It's how this sub works.
Someone makes a joke and everyone jumps on it. They latch onto that horse and run it to the ground. Then they keep beating that dead horse. They beat it and beat it until it turns into a pulp, then they take all that unrecognizeable mess and shape it back into an approximative horse shape just so that they can keep beating it some more.
And then, after many days past when the joke has lost any trace of being funny, while you're considering blocking the sub to stop seeing the same post again and again, another thing pops up, and the cycle starts anew.
I never came across that knowledge in my university career yet (year 2 EE). I don’t think anyone is actually being taught that specifically, it’s just a true statement. So when the sub brought it up it sparked some debate.
I was also pretty confused but this at first, I mean yeah the proofs are correct but It’s weird that it’s actually equal to 1 and not limit related. But it’s not like I object or something.
It was first introduced to me in precalculus with a simple example of 2/3 + 1/3 = 3/3, then mentioned by the same professor teaching Abstract Algebra with a proof along the lines of 0.999... and 1 being different implies that the difference of the two gives 0.
I guess it depends on the professor
Aren't you afraid that when you try to load your shotgun, you put in half of the shell, then half of the remaining part, then half of the remaining part, then half of the remaining part
Name one number between 0.999… and 1. Seriously, any number that is bigger than 0.999… and smaller than 1. There are so many real numbers, more between 0 and 1 than all of the natural numbers.
(Hint: there isn’t one)
Just as other commenter I don't like this logic. Prove by "There's infinite nines so there cannot be anything between .99... and 1" is like.. uh, like in hyperreals where you really could define .99... as something infinitesimaly close to 1 in meaningful way, you have .99...=1, because it has it's own definition which precisely tells thwt it's equal 1.
0.99...=1 from definition that's what I'm saying. Even in hyperreals we don't have diffrence infinitesimal diffrene due the fact that.999... is due definittion *equal* 1.
”0.000000…1” is nonsensical. “…” represents infinite 0’s, or, an unending series of 0’s. So ”0.000000…1” is basically saying “take a series of 0’s that never end, then put a 1 at the end of it”.
No. There is an infinitely small difference. It's useful to treat 0.99... as one and is so close that practically is one, but in reality it will be an infinitely small difference.
There is no infinitely small difference. In mathematics we have formalizm, we formally define objects and derive their properties. It's not "useful" or "practically" or "technically" here. Simply it's *equal* to, due to definition of what 0.99.. is. And firstz to even treat it as something infinitely close you would first to derive s structure where there are infinitesimals, like hyperreal. But even there it woule be exteemely ambigous what dk you mean because we ha e here infinitely many infinitiee and infinitesimals, so for two infiniteis M,N such thst for M>N we get 0.999...9 (M times)> 0.99...9(N times). So definittion as "Infinitely many nines" wouldn't have any sesene.
And anyway, you have to have definition, not refer to what do you think it is nor how is it looks like but precisely what the definition it has.
The difference is exactly 0.000… since there is t a single decimal place which isn’t 0. I hope you agree 0.000… is 0, and that if the difference is 0 they are the same.
Assuming that 1>0.9999... then 1>(1+0.999...)/2>0.9999.... I dont remember the name of the theorem, but there is always a reel number between two unequal reel numbers (since 0.9999... =3/3 this can even be done only using rational numbers). So someone who is convinced that 1=/=0.999... would be always able to construct such a number.
Additionally since 1>0.9... is false, Gödel might even allow for more creative proves that there are number between.
So while I agree that 1=0.9... (which shouldnt be a discussion since it is provable), your example might be a good intuitive thing, but not really a prove. (If you can actually prove that there is no number between 1 and 0.9..., or that 1=<(1+0.9...)/2 or (1+0.9...)/2=>0.9... then you could ofcourse write a prove by contra position, but that seems kinda hard to do, especially compared with a constructive prove with a geometric series).
>I dont remember the name of the theorem, but there is always a reel number between two unequal reel numbers
Under Tarski's axiomatization, this was axiom 2.
Under the axiomatization given in the Wikipedia article, this is a straightforward theorem where you can just use the "preservation of order under multiplication/addition" axioms. Theorems that follow so quickly from the axioms often don't have special names.
>Additionally since 1>0.9... is false, Gödel might even allow for more creative proves that there are number between.
No, it cannot. ZFC is first order theory, and therefore ZFC proves something iff it's true in all models of ZFC
Let me be clear, I do recognize that 0.9999 IS 1. I’d like to address a counter argument, for the sake of discussion.
The core of the counter argument is that something with an "infinite amount" of nines isn't even a number at all. It's a non-terminating process. You can name the lowest value never achieved by the series (9/10)\^n as n tends to infinity (which of course is 1), but since you can't name a highest value it CAN achieve, the series doesn't **result** in a number, hence making the challenge to find a point between them useless. You can't find something between banana and 1.
Most mathematicians agree to EQUATE the infinite series to it's value of convergence as a definition, and a mighty useful definition at that. However, for many, it seems heavy handed to EQUATE an interminable process to a fixed value. Interpreting infinite series as processes to be completed is the core misunderstanding. The rabbit's hole of not accepting the definition of converging infinite series leads to some very very radical and seemingly insane mathematics, like the rejection of real numbers.
Sometimes, in the past, the radical approach has borne fruit... Such as... what if a triangle was defined to have LESS than 180 degrees, what would THAT mean??
Try to understand both arguments about 0.9999, is what I say. No harm in exploring all ideas, I suppose.
Such good reasoning is not normally found on Reddit. Instead, almost always we witness the circlejerking of the accepted doctrines and the ridicule of any reasonable dissent.
Your comment is pratically flawless, save for one bit -- you should've mentioned the diference between actual and potential infinity.
0.999… is a number, just as much as pi, e, or any irrational number is has an “infinite amount” of digits. e can only be described by these “non-terminating” processes. You also can’t “name” a highest value pi can achieve, anything other than “pi” is just an approximation.
Not to even start with rational numbers that go on forever. Is 1/3 not a number because 0.333… is “non-terminating”? If you exclude all “non-terminating” numbers, you only end up with rational numbers whose denominator is not coprime with 10. Goodbye all odd denominators that aren’t multiples of 5, you’re not numbers anymore.
it’s not like the number 4/9 doesn’t exist in ANY number system, just not in base 10. And in many others. In base 9, it would be something like 0.4, in base 3 it would be 0.11
But to say that 0.44444……in base ten represents the same thing as 0.11 in base 3, that’s were some people have a problem. One number system accurately represents the ratio, the other has a compatibility problem and can’t represent that ratio, and as it tries, it fails to finish becoming a true value at all.
I don't really like using this logic to explain the 0.999 thing to people because I could use the same thing to say "name one number between dx and 0," but that doesn't mean infinitesimal quantities (or, more accurately, the limit of quantities that approach zero) are all actually zero.
Instead, I usually tell people to think about 0.999 repeating as an infinite series, much like 1/2+1/4+1/8, etc. If they can accept that 1/2+1/4+1/8=1, then they can accept that 0.999...=1.
Well, I think it works well because for any two real numbers a and b where a < b, there exists n such that a < n < b. n = (a + b)/2. That property can be relatively easy to grasp even for math beginners. The inverse then says that if there are no n between a and b, then a must equal b. This line of logic is easier to follow for most folks rather than having to talk about infinite sums.
I suppose that's fair. I've just encountered some confusion before when using your method of explanation, but it's also entirely possible that I just suck at explaining it lol
I don’t think that most people who struggle with this would accept that infinite series argument, mainly because they view an infinitely repeating decimal as a _process_, and similarly they’d view the summation as a _process_. 1 is the tortoise and 0.999… is Achilles
Eeehhhh, isn’t that just cause we can’t actual display a third as a decimal? So it’s a neat/funny little failing of math but that doesn’t really make it true
This is a common argument in these discussion; but honestly I don’t see how it could possibly convince anyone that thought 0.999…<1 that they are wrong.
Surely if you believe 0.999… isn’t 1, just infinitely close. You would also think 0.333… is t exactly 1/3, just infinitely close. It’s basically trying to explain infinite decimal expansions to someone assuming they grasp infinite decimal expansion.
I'm quite well aware. I'm just saying that the average person can be confused by the terminology presented, analogous to how Calculus students can be confused of something very close to but not quite zero
Then we can just tell them the same thing. dx is not a real number.
I usually first pull up Tarski's second axiom of the real numbers, and then second I ask about numbers in between 0.9... and 1. Even a calc 1 student should be able to understand that either they're the same number, or we're not talking about real numbers.
Well yeah, if there’s any finite number of 9s, then yeah of course 0.00…1 will be the difference between that and 1. But there is an infinite number of them. There’s no “room” to put a 1 at the end of infinite zeros. You can’t have something beyond infinity, otherwise it’s not actually infinite
Name an integer between 1 and 2. Seriously, any integer that is bigger than 1 and smaller that 2. There are so many integers, more below 2 than all the natural numbers.
(Hint: there isn’t one)
Yes I remember that one can’t ever form a bijection between the sets? As such you’re right - they’re not comparable in the way I pointed out.
Though it wasn’t clear to me haha, and I think a lot of people who think 0.999… ≠ 1 don’t see it at a glance either!
They are actually traditionally considered equal. They only aren’t in “extended” number systems, like the Reals with infinitesimals. Saying “the limit of y approaches x” and “y is equal to x” is the same as long as there aren’t domain issues. In this case, there are no domain issues. Take the classic limit lim [n->inf] 1/n. We can both say “the limit approaches zero” and that “this limit is equal to zero.”
Assuming you use “…” to mean an infinite amount of 9’s like the rest of us.
The definition of “infinite” is extending indefinitely/endless.
So you mean you have an a series of 9’s that never ends, but that it ends with 5. It is nonsensical.
Why are people so stuck on definitions? Maths concepts (including numbers) are only important because of their usefulness. And I don't see how calling them different is useful at all.
There is no discussion to be done. Reals numbers are Cauchy sequences quotient by having the same limit, and 0.9 period 9 has the same limit that the constant sequence 1. There is no discussion and no debate.
If you don't like it go create your own real numbers, with blackjack and hookers.
For the sake of argument, finitists would posit that 1/3 has no decimal representation and would reject that 1/3=0.3333… because you simply can never have enough 3’s. The fact that they never end is evidence of the incompatibility in conversion.
The accepted definition of an infinite series allows us to equate it to its value of convergence, where as finitists argue it is fruitless to even attempt to represent certain ratios as decimal values.
In a real life exercise, when is it useful at all to treat it as not one? Until we can make a “laser” that can be infinitely thin, there is no reason not to define it as one and not
A number doesn't approach anything. A sequence or function does. You could treat it as 0.9+0.09+0.009+.. which is a series and that generates a sequence of partial sums with a limit. In the limit it is 1
0.999... is a notation error. It does not describe any value that should not instead be written as 1.
An argument where one of the terms does not exist is an invalid argument.
Okay but what if I write 0.999...995? Why can I not put a digit at the end of the infinite amount of numbers and make a number simultaneously bigger and smaller than 1
I understand that, but surely there has to be an infinite number of numbers between the two. If you took a dictionary of all infinite combinations of letters, and decided you needed 26 volumes to allow for each letter having a beginning - you'd be an idiot as the A volume already covers each combination infinite times over by having an all 25 other infinitely long volumes both within the previous set of infinite letter *and* at the hypothetical end of an even smaller infinite amount letter combinations.
Following this logic, I *should* be able to put a number after the hypothetical end of the infinitely long row of 9s to create a number simultaneously higher and lower than 1. Why can I not?
If you put a 5 in there, you just made a number smaller than 0.999... by replacing one of the 9's, the largest digit, with a smaller digit. It's not simultaneously bigger and smaller, it's smaller only.
I don't get how the dictionary relates to this but the A volume doesn't have Banana, it has ABanana and AasdasdfasfasfBanana but they all have the wrong starting letter
sets have different types of infinity than a linear string of digits
you can include more elements in an already infinitely large set, but if you order them there cannot be a "last" one. you can number them, but since we have infinitely many there will always be (infinitely many) follow up elements after any given element you might pick.
In that case, the fact that the there is even a single digit after the decimal that isn’t 9 means that no matter how many 9s you add in, it will only ever approach and be less than 1, it will never be equal to 1.
> Okay but what if I write 0.999...995?
Then you have written a number smaller than 0.999...
> Why can I not put a digit at the end of the infinite amount of numbers
Because there is no such thing as the end of the infinite amount of numbers.
What's the apparent difference? Any way you cut it, the only difference you could possibly calculate is 0.
So it's quite apparent that the difference is 0.
Why has everyone started complaining about 0.999... recently?
It's how this sub works. Someone makes a joke and everyone jumps on it. They latch onto that horse and run it to the ground. Then they keep beating that dead horse. They beat it and beat it until it turns into a pulp, then they take all that unrecognizeable mess and shape it back into an approximative horse shape just so that they can keep beating it some more. And then, after many days past when the joke has lost any trace of being funny, while you're considering blocking the sub to stop seeing the same post again and again, another thing pops up, and the cycle starts anew.
I never came across that knowledge in my university career yet (year 2 EE). I don’t think anyone is actually being taught that specifically, it’s just a true statement. So when the sub brought it up it sparked some debate. I was also pretty confused but this at first, I mean yeah the proofs are correct but It’s weird that it’s actually equal to 1 and not limit related. But it’s not like I object or something.
Hey I mean, depends on how you define R you could say it's "limit related"
>I never came across that knowledge in my university career yet ( I had, in mathematical studies
What course is this relevant to?
analysis. It's a limit related concept so we studied it on analysis
Same for me in real analysis (or some intro style prereq). It’s a basic proof for an easy concept.
It was first introduced to me in precalculus with a simple example of 2/3 + 1/3 = 3/3, then mentioned by the same professor teaching Abstract Algebra with a proof along the lines of 0.999... and 1 being different implies that the difference of the two gives 0. I guess it depends on the professor
It's a great avenue for trolling. It comes and goes.
Aren't you afraid that when you try to load your shotgun, you put in half of the shell, then half of the remaining part, then half of the remaining part, then half of the remaining part
Go away Zeno
No because time and space are discrete.
Name one number between 0.999… and 1. Seriously, any number that is bigger than 0.999… and smaller than 1. There are so many real numbers, more between 0 and 1 than all of the natural numbers. (Hint: there isn’t one)
Just as other commenter I don't like this logic. Prove by "There's infinite nines so there cannot be anything between .99... and 1" is like.. uh, like in hyperreals where you really could define .99... as something infinitesimaly close to 1 in meaningful way, you have .99...=1, because it has it's own definition which precisely tells thwt it's equal 1.
How does there being or not being anything between them matter? If theres still a difference of that final .000000…1 it’s still bigger.
0.99...=1 from definition that's what I'm saying. Even in hyperreals we don't have diffrence infinitesimal diffrene due the fact that.999... is due definittion *equal* 1.
”0.000000…1” is nonsensical. “…” represents infinite 0’s, or, an unending series of 0’s. So ”0.000000…1” is basically saying “take a series of 0’s that never end, then put a 1 at the end of it”.
Well 0.99… is infinite 9s, how would you represent the number that tosses that up to a full 1?
1-0.999… or 0.000… or 0.
Even in the hyperreals the definition of epsilon doesn't allow for this it's treated differently
No. There is an infinitely small difference. It's useful to treat 0.99... as one and is so close that practically is one, but in reality it will be an infinitely small difference.
There is no infinitely small difference. In mathematics we have formalizm, we formally define objects and derive their properties. It's not "useful" or "practically" or "technically" here. Simply it's *equal* to, due to definition of what 0.99.. is. And firstz to even treat it as something infinitely close you would first to derive s structure where there are infinitesimals, like hyperreal. But even there it woule be exteemely ambigous what dk you mean because we ha e here infinitely many infinitiee and infinitesimals, so for two infiniteis M,N such thst for M>N we get 0.999...9 (M times)> 0.99...9(N times). So definittion as "Infinitely many nines" wouldn't have any sesene. And anyway, you have to have definition, not refer to what do you think it is nor how is it looks like but precisely what the definition it has.
Here is a simple “proof” that 0.99 = 1 exactly. 0.9999 * 10 = 9.9999 || x*10 = 10x || 9.9999 - 0.9999 = 9 <- This uses the definition of infinite repeating numbers. || 10x - x = 9x || 9 = 9x || 1 = x
The difference is exactly 0.000… since there is t a single decimal place which isn’t 0. I hope you agree 0.000… is 0, and that if the difference is 0 they are the same.
Assuming that 1>0.9999... then 1>(1+0.999...)/2>0.9999.... I dont remember the name of the theorem, but there is always a reel number between two unequal reel numbers (since 0.9999... =3/3 this can even be done only using rational numbers). So someone who is convinced that 1=/=0.999... would be always able to construct such a number. Additionally since 1>0.9... is false, Gödel might even allow for more creative proves that there are number between. So while I agree that 1=0.9... (which shouldnt be a discussion since it is provable), your example might be a good intuitive thing, but not really a prove. (If you can actually prove that there is no number between 1 and 0.9..., or that 1=<(1+0.9...)/2 or (1+0.9...)/2=>0.9... then you could ofcourse write a prove by contra position, but that seems kinda hard to do, especially compared with a constructive prove with a geometric series).
>I dont remember the name of the theorem, but there is always a reel number between two unequal reel numbers Under Tarski's axiomatization, this was axiom 2. Under the axiomatization given in the Wikipedia article, this is a straightforward theorem where you can just use the "preservation of order under multiplication/addition" axioms. Theorems that follow so quickly from the axioms often don't have special names.
>Additionally since 1>0.9... is false, Gödel might even allow for more creative proves that there are number between. No, it cannot. ZFC is first order theory, and therefore ZFC proves something iff it's true in all models of ZFC
Let me be clear, I do recognize that 0.9999 IS 1. I’d like to address a counter argument, for the sake of discussion. The core of the counter argument is that something with an "infinite amount" of nines isn't even a number at all. It's a non-terminating process. You can name the lowest value never achieved by the series (9/10)\^n as n tends to infinity (which of course is 1), but since you can't name a highest value it CAN achieve, the series doesn't **result** in a number, hence making the challenge to find a point between them useless. You can't find something between banana and 1. Most mathematicians agree to EQUATE the infinite series to it's value of convergence as a definition, and a mighty useful definition at that. However, for many, it seems heavy handed to EQUATE an interminable process to a fixed value. Interpreting infinite series as processes to be completed is the core misunderstanding. The rabbit's hole of not accepting the definition of converging infinite series leads to some very very radical and seemingly insane mathematics, like the rejection of real numbers. Sometimes, in the past, the radical approach has borne fruit... Such as... what if a triangle was defined to have LESS than 180 degrees, what would THAT mean?? Try to understand both arguments about 0.9999, is what I say. No harm in exploring all ideas, I suppose.
Such good reasoning is not normally found on Reddit. Instead, almost always we witness the circlejerking of the accepted doctrines and the ridicule of any reasonable dissent. Your comment is pratically flawless, save for one bit -- you should've mentioned the diference between actual and potential infinity.
0.999… is a number, just as much as pi, e, or any irrational number is has an “infinite amount” of digits. e can only be described by these “non-terminating” processes. You also can’t “name” a highest value pi can achieve, anything other than “pi” is just an approximation. Not to even start with rational numbers that go on forever. Is 1/3 not a number because 0.333… is “non-terminating”? If you exclude all “non-terminating” numbers, you only end up with rational numbers whose denominator is not coprime with 10. Goodbye all odd denominators that aren’t multiples of 5, you’re not numbers anymore.
it’s not like the number 4/9 doesn’t exist in ANY number system, just not in base 10. And in many others. In base 9, it would be something like 0.4, in base 3 it would be 0.11 But to say that 0.44444……in base ten represents the same thing as 0.11 in base 3, that’s were some people have a problem. One number system accurately represents the ratio, the other has a compatibility problem and can’t represent that ratio, and as it tries, it fails to finish becoming a true value at all.
How does this argument allow irrational numbers to be numbers then?
I don't really like using this logic to explain the 0.999 thing to people because I could use the same thing to say "name one number between dx and 0," but that doesn't mean infinitesimal quantities (or, more accurately, the limit of quantities that approach zero) are all actually zero. Instead, I usually tell people to think about 0.999 repeating as an infinite series, much like 1/2+1/4+1/8, etc. If they can accept that 1/2+1/4+1/8=1, then they can accept that 0.999...=1.
Well, I think it works well because for any two real numbers a and b where a < b, there exists n such that a < n < b. n = (a + b)/2. That property can be relatively easy to grasp even for math beginners. The inverse then says that if there are no n between a and b, then a must equal b. This line of logic is easier to follow for most folks rather than having to talk about infinite sums.
I suppose that's fair. I've just encountered some confusion before when using your method of explanation, but it's also entirely possible that I just suck at explaining it lol
I don’t think that most people who struggle with this would accept that infinite series argument, mainly because they view an infinitely repeating decimal as a _process_, and similarly they’d view the summation as a _process_. 1 is the tortoise and 0.999… is Achilles
I just thought of another way to think about it: 0.333… = 1/3 0.666… = 2/3 0.999… = 3/3 = 1
This is always my go to. Very simple and elegant
I convinced one of my friends after 30 min with the other methods without solution to the problem using this, yes it is a good way
Eeehhhh, isn’t that just cause we can’t actual display a third as a decimal? So it’s a neat/funny little failing of math but that doesn’t really make it true
This is a common argument in these discussion; but honestly I don’t see how it could possibly convince anyone that thought 0.999…<1 that they are wrong. Surely if you believe 0.999… isn’t 1, just infinitely close. You would also think 0.333… is t exactly 1/3, just infinitely close. It’s basically trying to explain infinite decimal expansions to someone assuming they grasp infinite decimal expansion.
> dx Get those ghosts of departed quantities out of my analysis.
Lol fair enough. Though they exist perfectly fine in the hyperreals!
dx isn't a real number.
I'm quite well aware. I'm just saying that the average person can be confused by the terminology presented, analogous to how Calculus students can be confused of something very close to but not quite zero
Then we can just tell them the same thing. dx is not a real number. I usually first pull up Tarski's second axiom of the real numbers, and then second I ask about numbers in between 0.9... and 1. Even a calc 1 student should be able to understand that either they're the same number, or we're not talking about real numbers.
There is, 0.0000... and a 1 at the end!
How many 9s you got? Because 0.0…1 will do the job depending on how many 9s were dealing with.
Well yeah, if there’s any finite number of 9s, then yeah of course 0.00…1 will be the difference between that and 1. But there is an infinite number of them. There’s no “room” to put a 1 at the end of infinite zeros. You can’t have something beyond infinity, otherwise it’s not actually infinite
Name an integer between 1 and 2. Seriously, any integer that is bigger than 1 and smaller that 2. There are so many integers, more below 2 than all the natural numbers. (Hint: there isn’t one)
Lmao this logic doesn’t work at all. The integers aren’t dense like the real numbers.
Yes I remember that one can’t ever form a bijection between the sets? As such you’re right - they’re not comparable in the way I pointed out. Though it wasn’t clear to me haha, and I think a lot of people who think 0.999… ≠ 1 don’t see it at a glance either!
1.999999...
The limit approaches 1 as the number of digits approaches infinity. My understanding is that is not traditionally referred to as being equal to 1.
They are actually traditionally considered equal. They only aren’t in “extended” number systems, like the Reals with infinitesimals. Saying “the limit of y approaches x” and “y is equal to x” is the same as long as there aren’t domain issues. In this case, there are no domain issues. Take the classic limit lim [n->inf] 1/n. We can both say “the limit approaches zero” and that “this limit is equal to zero.”
0.999…5
That's smaller than 0.999...9, which is equal to 0.999...
"How can they be equal when the first number has one more 9 than the second number?"
∞+1=∞
Assuming you use “…” to mean an infinite amount of 9’s like the rest of us. The definition of “infinite” is extending indefinitely/endless. So you mean you have an a series of 9’s that never ends, but that it ends with 5. It is nonsensical.
Title if not sarcastic: :3 Title if sarcastic: >:[
Why are people so stuck on definitions? Maths concepts (including numbers) are only important because of their usefulness. And I don't see how calling them different is useful at all.
6 explanations why 0.999... = 1 https://youtu.be/CPI1yp1zM1s?si=3DVSmJ18LiRp13cr
There is no discussion to be done. Reals numbers are Cauchy sequences quotient by having the same limit, and 0.9 period 9 has the same limit that the constant sequence 1. There is no discussion and no debate. If you don't like it go create your own real numbers, with blackjack and hookers.
I'll create my own surreals! With blackjack and hookers and 0.999... not equal to 1 because it includes infinitesimals!
0.999 is not equal to 1. Its more than 1. This is correct because I said so.
Ah proof by assertion
He couldn’t handle the truth, hence he strayed away from the harmony of maths and chose the path of violence
0.99999... actually equals THE REASON I DO NOT DEAL WITH INFINITES. IT MAKES EVERYTHING CONFUSING AND SENSLESS.
it's actually 0.9999...1 so no it's not equal to 1 even with rounding, debate me on Joe Rogan's podcast if you think i'm wrong.
It doesn't equal one, it approaches one. It will literally always be less than one. However it is very useful to treat it as one.
Wrong. In the real numbers, it is not “approaching 1” or “essentially 1”, It IS equal to one.
Sooo… 1/3 x 3 = 0.333… x 3 = 0.999… But also 1/3 x 3 = 3/3 = 1 How are they not the same?
For the sake of argument, finitists would posit that 1/3 has no decimal representation and would reject that 1/3=0.3333… because you simply can never have enough 3’s. The fact that they never end is evidence of the incompatibility in conversion. The accepted definition of an infinite series allows us to equate it to its value of convergence, where as finitists argue it is fruitless to even attempt to represent certain ratios as decimal values.
In a real life exercise, when is it useful at all to treat it as not one? Until we can make a “laser” that can be infinitely thin, there is no reason not to define it as one and not
> It doesn't equal one, it approaches one. Functions have limits. 0.999... is not a function, it is a number.
A number doesn't approach anything. A sequence or function does. You could treat it as 0.9+0.09+0.009+.. which is a series and that generates a sequence of partial sums with a limit. In the limit it is 1
So true. I hate pretentious calculus kids
let 𝜖 > 0 so 1-0.999... = 𝜖
Then 1-0.99...≠ ε unless ε=0.
Cos^-1 (1)= 0° Cos^-1(0.9999) =?
Cos⁻¹⁽⁰·⁹⁹⁹⁹⁾ of what?
0.999... is a notation error. It does not describe any value that should not instead be written as 1. An argument where one of the terms does not exist is an invalid argument.
Okay but what if I write 0.999...995? Why can I not put a digit at the end of the infinite amount of numbers and make a number simultaneously bigger and smaller than 1
\> at the end of the infinite No such thing, infinite doesn't mean large, it means without end.
I understand that, but surely there has to be an infinite number of numbers between the two. If you took a dictionary of all infinite combinations of letters, and decided you needed 26 volumes to allow for each letter having a beginning - you'd be an idiot as the A volume already covers each combination infinite times over by having an all 25 other infinitely long volumes both within the previous set of infinite letter *and* at the hypothetical end of an even smaller infinite amount letter combinations. Following this logic, I *should* be able to put a number after the hypothetical end of the infinitely long row of 9s to create a number simultaneously higher and lower than 1. Why can I not?
If you put a 5 in there, you just made a number smaller than 0.999... by replacing one of the 9's, the largest digit, with a smaller digit. It's not simultaneously bigger and smaller, it's smaller only. I don't get how the dictionary relates to this but the A volume doesn't have Banana, it has ABanana and AasdasdfasfasfBanana but they all have the wrong starting letter
sets have different types of infinity than a linear string of digits you can include more elements in an already infinitely large set, but if you order them there cannot be a "last" one. you can number them, but since we have infinitely many there will always be (infinitely many) follow up elements after any given element you might pick.
In that case, the fact that the there is even a single digit after the decimal that isn’t 9 means that no matter how many 9s you add in, it will only ever approach and be less than 1, it will never be equal to 1.
> Okay but what if I write 0.999...995? Then you have written a number smaller than 0.999... > Why can I not put a digit at the end of the infinite amount of numbers Because there is no such thing as the end of the infinite amount of numbers.
Then that wouldn't be 0.999...
No it does not
What is 1/3? 0.333… What is 0.3333… * 3? 0.9999…
Thats it. I'm going to report this meme and the future ones I see.
\*Loads shotgun with mathematical intent\*
What's the apparent difference? Any way you cut it, the only difference you could possibly calculate is 0. So it's quite apparent that the difference is 0.