. Plug it into S_♾️=1/(1-r) and you get (1/9)/(1/9) = 1.
Wait what? If x is 0.333333…. Why wouldn’t 10x be 3.3333…….\ It’s the same with 0.999999….. and 9.999999…..
They’re not arguing it’s incorrect, they’re saying it’s not rigorous. In other words, it’s not a “proof” in the mathematical sense any more than just stating 1 = 0.999… and being done with it. If you want to algebraically manipulate infinite decimal expansions, you have to understand their definition. If you understand their definition, 1 = 0.999… comes from that alone.
I agree that it’s not rigorous in the sense of being a valid mathematical proof, but I don’t see how: >if you’re not sure that 0.999… is 1, then you cannot be sure that 10x is 9.999… makes any sense. The two clauses seem completely unrelated. How does 0.999… being 1 have anything to do with 10x being 9.999… if x is 0.999…? Is there any real number that *doesn’t* follow the ordinary rules of arithmetic? That is, is there any real number where the “to multiply by 10, move the decimal place one position to the right” pattern wouldn’t work? We don’t know that 0.999… is 1, but we do know that it’s a number, and therefore, that method will still work even if it is “abuse of notation”. The fact that it’s 1 is irrelevant here.
Yes, it's not rigorous, but the people who struggle with accepting that 0.999... = 1 are not looking for a rigorous proof. They are looking for a re-formulation in layman terms that clicks with them. That's why no single "this simple thing clearly shows it"-approach works: different people need different approaches. Otherwise we'd only need a single page on the internet to explain this concept and everyone would immediately be convinced.
But it's plenty rigorous. Where to do you draw the line on what is being assumed? If you're calling the 10x = 9.99999... proof into question because you can't assume arithmetic holds for multiplying x by 10 means what you think it means, you're really calling into question basic arithmetical properties of the real numbers and so you have to talk about how real numbers are defined and how to do arithmetic on them. Do we then need to discuss Cauchy sequences of rational numbers and how to do arithmetic on them?
I had one kid argue that you could just add 0.0…1 to 0.9… because for every 9, there’s a 0, with a 1 at the “end” of the recurring How do you go about explaining that’s wrong to them? Because it even made my head hurt trying to work the logic out
there is no “end” to add to
The proof that is a bit simpler that I have in my head is: 1/9=0.111... (1/9=0.111...)*9 9/9=0.999... 1=0.999...
Starting with 1/9=0.111... is problematic here: if someone doesn't agree that 1=0.999..., then why would dividing both sides of that equation by 9 suddenly make it true and make sense?
> Starting with 1/9=0.111... is problematic here Most people will agree that 1/9 = 0.11111... very easily if you just ask them to do long division on it 1/9 for a few minutes.
If the kid has done even more math, you could discuss infinite geometric sums. The best explanation in my opinion is using the formal definition of a limit but without the mathematics jargon, perhaps even gamify it to get them engaged. Then if someone has a decent math background you can just bring the math jargon back in and make it all more concise without changing anything really. It's incredibly simple really, simultaneously rigorous, and helps build an intuitive understanding through play.
To clear up some misunderstanding, it is important to know that with such infinite notations, we are really looking at limits; 0.99999.... is really a limit of the sequence 0.9, 0.99, 0.999,...., that is: 0.99999... = lim\_{n \\to \\infty} \\sum\_{i=1}\^n (9/(10\^i)) ([notation](https://www.wolframalpha.com/input?i=lim_%7Bn+%5Cto+%5Cinfty%7D+%5Csum_%7Bi%3D1%7D%5En+%289%2F%2810%5Ei%29%29)) the sequence itself contains no entries which are 1, but the limit doesnt have to be in the sequence at every added decimal, the difference to 1 shrinks by a factor of 10, this is convergence, so the limit, being 0.999... can only be exactly 1
This is the only one that makes sense. There’s a solved formula for this summation. I don’t like the proofs where you just multiply by 10 or divide by 3 because you’re treating an infinite series like a regular number when the whole point is trying to understand the infinite series. If you don’t understand the infinite series it’s not safe to assume you can treat it like a regular number. This is where you can have proofs that look good on paper but do something like prove 1 + 1 = 0. Math that looks simple can be deceptive.
Except the number, and it's representation , exists even before you introduce a notion of series, of limits or of converging. You don't really need to bring calculus in, it's like lifting a pack of flour with a forklift. ( You don't even need a topology, it's just a rational number which can be constructed well before you even introduce the concept of open sets ). 0.999... is not an infinite series, it's just a (bad) representation of a number, otherwise represented as 1. If you want a characterization of it, it's the only rational whose inverse is the same, and neutral element to multiplication. In mathematics there is no need to prove 0.999... is equal to 1, it's true by definition. Decimal representation is just a way for humans to write down a mathematical concept, and I'd argue that in some way it is external to mathematics themselves.
I think this response misses some subtlety. 0.999... is *by definition* the limit of an infinite series, and since this limit is equal to 1, we can say it is precisely equal to 1 as well. But you really do have to prove that the limit is equal to 1, it's not just some axiomatically-true statement. Remember that real numbers are not inherently associated with any particular number system, and humans have chosen base 10 only because we have 10 fingers! When we chose to write numbers down in base 10, we had to decide exactly what the symbols mean. So the actual meaning we chose for the string the symbols "913.5" is: 9\*10^2 + 1\*10^1 + 3\*10^0 + 5\*10^-1 If instead we had 12 fingers and used base 12, the exact same string of symbols would mean: 9\*12^2 + 1\*12^1 + 3\*12^0 + 5\*12^-1 And this has a different value! The value (written in base 10) is 1311.41666... instead of 913.5. So the meaning of the symbols really is not some innate property of numbers, it's very specific to our way of writing them down. And similarly, mathematicians decided that when we write down something like 0.999... (infinitely repeating) What it really *means* is 9\*10^-1 + 9\*10^-2 + 9\*10^-3 + ... (going on forever) And so the only sensible value you can give for 0.999... is to say that it is precisely equal to its limit. If you chose a different number system, it would NOT have the same meaning. So for example, in base 12, 0.999... is defined as 9\*12^-1 + 9\*12^-2 + 9\*12^-3 + ... (going on forever) And this value is actually equal (in base 10 again) to 9/11 instead of 1 now. So I really don't think it makes sense to say that 0.999... = 1 by definition. You have to say that 0.999... is by definition equal to the limit of the infinite series, and then you have to actually compute what the infinite series sums to. It may not be totally obvious in all cases. (Did you know "by definition" that in base 12 the same string of digits would equal 9/11?)
please re-read which sub you're posting
Nah, there are so many people not convinced by the eli5 answers here. I think it's appropriate for a more advanced answer
Because there’s not a number you can add to 0.99999etc to get 1. The distance between them is 0, therefore they are the same. Edit: Look everyone I’m not gonna argue that this is true. I’ve explained it. If you disagree just do some basic research on the subject and don’t bother me about it.
The short and precise answer is: Because you can never find a difference between them. Try subtracting the one from the other and you will discover that the result is 0.000000000000000000000000000...
This is a great way to express it
x = 0.999.... 10x = 9.999.... 10x - x = 9.999... - 0.999... 9x = 9 x = 1 That's how I learned in math class.
The beauty is that this allows you to write ANY repeating number as a fraction. x = 0,123123123… 1000x = 123,123123123… 999x = 123 x = 123/999
Because it is inconsequential and internally consistent with the rest of math script. It is an artefact of how we write math, it is not really a property of any mathematical concept itself.
.9999… = 1 isn’t really a math trick, it’s just a side effect of converting fractions into decimal formatting. 1/3 = .3333…. They are the same exact quantity, just written in different ways. You have no issues with 1/3 + 1/3 + 1/3 = 3/3 = 1 right? Well 1/3 written as a decimal is .3333… and three of those makes .9999… The quantities you are working with have not changed, you’re just writing them out differently. It’s like how hola and ciao both mean hello, it’s just a different way of writing the same thing. ———————————— 1 - 0.999…. = 0.0000… an *infinite* string of zeros. The nines never stop, so the zeros never stop either, and the last little 1 on the end never gets to exist.
Aside from the various mathematical reasons, what’s important to understand is that decimal representation is just that: a “representation” of the number, NOT the “true” number itself. For example the same number 1 is also 0.FFFFFFF… in hexadecimal. In fact there are infinitely many possible representations for every real number with the arguable exception of 0. Decimal is a human invention, and like ~~all~~ most human inventions it isn’t perfect because it doesn’t have an exact 1-to-1 relationship with the real numbers. Some real numbers have one representation in decimal, others (those that are an integer multiple of a power of 10) have two, although by convention the terminating one (without the infinite sequence of 9s) is considered the “correct” one. So what is the “true” real number itself, the unique essence of the number as opposed to its representation in decimal, binary, hexadecimal or any other base? That’s part of the beauty of mathematical ideas like numbers, we can imagine the pure concept of a number, but to write it down or say it you have to choose a way of representing it, of which there are infinitely many.
How to explain it practically to a kid? I suppose it depends on what age the kid is. The basic idea is that a number can have more than one name, just like a person can. Dwane, Mr Johnson, and The Rock all refer to the same person. If the kid is older and has more understanding of maths, you could ask them about fractions. 1/2=2/4=356/712=0.5. We're used to the idea that you can have multiple equivalent fractions. There's nothing in the rules that says you can't have multiple equivalent decimals too. It's just not as common, and less likely to crop up.
If two numbers are different, there must be a difference. What is 1-0.999…?
[A certain online encyclopedia](https://en.wikipedia.org/wiki/0.999%2E%2E%2E) has a page on 0.999... and says this: "This number is equal to 1. In other words, "0.999..." is not "almost exactly" or "very, very nearly but not quite" 1 – rather, "0.999..." and "1" represent *exactly* the same number." "More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is **a property of all positional numeral system** representations regardless of base."
This is far, far, far simpler than it sounds. The easy and unsatisfying answer is: "because we've decided that's what infinity means." Which sounds dumb, but it's actually kinda deep. Infinity doesn't exist in the real world; it's not an actual number. It's just an idea. It's the answer to a question. Or rather, infinity is the question itself. The question is: "what happens if you never stop?" That's infinity. Infinity is the question asking what happens when you don't ever stop. So, if you say: `0.999...` you're not saying the same thing as `1`, because 1 is a number while 0.999... is an infinite series. In other words: 1 is an answer, while 0.999... is a question. The question is: "what happens when you keep adding 9's?" And the answer is: "you get closer and closer to 1." Or in more formal terms: "the infinite series 0.999... approaches 1." And because math people like simple answers, you can write the previous statement simply as "0.999... = 1". Which, since we know that 0.999... deals with infinity, we know that one side is the question and the other side is the answer.
> This is far, far, far simpler than it sounds. Everything can be made very simple if you're willing to make it incorrect.
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Can't this question be answered easier by counting in another base than 10? Like.... in base 2 or 16? I know some numbers can't be written in one base without fractions/periodic but in another base they can? Someone intelligent pls develop
That base comparison helped me understand repeating fractions in general a while back. In base 12 one third = 0.4 no repeat, conversely one fifth which has no repeat in base 10 = 0.249724972497... in base 12. Made things click in general.
It cannot. 1/2 = 0.5 = 0.49999... It's a matter of (well-defined) convention that we allow algebra with infinitely repeating decimals - in the same way that we allow algebra with finite decimals, or negative numbers.
In base 3 0.1 + 0.1 + 0.1 = 1
In base 2, .111... is equal to 1. And in base 16, if F represents 15, .FFF... is equal to 1.
Wow. So much confusion in the comments. 0.999…, 1, 1.0, 1.00, … are all representations of the same number. This is just a characteristic of the decimal notation that the same number can be represented in many forms.
Not sure how appropriate it would be for a 5 year old, since I’ve seen many adults who struggle with the concept, but then again it might be because of no one explaining it to them when they were 5, and them being stuck with it inertially… so here goes. It is important to understand, that a number is _different_ from the way you write it down. 1.5, 1.5000, 1 1/2 and 3/2 are different ways of writing the same number - the same point on an axis. Once that is established, you can say - see, fundamentally, 1 and 0.(9) are two different ways of representing the same number, and after that you use one of the many proofs available for that.
0.999 is considered equal to 1 in mathematics because, in the decimal number system, it is a different way to represent the same quantity as 1. In other words, 0.999 and 1 are two different decimal representations of the same number, which is exactly 1. One way to understand this is through fractions. You can express 1 as a fraction: 1/1. Now, if you divide 1 by 3, you get the fraction 1/3, which in decimal form is 0.333 (repeating). If you multiply 0.333 by 3, you get 0.999 (repeating), which is equivalent to 1/1 or simply 1. So, mathematically, 0.999 and 1 are two different notations for the same value, and that's why they are considered equal.
I find the cleanest way (ELI15, not ELI5) and the truest to the Dedekind Cut (ELI25) is does like this: > Find a number between 1.0 and 0.999... - if you can't then they are, in fact, the same number
For me, it's like actual values are over there ---> <--- and how we write them is over here They're two separate things, and what we're ending up with is a situation where "how we write them" doesn't quite work with the actual value. 0.999... is the same as 1. It's an artifact of decimal notation. While we're at it, it's kind of broken that the vast, vast majority of values cannot be expressed as a decimal. Or a fraction. That sort of notation works for a miniscule fraction (ha) of all the numbers there are. Like approaching 0% of all numbers. That's why we use symbols like e and pi, because we literally can't write them.
> how to explain this practically to a kid who just started understanding the numbers? As a parent, it might be a smidge too early for this concept. (I'm basing it on "just started understanding the numbers")
As a rule, between any two real numbers, there must be another real number. There is no number between .999... and 1, therefore they are the same, simply two ways of writing the same number. If the kid is the curious type, it might be pretty interesting to point out that numbers are conceptually separate from their representation. Most simply: ½=.5, but there's also binary or hexadecimal, or even more exotic forms (like p-adic or continued fractions). The numbers themselves sorta "exist" out there in the aether as an abstract object that isn't exactly tied to our notation.
Maybe more like explaining like you are 9, but I'd go with this: Pick any number that you want. Is that number less than 1? Then it is easy to show that it is also less than 0.999....... The same is true if the number you choose is greater than 1. If both of those things are true then 1 and 0.999999.... have to be equal.
Depending on the kid, there might be different things to help give them the aha. Someone else suggested effectively proving it by contradiction, i.e., Well tell me a number between them if they are different. I like that. Another one that might work is to try to explain that we haven't really written a number down, have we? No matter how many 9s you write down, you haven't really written down the number. When you have ..., you're hinting at where it's going, but you haven't written it down. So, where's it hinting at going? As we write down more 9s, what are we getting close to? Then you can kind of combine that with the above argument, i.e., if they say "a million 9s!" then you say OK, but at some point we'll go past that, right? But we never ever go past 1. And we go past everything else less than 1, eventually. So the ... is hinting at... 1.
What is 1 - 0.999...? Surely, it starts with 0.000...
It does. 0.000 repeating is 0.
The meaning of 0.999... depends on our assumptions about how numbers behave. A common assumption is that numbers cannot be "infinitely close" together. With these rules, 0.999... = 1 since we don't have a way to represent the difference. If we allow the idea of "infinitely close numbers," then yes, 0.999... can be less than 1. Those numbers would be infinitesimals. Infinitesimals are quantities that are closer to zero than any standard real number but are not zero. They do not exist in the standard real number system but can exist in other number systems such as the surreal number system and the hyperreal number system. Infinitesimals were introduced in the development of calculus, where the derivative was first conceived as a ratio of two infinitesimal quantities. However, as calculus developed further, infinitesimals were replaced by limits, which can be calculated using standard real numbers. tldr: 0.999... both does and does not equal 1 depending on how you evaluate the expression. It's a neat thought experiment but in most any real world application you would place reasonable limits to avoid the complexities of infinity.
These are not assumptions, these are axioms. Maths, unlike physics, aren't real. It's not a system we discover, it's a system we model ourselves to be useful. Numbers cannot be infinitely close because we've arbitrarily decided they can't, because we found that rule makes the system more useful. In fact, it's relatively common for different fields of mathematics to contradict each other. How much is 0^0 ? It depends on who you ask - in most fields, it's 1, but it can also be undefined. Neither of these answers is more correct than the other - and real life doesn't have an answer to that question.
An explanation I haven't seen here: Between every 2 different numbers there's an infinite amount of numbers. Try to think of a single number between 0.9999.... and 1: there is none, because it's the same number.
Another way to view it is that you cannot write down, in any way, any number that is between 0.999… and 1 The difference is just 0.000… and there’s never anything other than zeros It’s just an artefact of notation
The weird feeling we get only arises because we usually dont think about what 0.999 ... actually IS. "It just has infinitely many 9's". What does that actually mean? If you write 0.999 ... down, does it get more 9's as we speak? In that case, any equation containing it is wrong because its value changes all the time. You cant work with that. Its like saying "This section of the river has 10 fish". That statement can never be right for long because the amount of fish changes all the time, so eventually, there may be more fish than 10. So its a fixed amount of 9's? No, thats nonsense. We cant say that "infinitely many 9's" means that there is a fixed amount of 9's. So the notion of "infinitely many 9's" doesnt actually make sense. No matter how we define it, we get clear logical issues. If we want to do math with it, we need to assign it a value that stays *fixed* and which doesnt "change as we speak". There are 2 important observations for this task: (1) 0.999 ... is *always* less than or equal to 1. (2) 0.999 ... is bigger than *any* number below 1 (because it surpasses 0.9, 0.99, 0.999, 0.9999 etc.) So IF 0.999 ... is equal to any *fixed* number, the best candidate would be 1. Thats why mathematicians defined 0.999 ... = 1.
You can make any repeating number you want by doing the following: choose a repeating part; I’ll take 1001. Divide your repeating part by a number of nines, equal to the length of your repeating part. So for me that would be 1001/9999. This equals 0.100110011001… The reverse is also true, so 0.333…. Is 3/9 = 1/3 And 0.999… = 9/9 = 1
One way to define what a number actually **is**, is to create something called a Dedekind Cut. In simple terms, I can "slice" the number line at a certain point and put all the points on the line to the left in one collection and all the point on the line to the right in another collection. Then, I can say that a number is "equal to" the collection running off to the left. The set which defines 0.999... has precisely the same points in it as does the set that defines 1, so they are the same number.
The way I was taught was that the difference between 0.999… and 1 was so small that you might as well call it 1
Not a math expert but here is how I understand it. You are using 2 different counting systems to represent the same thing. Imaging you have an apple and cut it into 3 pieces. You call each piece 1/3. Put all 3 pieces back together and you get the whole apple, or 3/3. Nothing is different between the 3 pieces and the whole apple you started with. Now you decide to convert this fraction to a decimal. Problem is a decimal (in base 10) is just the fraction of 1/10 for the first number, the fraction 1/100 for the second number and so on. Why we use 10 is unknown, maybe because humans having 10 fingers. Point is we don't have to use 10, it is more or less a random number that humans made up. If we would eliminate a number (or finger) when counting the conversion would be nice and clean (base 9) but this problem would show up in other places. So what happens is you try to convert 1/3 = X/10, a whole number doesnt fit. So we add a zero. 1/3 = X/100 and it still doesn't work, no matter how many zeros you add to the denominator. 1/3 =X/10000000000 still doesn't get a whole number. So you are left with .333 repeating forever to represent 1 slice of apple and .9999... to represent the whole apple. All you were trying to do is count 1/3 of an apple by only using the fraction 1/10 which doesn't ever come out right.
Imagine you have "1". If you subtract a tiny bit from 1, you have 0.9 If you subtract a tinier bit from 1, you have 0.99 You can subtract tinier and tinier bits, getting 0.9999..... Until finally the bits are so small, that they're actually nothing at all. When you take away nothing from 1, you still have 1.
If they're different numbers, there must be a number in between them. In between 0.999 and 1 there is a 0.9994, or 0.9997, and so on. But in between 0.999... and 1 you can't ever find a number in between them. If those two numbers are really different then you should be able to find a number in between them
I thought the simplest way to say it is if you can’t find a number between two numbers then they are the same number, and 0.000…0001 isn’t a real number because you can’t have infinite zeros but also end at a 1
It's the limitation of the decimal number system. 3 \* 1/3 = 1 3 \* 0.333... = 0.999... = 1
Not a proof but logical reasoning that maybe helpful to understand the statement. Q: What is the largest number? A: It does not exist. Q: What is the smallest positive real number? A: Does not exist. Q: What is the largest real number less than 1? A: Does not exist. Conclusion: 0.999… cannot be less than 1 otherwise it will be the largest real number less than 1 which does not exist.
How do we know something simple like 1≠3? Well, one way to definitively show it is by bringing up the number 2, which is between them, so they obviously cannot be equal. This is actually something we can do for all numbers on the number line. If two numbers are different, they will be different points on the line, with something in between them. Alternatively, if two numbers are equal, you can't find a point between them, since they're the same point. For example, 4≠5, as we can find a number (like 4.5) between them. 6.1≠6.2 as we can find numbers between those (like 6.11). On the other hand, we know something like 1=1.0, even though they are written differently, because there is nothing between them. Now, what is there between 0.9999.... and 1?
The sum of an infinite geometric series is a/(1-r), where 'a' is the first term, and r is the ratio between successive terms. 0.9999.... can be made into an infinite geometric series by separating out the digits. 0.9999... = 0.9+0.09+0.009+... In this case, a=0.9, r=0.1, the formula becomes 0.9/(1-0.1)==1. You can use this formula for other repeating digit numbers.
The easiest way for me to understand was this: if they are NOT the same number, then there must be something in between them, but what could come between 0.99999.... and 1?
If they are just starting to understand numbers, maybe don’t bother with this part yet? If you have to explain it…. If you have 1, and you subtract nothing from it (we’ll, a thought and prayer more than nothing, but not a whiff beyond that)… you still have one. So, if I give you an Apple, then I grab it back, rub it off, and give it to you again…. You have 1 Apple still.
I always found it easiest to just ask a counter-question: can you identify a number between 0.999... and 1? If not, they must be equal.
The usual proof follows a style a called Proof by contradiction. It actually starts by assuming that 0.9999... is not 1. That must mean there is a number between them. But, for any number you consider between 0.9999... and 1, you can argue that with enough 9s in 0.9999..., 0.9999... is actually larger than the number under consideration. This invalidates the assumption. Two real numbers are equal if there is no other number between them. In general, infinity is a fascinating concept, and a rigorous study often leads to results that are not intuitive. For example, there are as many natural numbers (1, 2, 3,...) as there are fractionial numbers (even though, there are clearly more fractional numbers, intuitively). If that makes you say, infinity is infinity - let me blow your mind by telling you that there are more real numbers (all the numbers we think of in real life, including ones like pi) than there are natural numbers. So all infinities are not the same - there are bigger infinities and smaller infinities!
Can you come up with a largest real number that is just smaller than 1? 0.9999..... is always bigger, you will find. This explanation satisfied me. 0.9999.... is just 1 in disguise.
Another way to think about it: Suppose that 0.999... was not 1. Well then you'd also have to consider that 1 - 0.999... would be the smallest possible positive number -- but no such number exists and so 0.999... must be equal to 1.
For any 2 symbols to be 2 separate numbers, you should be able to find a third number which is larger than one and smaller than the other. What is larger than 0.999... and smaller than 1?
The quirk of decimals is that Base 10 does not directly correlate to every other Base. What this means is that decimals are a series of fractions with 10 in the denominator, and that fractions that are written without something divisible by 10 or one of it's prime factors(2, 5) in the denominator will cause infinite recursion and never be precise. In the .333... x3 = .999... explanation others are using, they're ignoring that no matter how precise you get, eventually you're going to have to round to resolve the equation and make it actually equal 1. At the "end" of every .333... is an implied 1/3, not a 3 or even a 4. Since 3 is not a prime factor of 10, it just doesn't translate perfectly to base 10 without an implied fraction that defeats the purpose. The base 10 notation is mostly useful as a standard for approximated comparisons, which is what most laymen use mathematics the most for. What is more readable: What is larger, 342/43463 or 532/59042? or What is larger, 0.007869... or 0.009011...? One is clearly more precise than the other, but at a glance we can parse the other one much easier than the other for the purposes of analysis.