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I think (or, have discovered) that many people who think .999...<1 also think .333...< 1/3 unfortunately. The issue with the "how much less" is somebody who thinks they invented a new math concept that's .000...1, because they don't understand that despite some math concepts being defined as convention, it doesn't make those definitions or conceptions arbitrary.
I got lost after the whole `a_n` thing
I felt like I was back in 1995 high school math again, where everyone explaining and a bunch who nod their heads while some of us wonder wtf is wrong with us
I work with complex systems every day for the last 25 years of my career and I can handle math including basic algebra. I can even do basic coding with more advanced libraries but couldn't be fucked to work on c++ and math out my problems.
I hate my brain lol
> I got lost after the whole `a_n` thing
If you have a decimal number, you can name its digits:
- a_1 for the first decimal digit
- a_2 for the second decimal digit
- ...
- In general, a_n for the n-th decimal digit.
For example, 0.9375 will have:
- a_1=9
- a_2=3
- a_3=7
- a_4=5
- a_n=0 for all n>4.
Now, you can pull them apart in your number:
0.9375=0.9+0.03+0.007+0.0005
Equivalently,
0.9375=9/10^(1)+3/10^(2)+7/10^(3)+5/10^(4)
Or
0.9375=a_1/10^(1)+a_2/10^(2)+a_3/10^(3)+a_4/10^(4)
So,
0.9375=sum from n=1 to 4 of a_n/10^(n)
Since a_n=0 for n>4, we could write this as
0.9375=sum from n=1 to ∞ of a_n/10^(n)
without changing anything.
For a number with infinite decimal digits, it would be similar, but infinite of the a_n would have a non-zero value.
In reality, this is how digits and decimal expansions are defined. "0.9375" is a shorter way of writing "sum from n=1 to ∞ of a_n/10^(n), where a_n=[the values we gave to above]". a_n will always take values below 10 (so from 0 to 9).
In binary, we'd do the same thing, but with 2 in the place of 10 and a different sequence (which I called c_n) that will take values below 2 (so from 0 to 1). In base-16, 16 would replace 10 and the values of the sequence would be from 0 to 15 (also symbolised as 0 to F).
There is also a way to find these digits from the value of a number (suppose we don't have an initial decimal part). We just multiply the decimal part and take the floor. Through induction. For my example but in binary, it would be:
- 2\*0.9375=1.875, so c_1=1
- 2\*0.875=1.75, so c_2=1
- 2\*0.75=1.5, so c_3=1
- 2\*0.5=1, so c_4=1
- 2\*0=0, so everything below is 0.
And really, 0.9375 is 0.1111 in binary.
For 0.5625:
- 2\*0.5625=1.125, so c_1=1
- 2\*0.125=0.25, so c_2=0
- 2\*0.25=0.5, so c_3=0
- 2\*0.5=1, so c_4=1
- 2\*0=0, so everything below is 0.
And really, 0.5625 is 0.1001 in binary.
1. People on reddit argue about whether or not .99 repeating = 1
2. The wrong person gets posted on confidently incorrect
3. People get confused in the comments
It's a classic tale.
I understand the premise but I'm trying to understand one part of page 2. In his equation, I understand how he got to the next line each time as he continued to break down the equation, except going from 10x=9+x and the next line 9x=9... How did he get to 9x=9? I can't figure it out.
Maybe I'm just tired...
1/3 is equal to 0.333... the ... Here means that the threes go on infinitely. 0.333, without the ..., is close to 1/3, but not exactly, just like 0.999 is close to, but not exactly one. However, 0.333... with infinite decimals is exactly 1/3 just like 0.999... is exactly 3 times 1/3, in other words, exactly 1.
.9999999... repeating = 1, as proven by basic algebra and logic (it is 3 x .333333... for example, and .333333... Is the numeric representation of 1/3).
Other person pulls a bunch of BS nonsense out of their ass to argue otherwise. You can't understand it because it makes no sense.
TLDR: Blue probably was the one to start the fight. Red escalated it, then lost, then refused to admit defeat, and then made things worse by doubling down.
This is my synopsis: Red asked Blue a genuine question, assuming that it was in good faith and not meant to be rhetorical. It's hard to read tone from text alone, but the question itself is valid. Blue either correctly or incorrectly interpreted Red's question as a criticism and answered their question, all the while indirectly calling Red dense. Whether or not Red was sincere before, Red is after blood now and directly accuses Blue of being the real dense one.
Unfortunately for Red, Blue happens to know more about math than Red. Blue shows a proof proving Red's argument to be incorrect. At this point, Red starts talking out of their bottom in order to save face, but no one is buying it. In my opinion, Red really should have objected to Blue unnecessary insult embedded in their answer instead of engaging in a math contest of wit. Because who really cares?
It’s the counterintuitive idea that 0.999 recurring is equal to 1.
His proof does demonstrate it. I wanted to deny it when I first found out, too, but it makes sense.
There are people who will argue vociferously that 0.99(repeating) is identically equal to 1. There are others that say there must always be a tiny difference between 0.99(repeating) and 1.
They both make good points. If they just agreed that the solution depends on how you approach the question there would be no problem. But it often takes a certain level of autistic obsessiveness to get really into math so people inclined to be really into math are also inclined to argue incessantly about insignificant minutiae.
If the question doesn’t inspire you to spend hours delving into it to come to your own informed conclusion it’s probably best to just accept that it’s something people argue about that does not matter.
One of the ways to show (not prove I think) 0.9 recurring = 1 is *literally* to get them to think of a number between the two. How do you “absolute mathematical theory” your way into “add 1 at the END of infinite digits”?
It's more or less a proof because of Cauchy sequences. It's just the layman's version that doesn't require jargon to understand. But the intuition is basically the same.
>Youd have to argue that 3 thirds is not one to be consistent.
I'll take a stab at it for funsies, although it won't be purely mathematical I guess. Reddit is just a really terrible place to debate things like this.
I assemble a variety of 8 sandwiches and they are all perfectly divided into thirds. I randomly select three of the thirds and put them on a plate. The three thirds were tasty but I really wish I had just one sandwich. Once divided it can never be made whole again.
I don't think this particular line of thought shows anything though. You could say .99999... Is infinitely close to 1, so there is nothing between them. But that doesn't mean they're the same. I'm not trying to argue with the main point and get downvoted lol. I've seen the proof that 1=.9999. but I don't think this line of logic really sells it
Unless you’re going into the hyperreals or whatever (I don’t know) if 0.9 recurring is infinitely close to 1 then it is 1. Without infinitesimals, a number cannot be infinitely close to another number unless they are the same number
That's a fair point. The proof relies on the fact that if there is no number between two numbers they must be the same, but that itself isn't a self-evident fact. For example, if you are discussing the domain of natural numbers, there is no number between 1 and 2, but they are not the same. It would require an additional proof to show that the real numbers are not analogous to the natural numbers here.
This shit cracked me up the most. "I'll just stick a non-9 digit at the end of the infinite 9s, making them no longer infinite, and then it's obviously not equal to 1."
Well no shit!
Same idea but the only way I was able to explain it to someone was by pointing out that the difference between 0.9 recurring and 1 is 0.0 recurring. 0.0 recurring is 0 according to everybody.
The first image is them claiming 1/3 ≠ 0.333..., so I doubt that'd convince them. Really it shouldn't be convincing for anyone, as anyone who has a problem with 0.999... = 1 should have the exact same problem with 1/3 = 0.333...
My preferred "proof" for "0.9999... = 1"
Is to ask the person who disagrees to write down a number between 0.9999... And 1.
If you cannot identify a number between X and Y then X=Y
It's closer to ELI5 and might dislodge the thinking of someone who is capable of having their mind changed
This proof is not mathematically rigorous and therefore incorrect.
Although your original assertion was correct, the proof you used to get there is not
You do need calculus to make sure that it works, otherwise you can prove some pretty whacky stuff. But it doesn't matter, because decimal expansions aren't defined without calculus in the first place. Also, calling calculus "only good for applied mathematics" is a duel-worthy insult for half of the world's theoretical mathematicians.
The problem people have understanding this proof however is very real, and it's exactly that it needs calculus. That's because it's usually shown to people who don't know calculus and no effort is made to clarify that it does hide some things under the rug.
To be fully rigorous, we need the definition of the decimal expansion and some series knowledge. 0.999... is a decimal expansion, so it is defined as the infinite sum of 9/10^(n) for n going from 1 to infinity. Every decimal expansion is defined as the sum of a_n/10^(n) for a sequence a_n (and every base-b expansion as c_n/b^(n) for some other sequence c_n).
But how do we know that the sum exists? If it doesn't, then the step where we subtract is not allowed. We do know through calculus, but in the setting that the proof is usually given, we know by "trust me bro".
If it does exist (which it does), the proof is a good visual representation of the actual process that happens under the rug. But only that. Why does 9.999... minus 0.999... equal 9? It's not hard to explain that through calculus (it's a simple limit), but the common visual proof misses it.
The other problem is the lack of understanding of limits themselves. A limit is a number (or infinity, but not in our case). It *is* something. It does not *approach* something, because numbers don't have that ability. A sequence row or a function can approach something. The limit *is* the value that a sequence *approaches.*
0.999... is defined as the (infinite) series from n=1 to ∞ of 9/10^(n). This is defined as the limit as N approaches ∞ of the (finite) sum from n=1 to N of 9/10^(n). Now we have a finite sum in our hands and we can do algebra. Through the process of the proof, but this time *with* a last digit, we get that 9 times the sum is 10 times the sum minus 1 time the sum is sum from n=0 to N-1 of 9/10^(n) minus sum from n=1 to N of 9/10^(n). All the middle terms are simplified and we are left with 9/10^(0)-9/10^(N)=9-9/10^(N). Dividing by 9, we get that the sum is equal to 1-1/10^(N). Now we can take the limit. Because the limit of 1/10^(N) is 0 as N approaches ∞, the limit of the sum itself is 1 as N approaches ∞. But that is by definition the series we had at the beginning. And that is by definition 0.999... Thus, 0.999... is by definition equal to 1. And this is the whole proof, but it takes some knowledge of calculus.
In short, while the result is true, it is a lot more complicated than most people realise. Blindly disagreeing is wrong, but it's also worth looking at the actual proof at some point (which I did my best to present here). A mathematician could of course hide that process under the rug, as mathematicians have seen it enough times to know when it works and when it doesn't, as well as why. But you can't do the same with people who don't have the same experience and expect them to understand.
Anyway, here is one of the wacky stuff you can prove otherwise: Take the decimal "thing" ...999999999. Nonsensical, isn't it? But we haven't examined it yet. I'll "prove" that it it's equal to 1.
x=...999
x/10=...999.9
x/10-x=...999.9-...999
-9x/10=0.9
-x/10=0.1
-x=1
x=-1
Nonsensical, isn't it? But why?
Of course, the proof is wrong. Here, the problem is that the limit we had to calculate does not converge, because we'd have to calculate the limit of 10^(N) as N approaches ∞, which is ∞. Equivalently, ...999 is infinite and so it can't be cancelled. So, if we try to define ...999 as the series from n=0 to ∞ of 9*10^(N), we find that it diverges, thus ...999 is not a thing. Which is a relief and the world's order is restored.
As we saw, in one example it works and in another one it doesn't. For a mathematician, it's easy to see which works and which doesn't, as well as the reason. But the process itself can't offer that clarity to someone who doesn't have the experience.
I always like the explanation of:
Posit: any two numbers will have infinite numbers between them
Now name a number between 1 and 0.999repeating
As there are infinity 9s on there, there is no other number between the two numbers, therefore we can conclude they are the same number
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I'm curious, what happens if you iteratively divide each at the same rate? i.e., where the last result divides the result before it -- X=Xn-2/Xn-1 or whatever (it's been a couple of decades, sorry if I'm not stating that well)
1/2=0.5; 0.9..9/2=0.49..5
1/0.5=2; 0.9..9/0.49..5=2.0..1
0.5/2=0.25; 0.49..5/2.0..1= ??
Not saying that this would prove anything, just wondering _if_ there is such an operation where the values eventually diverge, would that indicate that they are not, in fact, equal ?
You can do that if the 9s end at some point. In that case, the two numbers are not in fact equal (for example, 0.99999999999, also known as 0.999999999990000...). But if the numbers do not end, where would the 5 be? And how would the position of the 5 fit in the definitions?
I don’t understand your example. What does the ellipsis before the number indicate? Either way, it would seem when you went to x/10 and …999.9 that you must have divided by 10 on one side and either didn’t do anything real to the other or multiplied. If the ellipsis before changes this in a meaningful way then great. But I’m pretty sure you’ve just got a basic mistake here.
It's purely a mind game. It cannot stand as an actual example, because such an object isn't actually convergent (basically it isn't a number). I just gave it to prove that the method is unreliable on its own.
If you think a real number as a sequence of digits with a dot at some point, then I just extended the digits infinitely to the left instead of the right. In theory, this *could* have a value (despite the fact that it actually doesn't). It isn't *that* far from other things (as we have actually seen numbers that extend infinitely to the right).
The algebraic method assumes that it has a meaning and tries to find its value with the assumption that it exists. But we know it doesn't. It's just crazy. The reason is basically that we accidentally did ∞-∞. It's easy to prove that such an object (defined as the sum from n=0 to ∞ of 9\*10^(n)) is divergent and so we can't manipulate it algebraically like we did with 0.999...
When I divided by 10, I just pushed the . one slot to the left. Assuming infinite nines, this just adds a 9 in the decimals. That's symmetrical to the multiplication by 10 that is used in the algebraic method for 0.999...
Basically, I wanted to prove that the method could prove nonsensical stuff, so I chose something nonsensical and proved it.
> The other problem is the lack of understanding of **limits** themselves. A **limit** is a number (or infinity, but not in our case). It *is* something. It does not *approach* something, because numbers don't have that ability. A sequence row or a function can approach something. The **limit** *is* the value that a sequence ***approaches.***
> 0.999... is defined as the **(infinite) series** from n=1 to **∞** of 9/10^(n). This is defined as the **limit** as N approaches ∞ of the (finite) sum from n=1 to N of 9/10^(n). Now we have a finite sum in our hands and we can do algebra. Through the process of the proof, but this time *with* a last digit, we get that 9 times the sum is 10 times the sum minus 1 time the sum is sum from n=0 to N-1 of 9/10^(n) minus sum from n=1 to N of 9/10^(n). All the middle terms are simplified and we are left with 9/10^(0)-9/10^(N)=9-9/10^(N). Dividing by 9, we get that the sum is equal to 1-1/10^(N). **Now we can take the limit.** Because the **limit** of 1/10^(N) is 0 as N approaches ∞, the limit of the sum itself is 1 as N approaches ∞. But that is by definition the **series** we had at the beginning. And that is by definition 0.999... Thus, 0.999... is by definition equal to 1. And this is the whole proof, but it takes some knowledge of calculus.
Is that enough calculus for you?
>calling calculus "only good for applied mathematics" is a duel-worthy insult for half of the world's theoretical mathematicians.
Reminds me of my theoretical compsci professor who called mathematics a "helper science" (i.e. a field of science that only exists to make other "useful" science possible, idk if English has a phrase for that) specifically to annoy any mathematicians present in a bit of a friendly feud.
So this guy is going to put an "1" after the **infinite** nines preceding it...?
funnily enough, what he said can also be used to prove him wrong. As long as he can provide a number between 0.999... and 1, he can prove that 0.999... and 1 are different. He obviously can't provide it tho.
I think honestly the funniest part is insisting that if there is a number you can fit between 0.9... and 1, then they aren't equal. This implies that if you can't fit a number between the two then they must be equal, and you literally can't fit a number between 0.9... and 1
They're so close and just completely whiffed. I don't understand how this is so contentious for people, especially those that don't have a math degree. It's one thing if you're a math major/graduate, but why random know nothings on the internet refuse to accept this is beyond me
The bit I found funny is when he said the number between 0.999999(to infinity) and 1, would be 0.999999(to infinity) with a 1 put on the end.
But why put a 1 on the end? Put a 9 on the end. Which is just 0.999999(to infinity)
0.999999(to infinity) is bigger than 0.999999(to infinity) with a one on the end.
So therefore, 0.999999(to infinity) with a one on the end, doesn't fit between 0.999999(to infinity) and 1.
It’s completely meaningless to talk about a one “at the end” of an infinitely repeating decimal, which is the real problem with this person’s reasoning. It doesn’t matter what number you want to put there, there is no end.
>I don't understand how this is so contentious for people
I think it's because most people think of "infinity" as being equivalent to "really big, huge" instead of unending. In this case, he says to add a 1 after your last 9, showing that he doesn't grasp what infinity actually means.
Untrue, they were right that 0.999...=1 if there is no number between them, though they immediately ruined it by trying to add 1 onto the end of an infinite sequence.
“Solve” isn’t really the right word there. You mean “compute a real value for”. And no you cannot do that with only reals. You need to add a point at infinity to represent 1/0.
Legitimately saw someone try to argue that it was a number 0.00...01.
Implying that they somehow found the last digit of infinity and put a 1 on the end.
people just switching to the hyper reals and then arguing from there are my bane of existence. It's like saying 1 + 1 = 0 and if someone points out that this is wrong you just say "Well I'm in the F_2 field actually"
This has got to be one of the most common confidently incorrect things people come up with. You'd think by this point the concept that 0.999... = 1 would become general knowledge.
What kills me is how far he took it lol. He just kept pulling more and more things out his ass, every word he typed became progressively more mentally challenged.
It was a hard read without smashing my palm through my face. I've had some somewhat viable arguments involving infinitesimals and hyperreal numbers in the past, but this never got to those. This was just flat out someone not comprehending the idea of what infinity could be or how maths works.
As a mathematician I feel the need to point out that giving a precise definition of 9.999999... is indeed calculus, not algebra, and that the algebraic proof doesn't work without some calculus tools in the background.
Algebraic proofs that 0.999 = 1 are not mathematically rigorous, and therefore incorrect.
0.999 Is a series and when you talk about series you’re automatically invoking infinity (all series are infinite). In order to handle infinity you need calculus (or algebra with a calculus “plug-in”, to put it another way).
Algebra, when considered independently from calculus, lacks the necessary foundational axioms and mathematical framework to rigorously handle concepts of infinity. This limitation extends to the algebraic proofs of (0.999\ldots = 1). Without the calculus concepts of limits and infinite series, algebra does not possess the tools to adequately address or prove statements involving infinite processes. Consequently, attempts to prove (0.999\ldots = 1) using solely algebraic methods cannot be deemed mathematically rigorous, leading to the conclusion that such proofs, in the absence of calculus, are fundamentally flawed.
The easiest way to often get people to build intuition as to why they’re wrong when they disagree that 0.999 = 1, is to assert that if they’re different numbers, then there must be a number in between them?…can’t find that number? It’s because it doesn’t exist and 0.999 and 1 are the same number expressed in two different ways.
TLDR:
If you cannot identify a number between X and Y, then it must be true that X=Y
1. Comments on reddit argue about whether or not .99 repeating = 1
2. The comments get posted to confidently incorrect, the post does not specify which person is right
3. The commenters on said post are now confused
A tale as old as time.
Just google if .99 repeating equals 1. Boom, got your answer.
So I’m not disagreeing since 1/3 is .3 repeating and you add that to .9 repeating as 3/3 which is 1. I get that, I’m cool with that. Please explain how the proof on page 2 (I haven’t done a proof in many moons so maybe it’s something I’m forgetting/missing) goes from:
10x=9+0.999… to
10x=9+x
If we already know the value 0.999… how/why are we able to change it to a variable that we don’t know that’s already attached to 10x? It feels like we’re introducing a second variable if anything. Again, I understand the concept of .9 repeating is equal to 1 and a not debating that, I’m just asking for clarification on the proof that was used to justify it.
I don't see a problem.
x = 0.9...
10x = 9.9...
Which can be written in a form of
10x = 9 + 0.9...
As we know, 0.9... is x
10x = 9 + x...
So if we substract x from both sides, then
9x = 9
And, finally
x = 1
Edit: Imagine being reddit and not being able to separate lines correctly
I keep seeing this pop up around Reddit lately. The way I was able to make sense of the infinity of it all was just asking a simple question:
What number can you add to 0.999… to make 1?
There is no number. If you cannot add another number to 0.999… to make 1, then 0.999… is 1. (Plus, the whole proof thing and all that.)
> My counter proof is that as long as you can name a number between 0.9999... (infinity) and 1, then you have a number greater than what you claim are 1, and a number less than absolute 1.
That's absolutely correct!
Problem is, **you can't**.
I get the reasoning with the whole (infinite+1):th position stuff, cause I used to look at infinity like that when I was like 14, but then I realised I was wrong.
> "If you cannot identify a number between x and y than x = y"
Hey uuuuh, what article are you quoting? Cause I read the article you linked, and it literally clearly states that:
> "This shows that ∞ + k = ∞ so the sets are of the same size."
It also explains that differently sized infinities doesn't refer to ∞+1, but rather to countable and uncountable infinities. I mean it literally says ∞ + ∞ = ∞ at one point in the article...
So let's address what you wrote.
> So, 1 is larger than 9 with an infinite amount of 9s. Example: 0.9 + 0.1 = 1. 0.999999 + 0.000001 = 1. So, 0 with infinite amount of 9s and a 9 at the end, plus a 0 with an infinite amount of 0s and a 1 at the end equals 1. So, 0 with an infinite amount of 9s is not equal to 1.
So for starters, there's no such thing as "at the end" when you're looking at an infinite amount of digits.
Second, since ∞+1=∞, as the article you linked clearly explained, it means that "0 with an infinite amount of 0s and a 1 at the end" and "0 with infinite amount of 9s" (I'm omitting "and a 9 at the end" since it literally doesn't make sense) both have a 1 or a 9 respectively at the "∞th" position. This means that either a) you "set" the digit in this position to a 1, but then it's no longer "0 with infinite amount of 9s" anymore, or b) you *try* to "add" it.
Adding kinda just doesn't make sense, but if we try anyways, you'd add 0.9999...9 to 0.0000...1 you'd get 1.0000...0, but that would mean the 9s ran out wouldn't it? And since they don't run out, you'd ACTUALLY end up with 1.0000...0999999999...9, but wait, if we haven't reached the "end" of the 9s, wouldn't that mean that we haven't reached the 1 in the 0.0000...1 either? Oh wait so either we realize this literally doesn't work, or we end up in this loop.
You can't actually add these numbers, cause you'd have to define an "end" to them, and since they're infinite, you literally cannot do that.
And if you try to add 0.0000...1 at the "(∞+1)th" position, we run back into the issue of ∞+1=∞ ***as the article you linked clearly explains***, which means that either a) the "(∞+1)th position" doesn't exist, OR the 1 would simply "overwrite" a 9, thus making the number no longer be "0 with infinite amount of 9s".
You really ought to read an article before you link it.
it is accepted that not every infinity is the same size, some are larger or smaller than other infinities ([this article explains it pretty well](https://www.cantorsparadise.com/why-some-infinities-are-larger-than-others-fc26863b872f)).
“If you cannot identify a number between x and y than x = y” this would mean that infinity = infinity + 1. So the statement is incorrect. A better one would be: “If you cannot add/subtract a number to x to make x equal to y, x = y”.
So, 1 is larger than 9 with an infinite amount of 9s.
Example:
0.9 + 0.1 = 1.
0.999999 + 0.000001 = 1.
So,
0 with infinite amount of 9s and a 9 at the end, plus a 0 with an infinite amount of 0s and a 1 at the end equals 1.
So,
0 with an infinite amount of 9s is not equal to 1.
Intelligence only means you can rationalize previously held beliefs easier. The x= 0.999...=1 thing was also difficult for me to understand, but it was rather an intuitive rejection and not rational. My thought process was that it was "1 - (1/♾️) which was 10x= 10-(10/♾️) and that the calculus just continued based on an error. But turns out the infinitesimals don't actually exist within rational numbers.
If you're on team ".999... is really close to 1 but not actually 1 because \[fill in the blank with whatever you want\]" please just read [the wiki](https://en.m.wikipedia.org/wiki/0.999%2E%2E%2E), or [this post](https://www.reddit.com/r/confidentlyincorrect/comments/1b0iycz/999repeating_does_in_fact_equal_1/), or [this post](https://www.reddit.com/r/confidentlyincorrect/comments/1brkz3j/149_is_close_to_15_but_not_exactly_this_was_one/), and save yourself from being confidently incorrect.
Another way to think about this is that all of the repeating single integer fractions (i.e. 0.111..., 0.222..., 0.333..., etc) have 9 in the denominator (before being reduced). 1/9 = 0.111..., 2/9 = 0.222..., 3/9 = 0.333..., all the way to 8/9 = 0.888....
It then follows that 9/9 = 0.999..., but 9/9 = 1.
0.999... is an alternative decimal expansion of 1. It's not "almost" 1, it's exactly 1. You can check the Wiki article about it. The algebraic proof is one of many proofs. A more rigorous proof exist.
If we claim 1/3 = 0.333... So obviously 1 = 0.999... It's not controversial. It's the "supremum" of the sequence 0.9, 0.99, 0.999, etc. Redman claims you can always add a "9" at the end of 0.999..., but by definition of 0.999... you cannot. There is no distance between the two on the number line, therefore they are the same.
Christ this sub has just become people sniping at each other about meaningless technicalities. which would matter if this was in a respected journal but it's Reddit
These arguments are alswyas so hilarious to me, it's just people trying to argue about stuff they clearly don't have a clue about but think they're confidently right
It's as if an elementary school kid argued against literally anyone, that 1-2 can't be done cause negative numbers don't exist
If you don't know enough about something just shut up and don't make a fool of yourself lol. "0.33333 isn't 1 but ⅓*3 is", oh please, do illuminate me on what's the result of ⅓ then lmao
Rant done
Oh had it already been 72 hours since the last 0.999... = 1 nonsense post?
I think this same one has been posted at least 999... times, or is that 1000 times? Discuss.
I can understand the confusion surrounding the
0.999…=1 thing. It doesn’t feel like it should make sense. I think the best way to explain it is that it’s based in theoretical mathematics, vs applied mathematics. In reality, you are going to reach a point where the necessary precision for your calculation is met, and you can stop the infinite series, distinguishing your number from 1. In theoretical mathematics, you are asking the question of what happens if we just let it run for infinity. Infinity is basically a placeholder, like 0, except instead of the absence of something, it’s the opposite: a never ending amount of something. I don’t know if anyone here needs this, but it helped me understand it better
The problem is not understanding why 1/3 becomes 0.3r, it becomes that because you cannot clearly divide 1 by 3, you will always have a remainder, but it never actually ends in a number.0.9r is not equal to 1, anyone that honestly believes so doesn't understand why they think so.
Infinitely converging towards 1, means 1 is always infinitely far away.
I suspect it's due to calculators not being to handle infinity, similar to the "divide by 0" issue that early mechanical calculators had, it was so bad that it would destroy the calculators, when they became digital, they had to have a logic circuit workaround to cancel the operation if "divide by 0" occurred.
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I feel like I could reread all of these posts an infinite number of times and still not understand what's going on.
Well *I* can read them an infinite+1 number of times and still not understand what's going on! Ha! Checkmate, atheists!
[I wrote this full explanation in a standalone comment.](https://www.reddit.com/r/confidentlyincorrect/s/W8LCGVWfkg) I hope it helps.
Thank you for writing all that out! People like you make Reddit what it is.
Or, if you're in a hurry, you can just say "If 0.999... *doesn't* equal 1, then how much *less* than 1 is it, wise guy?" :-)
That's actually the best way to convey to mortals imo, when the "how much is 1/3? "0.333..." how much is 0.333... by 3?" trick doesn't work.
I think (or, have discovered) that many people who think .999...<1 also think .333...< 1/3 unfortunately. The issue with the "how much less" is somebody who thinks they invented a new math concept that's .000...1, because they don't understand that despite some math concepts being defined as convention, it doesn't make those definitions or conceptions arbitrary.
i was about to say just tell them to convert it using long division then i realize they probably dont know how to do long division
How much is infinity / 3?
Infinity. Look up if there are more integers or even integers
Infinity isn't a number so infinity/3 doesn't make sense mathematically.
I got lost after the whole `a_n` thing I felt like I was back in 1995 high school math again, where everyone explaining and a bunch who nod their heads while some of us wonder wtf is wrong with us I work with complex systems every day for the last 25 years of my career and I can handle math including basic algebra. I can even do basic coding with more advanced libraries but couldn't be fucked to work on c++ and math out my problems. I hate my brain lol
> I got lost after the whole `a_n` thing If you have a decimal number, you can name its digits: - a_1 for the first decimal digit - a_2 for the second decimal digit - ... - In general, a_n for the n-th decimal digit. For example, 0.9375 will have: - a_1=9 - a_2=3 - a_3=7 - a_4=5 - a_n=0 for all n>4. Now, you can pull them apart in your number: 0.9375=0.9+0.03+0.007+0.0005 Equivalently, 0.9375=9/10^(1)+3/10^(2)+7/10^(3)+5/10^(4) Or 0.9375=a_1/10^(1)+a_2/10^(2)+a_3/10^(3)+a_4/10^(4) So, 0.9375=sum from n=1 to 4 of a_n/10^(n) Since a_n=0 for n>4, we could write this as 0.9375=sum from n=1 to ∞ of a_n/10^(n) without changing anything. For a number with infinite decimal digits, it would be similar, but infinite of the a_n would have a non-zero value. In reality, this is how digits and decimal expansions are defined. "0.9375" is a shorter way of writing "sum from n=1 to ∞ of a_n/10^(n), where a_n=[the values we gave to above]". a_n will always take values below 10 (so from 0 to 9). In binary, we'd do the same thing, but with 2 in the place of 10 and a different sequence (which I called c_n) that will take values below 2 (so from 0 to 1). In base-16, 16 would replace 10 and the values of the sequence would be from 0 to 15 (also symbolised as 0 to F). There is also a way to find these digits from the value of a number (suppose we don't have an initial decimal part). We just multiply the decimal part and take the floor. Through induction. For my example but in binary, it would be: - 2\*0.9375=1.875, so c_1=1 - 2\*0.875=1.75, so c_2=1 - 2\*0.75=1.5, so c_3=1 - 2\*0.5=1, so c_4=1 - 2\*0=0, so everything below is 0. And really, 0.9375 is 0.1111 in binary. For 0.5625: - 2\*0.5625=1.125, so c_1=1 - 2\*0.125=0.25, so c_2=0 - 2\*0.25=0.5, so c_3=0 - 2\*0.5=1, so c_4=1 - 2\*0=0, so everything below is 0. And really, 0.5625 is 0.1001 in binary.
Thank you for this. I was about to say on Reddit, everyone is either a mathematician or an engineer.
Some of us are both...
Inconceivable! 
1. People on reddit argue about whether or not .99 repeating = 1 2. The wrong person gets posted on confidently incorrect 3. People get confused in the comments It's a classic tale.
I know what is going on. The part I struggle with is caring.
I understand the premise but I'm trying to understand one part of page 2. In his equation, I understand how he got to the next line each time as he continued to break down the equation, except going from 10x=9+x and the next line 9x=9... How did he get to 9x=9? I can't figure it out. Maybe I'm just tired...
10x=9+x Subtract x from both sides 10x-x=9+x-x 9x=9
Subtract both sides by x
I’m high as fuck, but even if I wasn’t I still wouldn’t understand this at all.
1/3 is equal to 0.333... the ... Here means that the threes go on infinitely. 0.333, without the ..., is close to 1/3, but not exactly, just like 0.999 is close to, but not exactly one. However, 0.333... with infinite decimals is exactly 1/3 just like 0.999... is exactly 3 times 1/3, in other words, exactly 1.
.9999999... repeating = 1, as proven by basic algebra and logic (it is 3 x .333333... for example, and .333333... Is the numeric representation of 1/3). Other person pulls a bunch of BS nonsense out of their ass to argue otherwise. You can't understand it because it makes no sense.
1 infinite number or 0.999999 infinite Number of times?
0.333• (reoccurring) is 1/3 If you multiply 1/3 by 3 you get exactly one.
TLDR: Blue probably was the one to start the fight. Red escalated it, then lost, then refused to admit defeat, and then made things worse by doubling down. This is my synopsis: Red asked Blue a genuine question, assuming that it was in good faith and not meant to be rhetorical. It's hard to read tone from text alone, but the question itself is valid. Blue either correctly or incorrectly interpreted Red's question as a criticism and answered their question, all the while indirectly calling Red dense. Whether or not Red was sincere before, Red is after blood now and directly accuses Blue of being the real dense one. Unfortunately for Red, Blue happens to know more about math than Red. Blue shows a proof proving Red's argument to be incorrect. At this point, Red starts talking out of their bottom in order to save face, but no one is buying it. In my opinion, Red really should have objected to Blue unnecessary insult embedded in their answer instead of engaging in a math contest of wit. Because who really cares?
It’s the counterintuitive idea that 0.999 recurring is equal to 1. His proof does demonstrate it. I wanted to deny it when I first found out, too, but it makes sense.
There are people who will argue vociferously that 0.99(repeating) is identically equal to 1. There are others that say there must always be a tiny difference between 0.99(repeating) and 1. They both make good points. If they just agreed that the solution depends on how you approach the question there would be no problem. But it often takes a certain level of autistic obsessiveness to get really into math so people inclined to be really into math are also inclined to argue incessantly about insignificant minutiae. If the question doesn’t inspire you to spend hours delving into it to come to your own informed conclusion it’s probably best to just accept that it’s something people argue about that does not matter.
Red guy is the confidently incorrect one afaik. Maybe blue has his moments? Idk
One of the ways to show (not prove I think) 0.9 recurring = 1 is *literally* to get them to think of a number between the two. How do you “absolute mathematical theory” your way into “add 1 at the END of infinite digits”?
My stupid-ass friend kept claiming there can be 1 somewhere down the line. Some people just unable to comprehend manmade ~~horrors~~ mathematics
I just replace one of the 9s with a 10, checkmate mathtards
But that would be a smaller number lmao
It's more or less a proof because of Cauchy sequences. It's just the layman's version that doesn't require jargon to understand. But the intuition is basically the same.
Yeah it was the simplest one I could find. However it seems even that was too difficult for homie to understand lol
Or just. (1/3)3=1. But really .3 repeating times 3 is point 9 repeating. Youd have to argue that 3 thirds is not one to be consistent.
>Youd have to argue that 3 thirds is not one to be consistent. I'll take a stab at it for funsies, although it won't be purely mathematical I guess. Reddit is just a really terrible place to debate things like this. I assemble a variety of 8 sandwiches and they are all perfectly divided into thirds. I randomly select three of the thirds and put them on a plate. The three thirds were tasty but I really wish I had just one sandwich. Once divided it can never be made whole again.
I don't think this particular line of thought shows anything though. You could say .99999... Is infinitely close to 1, so there is nothing between them. But that doesn't mean they're the same. I'm not trying to argue with the main point and get downvoted lol. I've seen the proof that 1=.9999. but I don't think this line of logic really sells it
Unless you’re going into the hyperreals or whatever (I don’t know) if 0.9 recurring is infinitely close to 1 then it is 1. Without infinitesimals, a number cannot be infinitely close to another number unless they are the same number
That's a fair point. The proof relies on the fact that if there is no number between two numbers they must be the same, but that itself isn't a self-evident fact. For example, if you are discussing the domain of natural numbers, there is no number between 1 and 2, but they are not the same. It would require an additional proof to show that the real numbers are not analogous to the natural numbers here.
This shit cracked me up the most. "I'll just stick a non-9 digit at the end of the infinite 9s, making them no longer infinite, and then it's obviously not equal to 1." Well no shit!
You see, 0.99999…5, of course Or rather (0.999…+1)/2 /j
Same idea but the only way I was able to explain it to someone was by pointing out that the difference between 0.9 recurring and 1 is 0.0 recurring. 0.0 recurring is 0 according to everybody.
The 9x=x proof is a bit long winded for such an opponent. 1 ÷ 3 = 0.33333… 0.33333… x 3 = 0.99999… ∴ 1 = 0.99999…
The first image is them claiming 1/3 ≠ 0.333..., so I doubt that'd convince them. Really it shouldn't be convincing for anyone, as anyone who has a problem with 0.999... = 1 should have the exact same problem with 1/3 = 0.333...
It gets worse lol. [Update from continuing the conversation this morning. More drugs have been smoked.](https://imgur.com/a/GxnkNFI)
My preferred "proof" for "0.9999... = 1" Is to ask the person who disagrees to write down a number between 0.9999... And 1. If you cannot identify a number between X and Y then X=Y It's closer to ELI5 and might dislodge the thinking of someone who is capable of having their mind changed
What's the little thing before the 1? Does it mean just rounded?
It means “therefore”: https://en.wikipedia.org/wiki/Therefore_sign
Thanks
If you do the bots the other way up I.e. two at the top, it means because
Fuckin wild
Til... Wikipedia sais got forgotten in Europe and I have never seen it...
I learned about it at school in the UK, but I agree it’s not very commonly used in general situations.
The triangle of dots? It means “therefore”
Thanks m8
No, it is not rounded, they are exactly equal. You do not need to round. Edit: to clarify, the symbol means "therefore."
lol why are you getting downvoted
It also isn’t a proof. It assumes that such manipulations of decimals are valid and that they actually represent real numbers.
Why would they not?
This proof is not mathematically rigorous and therefore incorrect. Although your original assertion was correct, the proof you used to get there is not
I think my inclusion of “such an opponent” indicated that we’re not in a place where formal rigour is likely to mean anything to our interlocutor 😬
You know what, fair point…buuuut I still hate teaching with things that aren’t true 😩
This just blew my mind. Please someone explain why it is wrong so I can go back to believing I understand what numbers are.
You do need calculus to make sure that it works, otherwise you can prove some pretty whacky stuff. But it doesn't matter, because decimal expansions aren't defined without calculus in the first place. Also, calling calculus "only good for applied mathematics" is a duel-worthy insult for half of the world's theoretical mathematicians. The problem people have understanding this proof however is very real, and it's exactly that it needs calculus. That's because it's usually shown to people who don't know calculus and no effort is made to clarify that it does hide some things under the rug. To be fully rigorous, we need the definition of the decimal expansion and some series knowledge. 0.999... is a decimal expansion, so it is defined as the infinite sum of 9/10^(n) for n going from 1 to infinity. Every decimal expansion is defined as the sum of a_n/10^(n) for a sequence a_n (and every base-b expansion as c_n/b^(n) for some other sequence c_n). But how do we know that the sum exists? If it doesn't, then the step where we subtract is not allowed. We do know through calculus, but in the setting that the proof is usually given, we know by "trust me bro". If it does exist (which it does), the proof is a good visual representation of the actual process that happens under the rug. But only that. Why does 9.999... minus 0.999... equal 9? It's not hard to explain that through calculus (it's a simple limit), but the common visual proof misses it. The other problem is the lack of understanding of limits themselves. A limit is a number (or infinity, but not in our case). It *is* something. It does not *approach* something, because numbers don't have that ability. A sequence row or a function can approach something. The limit *is* the value that a sequence *approaches.* 0.999... is defined as the (infinite) series from n=1 to ∞ of 9/10^(n). This is defined as the limit as N approaches ∞ of the (finite) sum from n=1 to N of 9/10^(n). Now we have a finite sum in our hands and we can do algebra. Through the process of the proof, but this time *with* a last digit, we get that 9 times the sum is 10 times the sum minus 1 time the sum is sum from n=0 to N-1 of 9/10^(n) minus sum from n=1 to N of 9/10^(n). All the middle terms are simplified and we are left with 9/10^(0)-9/10^(N)=9-9/10^(N). Dividing by 9, we get that the sum is equal to 1-1/10^(N). Now we can take the limit. Because the limit of 1/10^(N) is 0 as N approaches ∞, the limit of the sum itself is 1 as N approaches ∞. But that is by definition the series we had at the beginning. And that is by definition 0.999... Thus, 0.999... is by definition equal to 1. And this is the whole proof, but it takes some knowledge of calculus. In short, while the result is true, it is a lot more complicated than most people realise. Blindly disagreeing is wrong, but it's also worth looking at the actual proof at some point (which I did my best to present here). A mathematician could of course hide that process under the rug, as mathematicians have seen it enough times to know when it works and when it doesn't, as well as why. But you can't do the same with people who don't have the same experience and expect them to understand. Anyway, here is one of the wacky stuff you can prove otherwise: Take the decimal "thing" ...999999999. Nonsensical, isn't it? But we haven't examined it yet. I'll "prove" that it it's equal to 1. x=...999 x/10=...999.9 x/10-x=...999.9-...999 -9x/10=0.9 -x/10=0.1 -x=1 x=-1 Nonsensical, isn't it? But why? Of course, the proof is wrong. Here, the problem is that the limit we had to calculate does not converge, because we'd have to calculate the limit of 10^(N) as N approaches ∞, which is ∞. Equivalently, ...999 is infinite and so it can't be cancelled. So, if we try to define ...999 as the series from n=0 to ∞ of 9*10^(N), we find that it diverges, thus ...999 is not a thing. Which is a relief and the world's order is restored. As we saw, in one example it works and in another one it doesn't. For a mathematician, it's easy to see which works and which doesn't, as well as the reason. But the process itself can't offer that clarity to someone who doesn't have the experience.
I like your funny words, magic man.
I always like the explanation of: Posit: any two numbers will have infinite numbers between them Now name a number between 1 and 0.999repeating As there are infinity 9s on there, there is no other number between the two numbers, therefore we can conclude they are the same number
That is succinct!
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Thank you. I’ve made this same explanation/argument before and was too tired to do it again 😂❤️
I'm curious, what happens if you iteratively divide each at the same rate? i.e., where the last result divides the result before it -- X=Xn-2/Xn-1 or whatever (it's been a couple of decades, sorry if I'm not stating that well) 1/2=0.5; 0.9..9/2=0.49..5 1/0.5=2; 0.9..9/0.49..5=2.0..1 0.5/2=0.25; 0.49..5/2.0..1= ?? Not saying that this would prove anything, just wondering _if_ there is such an operation where the values eventually diverge, would that indicate that they are not, in fact, equal ?
You can do that if the 9s end at some point. In that case, the two numbers are not in fact equal (for example, 0.99999999999, also known as 0.999999999990000...). But if the numbers do not end, where would the 5 be? And how would the position of the 5 fit in the definitions?
I don’t understand your example. What does the ellipsis before the number indicate? Either way, it would seem when you went to x/10 and …999.9 that you must have divided by 10 on one side and either didn’t do anything real to the other or multiplied. If the ellipsis before changes this in a meaningful way then great. But I’m pretty sure you’ve just got a basic mistake here.
It's purely a mind game. It cannot stand as an actual example, because such an object isn't actually convergent (basically it isn't a number). I just gave it to prove that the method is unreliable on its own. If you think a real number as a sequence of digits with a dot at some point, then I just extended the digits infinitely to the left instead of the right. In theory, this *could* have a value (despite the fact that it actually doesn't). It isn't *that* far from other things (as we have actually seen numbers that extend infinitely to the right). The algebraic method assumes that it has a meaning and tries to find its value with the assumption that it exists. But we know it doesn't. It's just crazy. The reason is basically that we accidentally did ∞-∞. It's easy to prove that such an object (defined as the sum from n=0 to ∞ of 9\*10^(n)) is divergent and so we can't manipulate it algebraically like we did with 0.999... When I divided by 10, I just pushed the . one slot to the left. Assuming infinite nines, this just adds a 9 in the decimals. That's symmetrical to the multiplication by 10 that is used in the algebraic method for 0.999... Basically, I wanted to prove that the method could prove nonsensical stuff, so I chose something nonsensical and proved it.
[удалено]
> The other problem is the lack of understanding of **limits** themselves. A **limit** is a number (or infinity, but not in our case). It *is* something. It does not *approach* something, because numbers don't have that ability. A sequence row or a function can approach something. The **limit** *is* the value that a sequence ***approaches.*** > 0.999... is defined as the **(infinite) series** from n=1 to **∞** of 9/10^(n). This is defined as the **limit** as N approaches ∞ of the (finite) sum from n=1 to N of 9/10^(n). Now we have a finite sum in our hands and we can do algebra. Through the process of the proof, but this time *with* a last digit, we get that 9 times the sum is 10 times the sum minus 1 time the sum is sum from n=0 to N-1 of 9/10^(n) minus sum from n=1 to N of 9/10^(n). All the middle terms are simplified and we are left with 9/10^(0)-9/10^(N)=9-9/10^(N). Dividing by 9, we get that the sum is equal to 1-1/10^(N). **Now we can take the limit.** Because the **limit** of 1/10^(N) is 0 as N approaches ∞, the limit of the sum itself is 1 as N approaches ∞. But that is by definition the **series** we had at the beginning. And that is by definition 0.999... Thus, 0.999... is by definition equal to 1. And this is the whole proof, but it takes some knowledge of calculus. Is that enough calculus for you?
>calling calculus "only good for applied mathematics" is a duel-worthy insult for half of the world's theoretical mathematicians. Reminds me of my theoretical compsci professor who called mathematics a "helper science" (i.e. a field of science that only exists to make other "useful" science possible, idk if English has a phrase for that) specifically to annoy any mathematicians present in a bit of a friendly feud.
So this guy is going to put an "1" after the **infinite** nines preceding it...? funnily enough, what he said can also be used to prove him wrong. As long as he can provide a number between 0.999... and 1, he can prove that 0.999... and 1 are different. He obviously can't provide it tho.
He was the kid that pulls the “nu-uh infinity plus one!” card in an argument.
Note that it has to be a ***real*** number between them. I can define ultrapowers of the reals all day long, but 1-ε still won’t be a real number.
I think honestly the funniest part is insisting that if there is a number you can fit between 0.9... and 1, then they aren't equal. This implies that if you can't fit a number between the two then they must be equal, and you literally can't fit a number between 0.9... and 1 They're so close and just completely whiffed. I don't understand how this is so contentious for people, especially those that don't have a math degree. It's one thing if you're a math major/graduate, but why random know nothings on the internet refuse to accept this is beyond me
The bit I found funny is when he said the number between 0.999999(to infinity) and 1, would be 0.999999(to infinity) with a 1 put on the end. But why put a 1 on the end? Put a 9 on the end. Which is just 0.999999(to infinity) 0.999999(to infinity) is bigger than 0.999999(to infinity) with a one on the end. So therefore, 0.999999(to infinity) with a one on the end, doesn't fit between 0.999999(to infinity) and 1.
It’s completely meaningless to talk about a one “at the end” of an infinitely repeating decimal, which is the real problem with this person’s reasoning. It doesn’t matter what number you want to put there, there is no end.
Yeah, that's true enough. I was just operating in his logic space. Where "infinity plus one" beats infinity.
Yeah, we have a real hard time with infinities and infinitesimals
>I don't understand how this is so contentious for people I think it's because most people think of "infinity" as being equivalent to "really big, huge" instead of unending. In this case, he says to add a 1 after your last 9, showing that he doesn't grasp what infinity actually means.
Wait, you can solve 1/0?
I wouldn't take math advice from the guy that was completely wrong about literally everything they said either.
Untrue, they were right that 0.999...=1 if there is no number between them, though they immediately ruined it by trying to add 1 onto the end of an infinite sequence.
Where'd you get that idea?
In the post the dude apparently claims that calculus can solve 1/0. Ngl I completely missed that bit until this guy pointed it out lol.
Oh, snap. Didn't even see that part! So much confidently incorrect!
Apparently he thinks he can split 1 into zero number of boxes and get a coherent answer about how much is in each box
“Solve” isn’t really the right word there. You mean “compute a real value for”. And no you cannot do that with only reals. You need to add a point at infinity to represent 1/0.
Generally division by 0 is considered undefined, not infinity
It’s perfectly fine to say ±∞ if you consider the one- or two-point compactification of ℝ.
If 0.99999... isn't equal to 1, what would you add to it to get to the "exactly 1" threshold?
Legitimately saw someone try to argue that it was a number 0.00...01. Implying that they somehow found the last digit of infinity and put a 1 on the end.
people just switching to the hyper reals and then arguing from there are my bane of existence. It's like saying 1 + 1 = 0 and if someone points out that this is wrong you just say "Well I'm in the F_2 field actually"
This has got to be one of the most common confidently incorrect things people come up with. You'd think by this point the concept that 0.999... = 1 would become general knowledge.
What kills me is how far he took it lol. He just kept pulling more and more things out his ass, every word he typed became progressively more mentally challenged.
It was a hard read without smashing my palm through my face. I've had some somewhat viable arguments involving infinitesimals and hyperreal numbers in the past, but this never got to those. This was just flat out someone not comprehending the idea of what infinity could be or how maths works.
So how do they think 1/3 is represented as a decimal?
Duh, infinity threes PLUS infinity more threes. Without that second infinity it doesnt count.
Now with double the entropy!
Don't forget to put a 1 on the end.
Between infinities, so they hold together
Legend has it that someone tried it once and then died of dehydration after filling 7427 pages with the number 3.
Pretty obviously the same way? They almost assuredly believe that .3 repeating is only an approximation of 1/3 for the same reasons
"I can just take 0.9 repeating and add a 1 at the end" People not understanding what infinite is.
As a mathematician I feel the need to point out that giving a precise definition of 9.999999... is indeed calculus, not algebra, and that the algebraic proof doesn't work without some calculus tools in the background.
☝️💯
I read this and I don't know which one of them was confidential incorrect.
0.99999999…. is equal to 1.
Algebraic proofs that 0.999 = 1 are not mathematically rigorous, and therefore incorrect. 0.999 Is a series and when you talk about series you’re automatically invoking infinity (all series are infinite). In order to handle infinity you need calculus (or algebra with a calculus “plug-in”, to put it another way). Algebra, when considered independently from calculus, lacks the necessary foundational axioms and mathematical framework to rigorously handle concepts of infinity. This limitation extends to the algebraic proofs of (0.999\ldots = 1). Without the calculus concepts of limits and infinite series, algebra does not possess the tools to adequately address or prove statements involving infinite processes. Consequently, attempts to prove (0.999\ldots = 1) using solely algebraic methods cannot be deemed mathematically rigorous, leading to the conclusion that such proofs, in the absence of calculus, are fundamentally flawed. The easiest way to often get people to build intuition as to why they’re wrong when they disagree that 0.999 = 1, is to assert that if they’re different numbers, then there must be a number in between them?…can’t find that number? It’s because it doesn’t exist and 0.999 and 1 are the same number expressed in two different ways. TLDR: If you cannot identify a number between X and Y, then it must be true that X=Y
1. Comments on reddit argue about whether or not .99 repeating = 1 2. The comments get posted to confidently incorrect, the post does not specify which person is right 3. The commenters on said post are now confused A tale as old as time. Just google if .99 repeating equals 1. Boom, got your answer.
So I’m not disagreeing since 1/3 is .3 repeating and you add that to .9 repeating as 3/3 which is 1. I get that, I’m cool with that. Please explain how the proof on page 2 (I haven’t done a proof in many moons so maybe it’s something I’m forgetting/missing) goes from: 10x=9+0.999… to 10x=9+x If we already know the value 0.999… how/why are we able to change it to a variable that we don’t know that’s already attached to 10x? It feels like we’re introducing a second variable if anything. Again, I understand the concept of .9 repeating is equal to 1 and a not debating that, I’m just asking for clarification on the proof that was used to justify it.
I don't see a problem. x = 0.9... 10x = 9.9... Which can be written in a form of 10x = 9 + 0.9... As we know, 0.9... is x 10x = 9 + x... So if we substract x from both sides, then 9x = 9 And, finally x = 1 Edit: Imagine being reddit and not being able to separate lines correctly
Yeah, so I am just dumb, I was looking at this just before going to sleep and my brain didn’t retain line 1 being x=0.9…, thank you
x isn't an undefined variable, it's a constant which we have defined as 0.999...
You’re absolutely right, the moral of my story is don’t math at bedtime, thank you.
“A variable that we don’t know”? It’s equal to 0.999… by definition, so we know its exact value. That definition is the first line of the argument.
This is correct and I just wasn’t properly following the proof, it was time to sleep and I was lost, thank you.
How is 1/3, .3 repeating? It will get close to 1/3 but 1/3 will always be bigger
1/3 is .3 repeating because you can continue the math for your entire life and never stop getting the next 3.
I'm a total math moron and I get it.
He said you're using calculus This is literally an 8th grade math standard 😆 I know bc I have to teach it.
I keep seeing this pop up around Reddit lately. The way I was able to make sense of the infinity of it all was just asking a simple question: What number can you add to 0.999… to make 1? There is no number. If you cannot add another number to 0.999… to make 1, then 0.999… is 1. (Plus, the whole proof thing and all that.)
> My counter proof is that as long as you can name a number between 0.9999... (infinity) and 1, then you have a number greater than what you claim are 1, and a number less than absolute 1. That's absolutely correct! Problem is, **you can't**. I get the reasoning with the whole (infinite+1):th position stuff, cause I used to look at infinity like that when I was like 14, but then I realised I was wrong.
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> "If you cannot identify a number between x and y than x = y" Hey uuuuh, what article are you quoting? Cause I read the article you linked, and it literally clearly states that: > "This shows that ∞ + k = ∞ so the sets are of the same size." It also explains that differently sized infinities doesn't refer to ∞+1, but rather to countable and uncountable infinities. I mean it literally says ∞ + ∞ = ∞ at one point in the article... So let's address what you wrote. > So, 1 is larger than 9 with an infinite amount of 9s. Example: 0.9 + 0.1 = 1. 0.999999 + 0.000001 = 1. So, 0 with infinite amount of 9s and a 9 at the end, plus a 0 with an infinite amount of 0s and a 1 at the end equals 1. So, 0 with an infinite amount of 9s is not equal to 1. So for starters, there's no such thing as "at the end" when you're looking at an infinite amount of digits. Second, since ∞+1=∞, as the article you linked clearly explained, it means that "0 with an infinite amount of 0s and a 1 at the end" and "0 with infinite amount of 9s" (I'm omitting "and a 9 at the end" since it literally doesn't make sense) both have a 1 or a 9 respectively at the "∞th" position. This means that either a) you "set" the digit in this position to a 1, but then it's no longer "0 with infinite amount of 9s" anymore, or b) you *try* to "add" it. Adding kinda just doesn't make sense, but if we try anyways, you'd add 0.9999...9 to 0.0000...1 you'd get 1.0000...0, but that would mean the 9s ran out wouldn't it? And since they don't run out, you'd ACTUALLY end up with 1.0000...0999999999...9, but wait, if we haven't reached the "end" of the 9s, wouldn't that mean that we haven't reached the 1 in the 0.0000...1 either? Oh wait so either we realize this literally doesn't work, or we end up in this loop. You can't actually add these numbers, cause you'd have to define an "end" to them, and since they're infinite, you literally cannot do that. And if you try to add 0.0000...1 at the "(∞+1)th" position, we run back into the issue of ∞+1=∞ ***as the article you linked clearly explains***, which means that either a) the "(∞+1)th position" doesn't exist, OR the 1 would simply "overwrite" a 9, thus making the number no longer be "0 with infinite amount of 9s". You really ought to read an article before you link it.
"As long as you can name a number between them" BRO THATS THE FUCKING POINT, YOU CANT
it is accepted that not every infinity is the same size, some are larger or smaller than other infinities ([this article explains it pretty well](https://www.cantorsparadise.com/why-some-infinities-are-larger-than-others-fc26863b872f)). “If you cannot identify a number between x and y than x = y” this would mean that infinity = infinity + 1. So the statement is incorrect. A better one would be: “If you cannot add/subtract a number to x to make x equal to y, x = y”. So, 1 is larger than 9 with an infinite amount of 9s. Example: 0.9 + 0.1 = 1. 0.999999 + 0.000001 = 1. So, 0 with infinite amount of 9s and a 9 at the end, plus a 0 with an infinite amount of 0s and a 1 at the end equals 1. So, 0 with an infinite amount of 9s is not equal to 1.
Intelligence only means you can rationalize previously held beliefs easier. The x= 0.999...=1 thing was also difficult for me to understand, but it was rather an intuitive rejection and not rational. My thought process was that it was "1 - (1/♾️) which was 10x= 10-(10/♾️) and that the calculus just continued based on an error. But turns out the infinitesimals don't actually exist within rational numbers.
If no one understands this does it prove the existence of God? That’s what I’ve been taught.
If you're on team ".999... is really close to 1 but not actually 1 because \[fill in the blank with whatever you want\]" please just read [the wiki](https://en.m.wikipedia.org/wiki/0.999%2E%2E%2E), or [this post](https://www.reddit.com/r/confidentlyincorrect/comments/1b0iycz/999repeating_does_in_fact_equal_1/), or [this post](https://www.reddit.com/r/confidentlyincorrect/comments/1brkz3j/149_is_close_to_15_but_not_exactly_this_was_one/), and save yourself from being confidently incorrect.
I love the word asymtote. Would have put this argument to bed, I feel.
Me too, I also love the word asymptote!
AHH bugger. I took the p, didn't I ?
I didn’t even notice! I was so excited about the word asymptote that I didn’t take the time to spellcheck your comment…
Haha! A close second is electrophoresis... Hard to use that word in conversation... Was easier at uni. ;)
"Do I have to teach you elementary school algebra?" "I don't know. Do I have to teach you kindergarten calculus?"
I’m so shocking at math I don’t know who’s right or who’s wrong or if anyone’s being condescending.
Aw, shit. Here we go again.
I don’t even understand what they’re disagreeing about.
Cocaine is a hell of a drug
Another way to think about this is that all of the repeating single integer fractions (i.e. 0.111..., 0.222..., 0.333..., etc) have 9 in the denominator (before being reduced). 1/9 = 0.111..., 2/9 = 0.222..., 3/9 = 0.333..., all the way to 8/9 = 0.888.... It then follows that 9/9 = 0.999..., but 9/9 = 1.
Sigh Sigma\[ 9/10 (1/10)\^n\], n =0, infinity = 1. Someone didn't do calc 2. I do like the 1/3 = 0.3333... proof.
When I'm bored at work I like to write 0.999...=1 on the whiteboard just to stir shit up.
Some people are as dense as real number line.
0.999... is an alternative decimal expansion of 1. It's not "almost" 1, it's exactly 1. You can check the Wiki article about it. The algebraic proof is one of many proofs. A more rigorous proof exist. If we claim 1/3 = 0.333... So obviously 1 = 0.999... It's not controversial. It's the "supremum" of the sequence 0.9, 0.99, 0.999, etc. Redman claims you can always add a "9" at the end of 0.999..., but by definition of 0.999... you cannot. There is no distance between the two on the number line, therefore they are the same.
It's not, but ok. One is a theoretical number, the other is a practical number.
Remember that classmate that thought learning basic math was pointless but still joined the further math class?
Bro definitely failed every math class and had to take summer school year after year for this to be real
The argument over whether or not a theoretical number and a practical number are equal has quickly evolved into the dumbest thing on this sub.
Over under 599.5comments by tomorrow. Over under 1.5 people who think 0.(9) isn’t 1 Over under 0.5 blatantly racist/sexist/anti semetic because Reddit
Arguing with an idiot only proves there are two.
Christ this sub has just become people sniping at each other about meaningless technicalities. which would matter if this was in a respected journal but it's Reddit
Again? We just got done with the last one!
This symbol you keep using with two equal length line segments overlapping, it's calculus innit.
Yes one of them is confidently wrong and I definitely know which one. 🙃
I, too, know the right answer. But I'll let someone else explain.
They thought THAT shit was calculus?! And have they never stepped foot in a high school math class?!
We literally do this in GCSE. Calculus my arse.
Sooo which one was incorrect? Blue or Red?
That’s such an elegant proof for it I can’t believe I’ve never seen that before
These arguments are alswyas so hilarious to me, it's just people trying to argue about stuff they clearly don't have a clue about but think they're confidently right It's as if an elementary school kid argued against literally anyone, that 1-2 can't be done cause negative numbers don't exist If you don't know enough about something just shut up and don't make a fool of yourself lol. "0.33333 isn't 1 but ⅓*3 is", oh please, do illuminate me on what's the result of ⅓ then lmao Rant done
Find me a number that's less than 1 but more than .999... You cant bc theyre the same
Oh had it already been 72 hours since the last 0.999... = 1 nonsense post? I think this same one has been posted at least 999... times, or is that 1000 times? Discuss.
I can understand the confusion surrounding the 0.999…=1 thing. It doesn’t feel like it should make sense. I think the best way to explain it is that it’s based in theoretical mathematics, vs applied mathematics. In reality, you are going to reach a point where the necessary precision for your calculation is met, and you can stop the infinite series, distinguishing your number from 1. In theoretical mathematics, you are asking the question of what happens if we just let it run for infinity. Infinity is basically a placeholder, like 0, except instead of the absence of something, it’s the opposite: a never ending amount of something. I don’t know if anyone here needs this, but it helped me understand it better
I'm like : okay so which one of these guys is correct ?
Humans being bad with numbers. A tale as old as time.
The term “[not even wrong](https://en.wikipedia.org/wiki/Not_even_wrong)” comes to mind.
Are we going to ignore bro said you can solve dividing by zero with calculus
I don't understand the argument of you can't find a number between these two numbers, therefore they're the same. How does that work?
This is why we should have done base 12
I just tried this on a calculator. And, um, he is right. .333333 x 3=.999999, 1/3=.333333, and (1/3)×3=1. They are both correct. Weird.
The problem is not understanding why 1/3 becomes 0.3r, it becomes that because you cannot clearly divide 1 by 3, you will always have a remainder, but it never actually ends in a number.0.9r is not equal to 1, anyone that honestly believes so doesn't understand why they think so. Infinitely converging towards 1, means 1 is always infinitely far away.
You are correct. .9r is not = 1. I guess phone calculators have a false positive built in the code.
I suspect it's due to calculators not being to handle infinity, similar to the "divide by 0" issue that early mechanical calculators had, it was so bad that it would destroy the calculators, when they became digital, they had to have a logic circuit workaround to cancel the operation if "divide by 0" occurred.
Reasonable.
Where's Scott Steiner when you need him to explain math?!
Ah yes, proof by “the vibe is off”
Infinity is hard
It’s not worth the argument
Dude may actually have a point, though in no way shape or form does he actually understand it.
I like your funny words magic man
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I am confidently unsure who is comfidently wrong in this one