Suppose there was a number between 0.999... and 1 such that 0.999...<x≤1. Then subtracting 1 from each side results in 0≤1-x<1/10^(∞). But since 1/10^(∞)=0 there couldn't be a number 1-x between 0 and 1/10^(∞). Therefore there is no number between 0.999... and 1 such that 0.999...<x≤1. Since 0.999... is greater than or equal to every element from 0 to 1, and since 1 is greater than or equal to every element in this same interval, then 0.999...=1. ??????????????
1/inf=0
>>169025040.999... != 1 because they're represented differently. They're two different things that convey two different ideas.
Suppose 0.999... was not equal to 1. Then what happens if you create a new number H = (0.999... + 1)/2. What is that number H equal to and how can it be larger than 0.999... at the same time as being smaller than one? 0.999... is already as close to one as any number can possibly be, so trying to squeeze another number there that is even closer makes no sense. So the existence of H makes no sense. And if 0.999... and 1 were not equal, H would exist. Therefore 0.999... and 1 must be equal.
>>16902504>suppose 1=2, then 1=2
>>16902656Why do there being no number between 0.999... and 1 make them equal? Why can't you just go from 0.999... to 1?
>>16902669If someone told you you had to go from point A to point B, made you stand on point A, didnt' make you move, and then told you you've reached point be, would you argue they are two different points? Would you argue that you went *from* point A *to* point B?
>>16902677Why cant it be discrete though? I could still move but that doesn't mean I have to move over anything, does it?
>>16902688>Why cant it be discrete though?Because it's proven not to be? At least read past the first page about real numbers in your first calculus book dude
>>16902677>would you argue they are two different points?yes. obviously the labeling of point B implies a new, unique point. nowhere else in mathematics does such alleged schizophrenia occur.
>>16902729>let x = 5>let y = 5>y is a different numberare you talking about schizophrenia out of personal experience?
>>16902688>>16902656Because 0.999... would be an upperbound in the interval 0.9, 0.99, 0.999, ... Now it's easy to see that 1 is the least upper bound for this interval, and obviously 0.999... cannot be greater than 1 so it's clear that 0.999... must be the same as 1.
>>16902800But you are still assuming it is continuous. How do you know it's continuous? Why can't it just be discrete?
>>16902819For any pair of real numbers we have a formula to calculate a number in between, so how can it be discrete?
i still don't fucking understand why .999... = 1 but .333... does not equal 1/3
>>16902827Why does 0.99999999... to 1 have to be continuous for them to be different?
>>16902846but it does
>>16902847they are not different
>>16902854no it does not i was corrected by some nerd some time ago you cannot prove .999... = 1 by saying .333... = 1/3 and .333... * 3 = 1. something about that proof is wrong but i forget what
>>16902654That's not what the equals sign means in math.
>>16902659/thread
>>16902866maybe stop using chud math then.
>>16902871Exsqueeze me?
>>16902858Yes they are. Number a =/= Number b.
Yeah basically. Except I think you meant 1 is the supremum and not just another upper bound. Also I think you should have specified that you were doing a limit instead of just plugging in infinity but whatever.
>>16902878So 2 + 2 =/= 4?
>>16902878= means that the values of a and b are the same, which they are. "3 minus 2" looks different than 1 and describes a different process. It's still correct to say 3 - 2 = 1.
>>16902504i read a small book called "a very short introduction to mathematics" and it told me about infinitesimals - not a single expert in here has mentioned them thoughlook up "non-standard analysis" - basically we just sweep it under the rug and say since .999... is infinitely close to 1, we say it may as well equal 1, but there is robust maths that uses infinitesimals to say it's still "an infinitesimal" away from 1
>>16902877Exsqueezed
>>16902669Then what is H equal to? Now it's neither 0.999... nor 1 because it's the average of two different numbers. And we assume there's nothing between 0.999... and 1. So what is it? Either way H makes no sense whether we assume there are or are no numbers between 0.999... and 1 while 0.999... and 1 being not equal.
>>16902909>anNot true, the author has misled you.
>>16902918How the fuck does there not being a number between 0.999... and 1 make them equal though? Lol no one has been able to sufficiently answer this. Why can't you just jump for 0.999... to 1 like it's a discrete value?>reee because the number line is continuous reeeeeOk but why should we assume it's continuous here? Also putting that aside nobody defines value by whether or not other number are between two other numbers. 5 is not equal to 6 because 5.5 is in between them lol. 5 is not equal to 6 because 5 =/=6, x=/=y, a=/=b. Simple identity here. Modern math is pseud quackery I'm telling you.
>0.999...=1Ah yes. The great filter. If you cannot understand then I'm sorry you can't be helped.
>>16902930>Ok but why should we assume it's continuous here?this has been answered multiple times
>>16902941No it hasn't. Explain why 0.9999999....... automatically equals 1 just because there isn't another number in between. It does not follow from this.
>>16902944We do not assume it's continuous, we have PROVEN it's continuous
>>16902952No you haven't.
>>16902866it pisses me off when people use that because != is asking a question, not declaring a statement. assume the person just finished cs101 when they do that pretentious shit >>16902504obviously. there aren't any number in between
>>16902958So you know more than me? Then you can fuck off with your questions
>>16902930Because there's nothing illogical about there being an average of two different numbers no matter what those two numbers are. And if we end up in a situation where you cannot have such a number, something is wrong. There always is an average. If the only way how (0.999... + 1)/2 makes sense is if 0.999..m = 1 then 0.999... = 1.
>>16902965>And if we end up in a situation where you cannot have such a number, something is wrong.Why?>If the only way how (0.999... + 1)/2 makes sense is if 0.999..m = 1 then 0.999... = 1.No because that makes even less sense. You can't just declare 2=1 because you ran into some weird problem with those two numbers. You're literally doing what physics-tards do with dark matter. It's all ad hoc Jewish bullshit.
>>16902969So you believe the real numbers are discreet?
>>16902863There's no mistake but the reasoning is weird. If I don't believe you when you say that 0.999...=1 and if i ask for a proof, why would you assume that I will agree with you writing 0.333...=1/3 ?The proof clearly stops too abruptly, it lacks a few steps. Then again, it usually convinces the average non-math guy so it reaches its goal.
>>169029441.0... - 0.9... = 0.0... by induction. The nth decimal digit is always 01 = 1.0... and 0 = 0.0... both follow from how integers are defined in decimal notation. So by substitution you have 1 - 0.9... = 01 = 0.9... follows from the additive identity.
>>16902504
1/9 = 0.111...+8/9 = 0.888...=9/9 = 0.999...
>0.333 = 1/3>0.666 = 2/3>0.999 = 3/3
the claim is that 0.999... and 1 are two representations of the same number. okay, fine. so 1/2 = 0.5. what is 0.999.../2? and no, don't just say 0.5, since that relies on a different representation. within the representation of 0.999..., show explicitly, and formally what 0.999.../2 simplifies to. it's not 0.4999...
>>16903406[math]\begin{align}0.999... = 9\times \sum_{n=0}^\infty \frac{1}{10^n}\end{align}[/math]ergo[math]\begin{align}\frac{0.999...}{2} &= 4.5\times \sum_{n=0}^\infty \frac{1}{10^n} \\ &= 0.45 + 0.045 + 0.0045 + \ldots\end{align}[/math]we now see that 0.999.../2, whatever this number is, must ALWAYS end in a 5. this is what 0.999 != 1 chads have been saying for decades, that 0.999... must always end in a 9, not a 1. however due to the infinite repetition of 9s, math schizos were able to justify this somehow equaling 1. i'd like to see how the schizos argue that an expression that always ends in a 5 must be identically equal to 0.5 kek
>>16903412You're literally retarded, there is no final digit in an infinitely repeating chains of 0.999..... Dividing it in half just gets you 0.4999... which is exactly equal to 1/2.
>>16903443>Dividing it in half just gets you 0.4999...proof?
>>16903412>must ALWAYS end in a 5source?
>>16903593see>>16903412
>>16903490You have 2 * a = b, with b = 0.9... Write the dth decimal digit of a or b as a(d) or b(d).Decimal digits are integers so 2 * a(d) must be even, meaning that b(d - 1) odd implies a(d) >= 5 and b(d - 1) even implies a(d) < 5. In this case, b(d - 1) is even for d = 1 and odd for d > 0 so a(d) < 5 for d = 1 and >= 5 for d > 1. Now if b(d) = 9, then a(d) can be either 4 or 9. Plug that into the lemma and you get a(1) = 4 and a(d) = 9 for d > 1.
>>16903621in english, schizo
>>16903651Essays and other degenerate nonproofs go in /lit/ along with your poetry about the Planck constants :)
>>16902863>>16903015The whole point of 0.333... was to represent 1/3 as a decimal. If you don't get 1 back when you multiply it by 3, then what was the point of 0.333... in the first place? This makes it clear to the layman that 0.999... needs to equal 1 for the math to make sense. Of course this is not good enough for the mathematician, who wants to verify that the math does make sense. A good way to do this is to pick a definition for what infinite decimals mean and verify that it does the things we expect, including making 0.333... = 1/3 and 0.999... = 1. See >>16903084 for one way of fleshing this out.
>>169037651/3 = 3/10 + 1/30= 0.3 + 1/30= 0.33 + 1/300= 0.333 + 1/3000:= 0.333... + 1/inf= 0.333... + 0= 0.333...
