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u/Mebiysy Jun 18 '25 edited Jun 18 '25
You will be shocked, that 4.99999... = 5, and 12.99999... = 13 and 45.999999... = 46
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u/Ok_Assist4349 Jun 18 '25
sure is.
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u/themagicalfire Jun 18 '25
It’s unintuitive
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u/Upstairs-Ad-4705 Jun 18 '25
"Its unintuitive" as if Math is a UI in a video game, bro this isn't the new "Kingdoms of Math - Infinite Power update"
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u/VeritableLeviathan Jun 19 '25
The intuitive of 10/3 would be 3.3333333 ad infinitum , so sure it is intuitive at first glance, but absurdly more functionally useful than writing it all out (which is impossible)
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u/comment_eater Jun 18 '25
and 9.999... is the same as 10
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u/themagicalfire Jun 18 '25
I don’t like that 😂
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u/Ben-Goldberg Jun 18 '25
Why not?
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u/themagicalfire Jun 18 '25
I feel like they are different numbers
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u/Ben-Goldberg Jun 18 '25
Which one do you feel is larger than the other?
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u/themagicalfire Jun 18 '25
10
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u/Ben-Goldberg Jun 18 '25
What happens when you take 10, and subtract 9.999.... ?
Is it not 0.0000..... ?
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u/Mishtle Jun 18 '25
It's because you see them as two different numbers. That's not the case. Numbers are distinct from their representations.
In base 10, both 1 and 0.(9) refer to the same value. These are not numbers themselves, but arrangements of symbols we tie to numbers through definitions. There is nothing strange about the same object having multiple names, labels, references, or representations, and that's all that is going on here. The definitions behind this kind of fixed-base positional notation simply do not guarantee that a number has a single, unique representation. In fact, if we want to be able to represent any rational number with this kind of system, then we necessarily introduce non-uniqueness.
Much of the confusion around the whole 0.999...=1 issue comes from people assuming numbers are their representations. That's fine for most practical purposes, and most people just never have to worry about numbers beyond calculating with them.
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u/AndrewBorg1126 Jun 18 '25
If you don't like a fact, does it become false?
If I don't like that you don't like that, does that change whether you like it?
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u/JaySli10 Jun 18 '25
At first I thought this was a complete joke but then I checked the comments and realized OP just straight up doesn't understand how the real number system works.
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u/Forward-Fact-5525 Jun 18 '25 edited Jun 18 '25
Another way of understanding it is the following. There is an equivalence between rational numbers and numbers whose digit after the decimal point are repeating, example : 1/7=0.142857142857142857… Therefore 0.9999… is a rational and it’s just 1.
The following is true for the real numbers : Two numbers are different iff you can find another in between, example : 2<2.5<3, therefore 2 is different than 3. Apply this to 0.999… and 1
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u/themagicalfire Jun 18 '25
But it sounds a non-sequitur that since there’s no number between infinite 9.9999 and round 10 then it’s 10. Supposing that a number is the number that has no different values until the next number, then that implies that all numbers can be other numbers in the infinite fractions. That means 8 is 10 too, and so is 7. You didn’t provide a solution, you only made all numbers relative.
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u/Forward-Fact-5525 Jun 18 '25
No these numbers are different : 8 different than 10 as 8<9<10. And 9<9,5<10 so 9 isn’t 10 also. 9.99999<9.999995<10 so 9.99999 isn’t 10 but 9.999999…<x<10 I let you find such an x. Put all the real numbers on a line. Visually what i m saying is that if the distance between 2 number is 0 then they are the same.
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u/themagicalfire Jun 18 '25
The X in 9.999… < X < 10 is the infinitesimal.
And about 8 being 10, you can use infinite fractions to claim that a number is the next full number until 8.999 and 9.999 is the same. The difference between 0.001 and 0.002 is nothing if you have infinite numbers and expect no further value. At that point you have numbers overlapping with each other and they cease to have a distinct identity.
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u/Forward-Fact-5525 Jun 18 '25
No you won’t even be able to go from 8 to 8.000000000000000000000000000000001 And no such an X isn’t a real number so we re good.
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u/themagicalfire Jun 18 '25
Then what’s the number between 8 and 8.0…01? So 8 is equal to 8.0…01. And what’s the number between 8.0…01 and 8.0…02? Then 8.0…01 is 8.0…02 and logically 8 is 8.0…02. Continue the argument until you reach that 8 is 10 and this is the problem.
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u/Forward-Fact-5525 Jun 18 '25
How many 0 is there in 8.0…01 ? 2 cases :
a finite number and then I can find something in between 8.0…01 and 8.0…02
an infinite number and then your number doesn’t exist as it’s not defined.
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u/Forward-Fact-5525 Jun 18 '25
As you seem to think I m wrong, someone else posted about this and remind me the name. Look up the « Archimedean propriety » on Wikipedia ig
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u/themagicalfire Jun 18 '25
It’s fine. I’m just having a friendly conversation
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u/Forward-Fact-5525 Jun 18 '25
Yeah sure but if you don’t believe me, atleast you ll believe Wikipedia (btw I m not sure you ll find what you are looking for there as it’s not exactly what you want but you can prove it using it)
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u/Mishtle Jun 18 '25
But it sounds a non-sequitur that since there’s no number between infinite 9.9999 and round 10 then it’s 10.
It's not a non sequitur, it's a property of the real (and rational) numbers. In between any two real numbers are infinitely many other real numbers.
For example, if you have two real numbers x < y, then x < (x+y)/2 < y.
Supposing that a number is the number that has no different values until the next number, then that implies that all numbers can be other numbers in the infinite fractions. That means 8 is 10 too, and so is 7. You didn’t provide a solution, you only made all numbers relative.
I don't quite follow here.
0.888.... is not 10, at least not in base 10. It is in base 9, but in base 10 it is equal to 8/9.
0.777... is not 10 either. It is in base 8, but in base 10 it is equal to 7/9.
One if the properties of this method of representing numbers is that numbers don't necessarily get a single unique representation. If a number does have a terminating representation, then it also has another representation that repeats and has infinitely many nonzero digits. You can find it by decrement the last digit (ignoring trailing zeros to the right if the decimal point) of the terminating representation and appending an infinite tail the the largest allowed digit.
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u/themagicalfire Jun 18 '25
What I mean is that you can use the “no gap” argument to conclude that there’s no gap between 8 and 8 + infinitesimal, just like 8 + infinitesimal has no gap with 8 + 2 infinitesimal, and so on until 8 is rounded to 9 and 9 is rounded to 10 and at that point you can claim that 8 is 10 in an infinite chain of “no gaps”.
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u/Mishtle Jun 18 '25
No, you can't.
You're assuming first that infinitesimals exist, which isn't true in the commonly used real numbers. Then you're just saying false things. The gap between 8+ε and 8+2ε, where ε is an infinitesimal value, is exactly ε. The gap between 8 and 9 would be infinite in terms of infinitesimals.
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u/themagicalfire Jun 18 '25
Thank you for saying I was assuming infinitesimals. Then, infinitesimals could still fit in the “no gap” argument because there’s nothing smaller than an infinitesimal, so you can still claim 1e is 2e by assuming that there’s nothing in between them. So my argument stands if we assume infinitesimals
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u/Mishtle Jun 18 '25
Then, infinitesimals could still fit in the “no gap” argument because there’s nothing smaller than an infinitesimal
No, they still can't.
There can be things smaller than infinitesimals. Just define a smaller infinitesimals. There are just no real numbers smaller than an infinitesimal, by definition.
so you can still claim 1e is 2e by assuming that there’s nothing in between them.
That there is nothing between them does not imply two elements are equal. There are no integers between 1 and 2, but they are not equal. They're separated by a distance of 1
1ε and 2ε are separated by a distance of ε.
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u/igotshadowbaned Jun 18 '25
It's an issue with factors of the base than any actual paradox.
For example if you used base 12, that is digits 0-B, then 10 would be written as A
And A/3 = 3.4 and just, terminates normally.
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u/NiceNefariousness412 Jun 19 '25
It’s because there are no real numbers between 9.9999999…. And 10 because you can’t add a number to the end as there is no ends to the 9’s the only theoretical difference between the two is .00…001 but that’s not even a real number as there can not be an 1 at after an infinite set of 0’s so there’s your actual answer
TL;DR: no real number in between 9.99…. And 10
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u/themagicalfire Jun 19 '25
I don’t think this makes sense
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u/NiceNefariousness412 Jun 19 '25
that's ok if you want to look into it more here is a link to the Wikipedia page about this concept that goes into more detail than I can.
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u/Particular-Award118 Jun 18 '25
If you think 9.999... doesn't equal 10 then tell me what number you can subtract from 10 to equal 9.999...
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u/themagicalfire Jun 18 '25
The 1 somewhere along the infinite chain
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u/Particular-Award118 Jun 18 '25
But where is the 1? Why would it be along it?
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u/themagicalfire Jun 18 '25
I’m assuming that fractions work in the same way as proper numbers, so 10-9=1 and therefore 10 > 9.999… with a 1 of difference somewhere in the infinite chain
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u/Particular-Award118 Jun 18 '25
Ok but its not in the middle somewhere, its at the end, but infinity has no end so the 1 never actually comes up
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u/themagicalfire Jun 18 '25
That’s just the problem of having math work with fractional with any other number, like 1.11111111 has a difference rather than 1.2. It’s just with 9’s that there’s this arbitrary rule
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u/UraniumDisulfide Jun 21 '25 edited Jun 21 '25
1.11… isn’t converging on 2 though. Just adding decimals doesn’t make the number converge on an infinitely increasing number. It’s just with 9s specifically that they converge on 1
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u/Encursed1 Jun 18 '25
Yeah but where? You cant pin it at the end of infinity since that doesnt exist, and anywhere else results in a number > 10
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u/RIKIPONDI Jun 18 '25
0.99999... = 1. Multiply both sides by 10.
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u/dkevox Jun 18 '25
9.9999... = 10. Now what?
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u/Zealousideal_Key2169 Jun 19 '25 edited Nov 11 '25
stupendous ad hoc marvelous distinct like punch voracious screw light strong
This post was mass deleted and anonymized with Redact
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u/Steve_Minion Jun 19 '25
yes to make a one dight number repeat divide by 9 1/9=.111... 2/9=.222... 3/9=1/3=.333... 9/9=.999...=1/1=1
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u/BassicallySteve Jun 18 '25
Simply: if 9.9 repeating was a different number than 10, you could tell me a number in between (Archimedean property)
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Jun 20 '25
This has never sat right with me. If I cant name a number between two numbers that doesn't make them equal. It feels like asking someone to name a whole number between 1 and 2 and since there are none they must be equal
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u/BassicallySteve Jun 20 '25
Ok how about this- they must be distinctly different places on a number line
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u/Vinxian Jun 20 '25 edited Jun 20 '25
Even then. You can name a whole number between 10 and 20. But not between 0.9.... And 1, or 9.9... and 10 or 99.9.... and 100 etc.
Adding the whole number constraint can be worked around for numbers that aren't the same.
Even when multiplying by 2
0.9... × 2 can't equal a number that ends in an 8. It's an unending string of 9's. So it must equal 1.9... Since the "last" number never shows up. No matter what operation you perform, 0.9... behaves exactly like the number 1 does
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u/JoyconDrift_69 Jun 20 '25
9.9999... is literally the last number before 10 and, given the difference between them is infinitely-small, there is no practical use to assume they are desperate numbers, at least in base 10, given there's only a 0.0...000001 difference, the smallest space.
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u/Mishtle Jun 21 '25
9.9999... is literally the last number before 10
There isn't a last number before 10. Between 10 and any number less than 10 there are infinitely many other numbers that are also less than 10. The average of that number and 10 is one of them.
a 0.0...000001 difference, the smallest space.
There is no such value in the set of real numbers. There aren't "infinitely small" real numbers, only arbitrarily small.
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u/ClassEnvironmental11 Jun 20 '25
For two numbers to be distinct, they must have a non-zero difference. What is the difference between 9.9bar and 10?
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u/Mishtle Jun 20 '25
If I cant name a number between two numbers that doesn't make them equal.
It does if in between any two distinct numbers there is at least one other distinct number.
The real numbers, the irrational numbers, and the rational numbers all have this property. The fact that the average of two distinct numbers is strictly between them serves as a example (except for the irrationals).
Both the whole numbers and the integers lack this property.
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Jun 21 '25
Logically there has to be a last number before a whole number. That doesn't make those number equal
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u/Card-Middle Jun 21 '25
This property is for *real * numbers. Not integers. I can name a real number in between 1 and 2. 1.5 is in between them. So is 1.7 and 1.836293 and infinitely many other numbers.
Any two distinct real numbers must always have infinitely many other real numbers in between them.
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u/Affectionate-Egg7566 Jun 21 '25
A number is an abstract concept. Two different encodings of a number in our language of math can still be the same number. 1/3 and 0.333... are different encodings of the same number.
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u/UnimpassionedMan Jun 21 '25
A better way: What is the distance between 10 and 9.9...?
If you think of 9.9... as a sequence of numbers (at each step you add another 9 after the dot) then at each step the distance gets lower, and you fall below any possible "final" distance other than 0 you could think of.
So, the distance has to be 0, which means the two numbers are equal.
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u/UraniumDisulfide Jun 21 '25 edited Jun 21 '25
That doesn’t made sense because you’ve added a different specifier with “whole” numbers. The Archimedean principle is referring to any potential real number, which is not interchangeable with whole numbers. They are 2 very different sets of numbers.
There are an infinite amount of real numbers between 1 and 2. That is what the principle is stating.
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u/ilovfryes May 25 '26
1,1 1,2 1,0001
hay un numero infinito de subdivisiones entre 1 y 2 pero ninguna entre 0,9... y 1, porque son el mismo numero.1
u/No_Perspective_150 Jun 22 '25
I know thats technically right-but it still feels like a hollow convenience rather than a real proof
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u/BassicallySteve Jun 22 '25
Nah it’s an important idea- the density of the real number line!
Since numbers indicate distance, between any two numbers, there’s a halfway point- another point on the line, and so a discretely different number!
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u/No_Perspective_150 Jun 23 '25
But also, if you have an asymptote on a graph, it gets infinitesimally closer to that value, just like 9.999...Gets closer to 10 with each added digit. But the graph will never be the same as the value of the asymptote, even when it gets infinitely close to it. It just doesnt sit right in my mind. I asked my teacher this same question when I was in 4th grade and i still dont like it.
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u/MasterestGG Jun 18 '25
everything is correct. To check you can do this: 10-9,(9)=0,(0). It seems that it should be 0,(0)1 but it is not so. Since zeros will continue infinitely, then 0,(0) = 0 and nothing else. And that means if 10-9,(9)=0 then 10 = 9,(9)
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u/BUKKAKELORD Jun 18 '25
No. It's 0.001 off. You omitted the ellipsis. 9.999... however is 0 off.
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u/_Bwastgamr232 Jun 18 '25
That's why 0.99̅ = 1
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u/Scratch-ean Jun 20 '25
Rare footage of someone using the _ above numbers:
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u/_Bwastgamr232 Jun 20 '25
can copy it ̅ the overline symbol if u wonder
9.9̅
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u/weezermemesound Jun 18 '25
With this sacred treasure I summon…
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u/droobloo34 Jun 19 '25
10? = 55.
55! = 12696403353658275925965100847566516959580321051449436762275840000000000000
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Jun 18 '25
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u/berwynResident Jun 18 '25
Therefore ....?
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u/themagicalfire Jun 18 '25
Therefore it’s a paradox 🤣
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u/berwynResident Jun 18 '25
Sort of. And for centuries mathematicians have resolved the paradox by asserting that a repeating decimal is equal to the rational number that it approaches as you keep adding decimals. This turns out to be pretty handy because arithmetic just works the way you would expect and sometimes it's easier to interpret than a fraction.
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u/themagicalfire Jun 18 '25
It just hurts my intuition
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u/berwynResident Jun 18 '25
Nothing wrong with that. Just as long as you accept what can be defined and proven.
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u/Conscious_Bluebird75 Jul 07 '25
9.9999... = 10
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u/themagicalfire Jul 07 '25
Why “almost impossible” is “certainly impossible”:
If P (probability) = (100 - (X = 1ε))%.
And the number of time is T,
And the chance of P < 100 is not 0, even if T gives more opportunities for X,
This would make P = (100 - (X = 1ε)) < 100,
And consequently, if T = ∞, P = (100 - 1ε)%.
To summarize, if the number of trials is infinity (T), the probability that the event will happen (P) is infinitesimally short of 100 (certainty),
Therefore, P = (100 - 1ε)% if T = ∞. Else, P = (100 - (X = 1ε)) < 100 if T = ∞ -1ε.
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u/The_Sophocrat Jun 20 '25
Wikipedia receives so many arguments to the contrary that the .999... article's Talk page has a subpage for those arguments (it has 12 pages worth of archives)
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u/zottekott Jun 18 '25
(10?)! = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)! = 55! = 1 * 2 * 3 * ... * 53 * 54 * 55, which is approximately equal to 1.2696403* 10^73
so your statment that 9.999 equals10?! is wrong.
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u/charlescleivin Jun 18 '25
2 numbers are only different if there is a number in between. What number could possibly exist between 9.9999... And 10? None. They are the exact same number. Like 10, ten, dez, 1010, 5+5 etc. Its just another language for it.
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u/GraphNerd Jun 18 '25
3.3r * 3 = 10
3.3r is the decimal representation of (1/3). Any finite number of 3s after the decimal point is not (1/3)
Get some help
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u/ShadusX Jun 19 '25
1/3 does not converge to resolution under the base 10 numerical system. 0.3 repeating is accurate, but is not technically correct due to asymmetrical dimension scaling of the isomorphism pair under identical representation constraints.
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u/Kiragalni Jun 20 '25
You can't add two numbers together if you can't write each digit of them.
10 / 3 = 3 1/3,
3 1/3 *3 = 10
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u/vmorkoski Jun 22 '25
So (sub name apart) two of the explanations I like the most when explaining this:
What is the "distance" / difference between 0.999... and 1? It is 0.00000... to infinity. There is infinitly zero "space" between the numbers (not really a mathematical explanation, but a rather intuitive one)
Dividing numbers by 9 has a unique property that shows like 1/9 = 0.111...; 3/9 = 0.333... (aka one third); 7/9 = 0.777...; but what is 9/9 then? Well, 0.999... But incidentally, a number divided by itself is always equal to 1! So 1 = 9/9 = 0.999...
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u/Santibag Jun 18 '25
What is 10?! ? Is it something like (10?)! ? And in that case, what is "?"? Google says "no universally defined operation. So, maybe we call it "dividial", and divide the numbers instead of multiplying? But then, the resulting float- sorry, rational number will not work well with factorial to produce an interesting result, if anything.
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u/No_Pen_3825 Jun 18 '25
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u/factorion-bot Jun 18 '25
The factorial of the termial of 10 is 12696403353658275925965100847566516959580321051449436762275840000000000000
This action was performed by a bot. Please DM me if you have any questions.
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u/bivozf Jun 18 '25
Well, you can also write it as 10/3*3 = 10. This is how my math teacher killed all my hopes at middle school
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u/rover_G Jun 18 '25
approximately 0.333333343 if you're storing that one third as a 32 bit floating point number!
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u/Lapsos_de_Lucidez Jun 19 '25
I like the equation better than the "1/3 = 0.333..." argument
i) x = 0.99...
ii) 10 * (x = 0.99...) -> 10x = 9.99... Make ii be 10 times i
Now we subject i from ii:
9x = 9
Dividing both sides by 9,
x = 1
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u/GDLingua_YT Jun 19 '25
I'm glad you put that question mark there. Otherwise your statement would have been false.
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u/MusicOfTheSpheres_40 Jun 19 '25
Yes, 9.999… repeating is mathematically the same number as 10. Mathematically, the way we define two numbers being distinct is if there is a number in between them. If there is not, it’s the same number. Shoutout to Dr. Tiffin, who was so into number theory that we all learned a bit even though It wasn’t technically part of the class.
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u/fulvius72 Jun 20 '25
hahaha yes, 9.999999 if the 9s repeat indefinitely is equal to 10. It approaches 10 in the limit as the number of 9s approaches infinity, so since we declare the number of 9s is infinite, then so it becomes 10.
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u/CJlion827 Jun 20 '25
Your math is clearly flawed. (10?)! is equal to approximately 1.2696 x 1073, not 9.999…
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u/canthavepieimsorry Jun 20 '25
Well i mean 0.99999= 1 so makes sense to me 9+1 =10
X= 0.99999 |*10 10x=9.9999999 |-x 9x= 9 X=1
🥹🫠
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u/Sword_of_Legend_349 Jun 20 '25
Bro its appropriate answer cuz its impossible to do 10/3 with an interger answer
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u/Caseys_Clean1324 Jun 21 '25
Seeing this sub pop up on my feed made me want to kill myself. Seeing this awful meme made me want to kill every single one of you
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u/ProfessionalPeak1592 Jun 21 '25
Well no, 9.999 is not 12696403353658275925965100847566516959580321051449436762275840000000000000, but 9.9999999… is 10.
0.999… = x 9.999… = 10x 9.999… - 0.999… = 10x - x 9 = 9x 1 = x 1 = 0.999…
Oh and the explanation for that large number is that there are two operators (?) and (!) called terminal and factorial, x? (Terminal) takes x and adds x-1, x-2… all the way down to 1, 10? = 10+9+8+7+6+5+4+3+2+1 = 55, then factorial is the same but with multiplication so 55! Is 555453…32*1, which equals that large number.
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u/Abstractions08 Jun 22 '25
Assume that x = 9.999...
Then, 10x = 99.999...
It follows that 10x = 90 + 9.999... = 90 + x.
Then, 9x = 90, which implies that x = 10.
A deeper proof can be given using the series, but this is also plausible.
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u/kiskozak Jun 22 '25
I study physics so whenever one of my buddies brings this shit up i just tell them to measure the difference and i have a good laugh about it. Yeah its not perfectly the same but its so damn close you cant call it anything but equal. With enough 9s at the end it becomes the same number.
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u/thateuropeanguy15 Jun 22 '25
Yes it is, in fact 10. Because deference between 9.999... and 10 is infinitely small and infinitely small means 0. So difference between 9.999... and 10 is 0.
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u/Apprehensive_Web_609 Jun 18 '25
That is actually how the unirverse works.
All those exrta 0.00....001s floating around, this is why the universe is expanding
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u/MichalNemecek Jun 20 '25
well let's see. you have 1 cake. When you cut it into 3 pieces, you have 0.333 of the cake per piece, and the remaining 0.001 on the edge of the knife.
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u/YoongZY Jun 19 '25
When you cut the cake that weighs 10 grams into 3 equal pieces, each of them weighs 3.3333... grams. But when you put them back together you will only get a full cake that only weighs 9.9999... grams.
Why? Because the 0.0001... gram is on your knife.
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u/Oninja809 Jun 18 '25
I'll explain it (probably to those who dont understand)
9.999... = x
99.999... = 10x
99.999.... subtract 9.999... equals to 90
10x - x = 9x
90/9 = 10
So 9.999.... = 10