Why is 3/3 equal to 1?Ok, so 1/3 is 0.33333 forever. 2/3 is 0.6666 forever. So, if that's the case, why is 3/3=1 and not 0.99999 forever? Where does the last little bit get added to 3/3 to have it equal 1?
>>16863469You have 3 apples to distribute equally among 3 people. How many apples does each person get?
>>168634690.9999... is equal to 1, so 3/3 could be equal to 1 or 0.999.. and both equations are the sameProof:x = 0.999...10x = 9.999...subtract x or 0.999... from both sides9x = 9x = 1
>>16863469What computers do is 2/3 = 0.666...667. They just eventually change the last digit to a 7 because it's the only way to make 1/3 + 2/3 = 1 work. You can see this in some programs like ms excel. It works like this because math is dumb and doesn't work unless you just make up rules to get around things that make no sense and then everyone has to agree on things that make no sense for it to be able to make sense.
>>16863488>subtract x or 0.999... from both sidesIsn't that relying on the result of the proof within the proof? Like some people would say that 10 - 0.999... doesn't equal 9 the same way 0.999... doesn't equal 1
>>16863506no because we defined x as being equal to 0.999... This is just so we have a representative variable for our number, it has nothing to do with the end result of the proof.Subtracting x from both sides is the same as subtracting 0.999... from both sides.>>16863500I would argue that its just because rational numbers have infinitely repeating decimals and calculators cannot actually represent this unless they round a digit or use certain notation. Hence the 7.
>>16863515Yes but you're using 0.999... = 1 to prove 0.999... = 1. For someone that disagrees that 0.999... = 1 then using 0.999... = 1 to prove to them its true isn't going to prove to them it's true
[math] \displaystyle1= \dfrac{3}{3}=3 \cdot \dfrac{1}{3}=3 \cdot 0. \bar{3}=0. \bar{9}[/math]
>known issue with the format becomes a memeFine and dandy.>known issue with the format stays a meme for several yearsI am determined to avoid abusing the catalog to overly-curate my experience, but I miss the days when memes could get old and die.
>>16863540>Yes but you're using 0.999... = 1 to prove 0.999Ummm no I didn't. Can you point out where I did that?
>>16863469no, 1/3 is not 0.33333 forever. it’s exactly 20 in base 60. decimal representation depends on base
>>16863710>10x = 9.999...>subtract x or 0.999... from both sides>9x = 910x became 9x. But x was 0.999.... Meaning it was using 0.999... = 1
>>16863872He subtracted x.10x - x = 9x no matter what x is.
The issues with 0.333... and 0.999... go away once you understand what they mean.The first mistake is thinking of a real number as a sequence of digits. If you think this way, then obviously you would think 0.999... is not 1 because the digits are different. Instead of just thinking of the digits, you should try to understand what the digits mean. 0.999... is a name for a point on the number line.Everyone who has mastered middle school mathematics knows that 0.9 is a point at 9 tenths of the distance from 0 to 1, 0.99 is a point 9 hundredths of that same unit distance to the right of 0.9, and 0.999 is a point 9 thousandths of a unit to the right of 0.99. We can continue this process as long as we want to find the point represented by "0." followed by any finite whole number of 9's. The second mistake people make is thinking that 0.999... means where we'll be after infinitely many steps. You could argue indefinitely about what point this would be, and whether the idea even makes sense. It's not a clear definition.To understand the correct meaning of 0.999..., consider the never-ending process of starting with "0." and repeatedly adding a "9" digit and finding the new point on the number line. We split the number line into two parts. The first part is the number points which will eventually be surpassed by this process of adding 9's, that is, points which lie to the left of some number made from "0." and finitely many 9's. The second part is all the number points which will never be surpassed. There is a point at the boundary between these two parts of the number line. This is the point we mean by "0.999...".We can ask which of the two parts our boundary point 0.999... belongs to. With a bit of thought, you can see that it must belong to the second part. So a snappy definition of 0.999... which you will find in many textbooks is that 0.999... is the smallest real number at least as large as every number made from "0." and finitely many 9's.
>>16863873>x = 0.999...>2x = 1.999...>3x = 2.999...>4x = 3.999...>5x = 4.999...>6x = 5.999...>7x = 6.999...>8x = 7.999...>9x = 9>10x = 9.999...?
>>16863891This isn't the line of reasoning for 10x=9.999...Multiplying 0.999... by 10 moves the decimal place to the right.This is the exact method we use to convert any infinitely repeating decimal to an integer or rational number.Example:x = 0.777...10x = 7.777...Subtract x from both sides9x = 7x = 7/9Plug 7/9 into a calculator and you get the same answer.
>>16863891>>16863896Also, the point is that subtracting 0.999... from 9.999... cancels out that repeating decimal, giving you 9x = 9 (as per the proof, this is equivalent to 8.999... as well)
>>16863896That's why it's bs. Because there's tricks and rules to make it work around the fact that 1 and 0.999... are fundamentally not equal. 0.999... infinitely repeating is an asymptote. 0.999... does not equal 1, however the limit of 0.999... equals 1.
>>16863579Being autistic is a choice.
>>16863878I like how you actually provided the rigorous formal explanation that puts this shit to rest and nobody even gives a fuck.Plus you can expect this exact same fucking retarded thread topic to show up again in less than a week. It's because they don't actually care about this subject, you realize don't you? This is what tourists believe is "board culture" and they're doing it to try and virtue signal that they belong here. They think they're being chads and not just acting like obnoxious tourists that need to fuck off back to /pol/ or /x/ or /g/ or whatever fucking cesspool they crawled out from.
>>16863899Here's another proof:Keep in mind that ALL rational numbers are infinitely repeating decimals.Let's look at:1/3 = 0.333... We know this is decimal infinitely repeats 3 because doing long division on this number will constantly yield remainders of 3 to infinity.If we multiply both sides by 3, we get:3/3 = 0.999...We know 3/3 is equal to 1, so once again we get a statement that says 1 = 0.999...The only alternative solution, as mentioned in the OP, is that a number just gets "added" somewhere randomly when you multiply 0.333... by 3. There is no evidence for this occurring.Math is a series of tricks and rules discovered to line up with the reality. That doesn't make it bs, it just makes it unintuitive sometimes.
>>16863469This is the problem with representing numbers typographically.>Just pretend that you could keep applying the division algorithm and writing 3s infinitely! That's 1/3!The ancient Greeks understood that all numbers were measurements, and that 1/3 is the result of applying Thales theorem to line segments of 1 and 3.
>>168634850.999999...
>>16863995Absolutely right, sir. They each get exactly 1.
0.999 = 1 because it's a convergent limit/thread
>>16863469https://en.wikipedia.org/wiki/Limit_of_a_sequencefucking idiot
>>16864000What do I win?
ok im bored, changing the rules of engagement:from this post forth EVERY argument must be posed from the rationals, i don't want to see any of you niggers so much as to even think of touching the reals, i ain't even letting you graze the algebraic reals, PURE [math]\mathbb{Q}[/math] FROM HERE ON OUT, you hear me?>but whycause this shit starts there, every system beyond the rationals merely inherits it from the rationals, if you can't defend it being in the rationals you are fucked, simple asfuck all you niggers and good luck!
>>16864401fuck off back to /x/
>>16864404what?
>>16863899Only functions can have asymptotes.0.999... is a graphical depiction. It's not a function.If it's a function, it must accept at least one variable. Where is that variable?
3/3 is like one third multiplied by three. It's like dividing and then undoing it, it is 1 by definition.
Claim: [math] 0.999_{\dots} \neq 1[/math]Proof: We use induction. The base case is trivial: [math] 0.9 \neq 1[/math]. Next we introduce the notation that [math]0.9_n = \underbrace{0.9999999}_{n-\text{many nines}}[/math] is the decimal with n-many 9s.Now the inductive step: we assume [math]0.9_n \neq 1[/math]. Then trivially [math]0.9_{n+1} \neq 1 [/math]. It might help to notice that [math] 1 - 0.9_{n+1} \neq 0[/math]. This implies that [math]0.9_n \neq 1 \qquad \forall n\in \mathbb{N}[/math]Finally, we define [math] 0.999_{\dots} := \lim_{n\to\infty} 0.9_n[/math].[math]\therefore 0.999_{\dots} \neq 1 \qquad \square [/math]
>>16864514But wouldn't that also mean 1/3 is not 0.333 forever and 2/3 is not 0.6666 forever meaning the original question is working on an incorrect assumption
>>168644010.999... still means the least upper bound of its finite truncations. 1 is an upper bound, and any rational number x < 1 is not an upper bound because [math]x < 0.\underbrace{999 \ldots 9}_{n}[/math] where n is the number of digits in [math]\left\lfloor \frac{1}{1-x} \right\rfloor[/math].
>>168634690.9_ and 1 are two different expressions of the same value. Just like 12dec and 1100bin are two different expressions of the same value.If you agree that 1/3 and 0.3_ represent the same value, then you already agree with this.
>>16863899Which tricks? List them.
>>16864979So, how do i prove in [math]\textbb{Q}[/math] that every correspoding set of finite truncations has a least upper bound without relying on [math]\textbb{R}[/math]'s axiom
For any two distinct real numbers, there must be at least one real number between them (that's what it means for two real numbers to be distinct).If 0.9_ and 1 are distinct, you can name at least one real number that is strictly greater than 0.9_ and strictly less than 1.You can't do this, so they're equal.
>>16866775What do you mean by "every correspoding set of finite truncations"?
>>16866905The underlying set corresponding to every periodic decimal
>>16866952First we show a rational number is an upper bound on all the truncations of a repeating decimal if and only if it is an upper bound on the subset of truncations which consist of the nonrepeating part followed by a whole number of repetends. The forward implication is trivial; for the backward case, for any truncation of the repeating decimal, append digits until you reach a decimal of the desired form, and since the extended decimal is less than or equal to the upper bound on the subset, so is the original decimal. If we can find a least upper bound on this subset, it is a least upper bound on the whole set.The truncations in the subset can be written in the form [math]A + \sum_{k=0}^{n-1} B \cdot 10^{-mk} = A + B \frac{1 - 10^{-mn}}{1 - 10^{-m}}[/math] where A and B are non-negative rational numbers and m is the number of digits in the repetend. The number [math]X = A + \frac{B}{1 - 10^{-m}}[/math] is an upper bound. Let y be a rational number smaller than X; to show y is not an upper bound, we need to find an n such that [math]A + B \frac{1 - 10^{-mn}}{1 - 10^{-m}} < y[/math]. This holds iff [math]B \frac{10^{-mn}}{1 - 10^{-m}} < X - y[/math]; if B = 0, it is true for every n. If B > 0, then it holds iff [math]10^{mn} > \frac{B}{(1 - 10^{-m}) (X - y)}[/math], so a sufficient n can be obtained by counting the digits in [math]\left\lfloor \frac{B}{(1 - 10^{-m}) (X - y)} \right\rfloor[/math], dividing by m, and rounding up. Thus X is the desired least upper bound.
>>16863995This is probably correct when you account for apple material lost on the blade of the knife.
>>16867365cool
1/7 = 0.142857.. why is this important, after 6 unique digits the number repeats the same 6 digits, this is the only clue we have of our numerical system's finality, therefore 3.333... is only 3.333333 and that is not equal to 1/3 it is fundamentally incorrect from our current numerical system
>>16868932oh, i know that one, the following is integers, the realizatio of what it meansfor the decimals is up to the reader:142857*1=142857142857*2=285714142857*3=428571142857*4=571428142857*5=714285142857*6=857142142857*7=999999
