Which is the point in using the word obvious, obviously. Namely, using it to feel superior or to not provide a better argument.
Anyone who uses the word differently is doing it wrong.
I will never accept that they are the same. The difference between 0.9 repeating infinitely and 1 is infinitely small, but it isn't zero.
Is 9999..... the same as infinity?
What is 1.0 - 0.99999.... = ?
What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer?
What is an infinitely large number?
> What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer?
By that logic, 0.99 repeating isn't a number at all, and therefore can't be equivalent to 1, because you can't subtract it from 1. So my understanding that they are different is correct.
> What is an infinitely large number?
Neither is a well-defined concept within the standard reals, and completely unnecessary for understanding that 0.999…=1.
> > What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer?
> By that logic, 0.99 repeating isn't a number at all, and therefore can't be equivalent to 1, because you can't subtract it from 1. So my understanding that they are different is correct.
0.99… is a real number. The sequence (a_n)_{n positive integer} with a_n = 9/10^1 + 9/10^2 + … + 9/10^n has a limit (do you want me to prove that?). 0.99… is defined as that limit. That limit is 1. Therefore 0.99… = 1.
I think you're struggling to grasp the definition here. The defintion of 0.ddd…, where d is an integer between 0 and 9, is the limit of the above sequence with 9 replaced by d. That limit always exists, and the definition is therefore OK. In the case of d=9, the limit is 1.
0.9 is not equal to 1,
0.99 is not equal to 1,
0.999 is not equal to 1,
0.9999 is not equal to 1,
0.99999 is not equal to 1,
0.999999 is not equal to 1,
Saying that if you add enough "9"s it suddenly equals 1.0 makes absolutely no sense to me, and I seriously doubt that anyone will be able to convince me that it does make sense. I've read every single post in this thread and none of you have gotten me any closer at all to believing or understanding that 0.9 repeating equals 1.
Maybe I'm too old to understand this "new math" where all numbers are equal to each other.
No finite representation of repeating 0.9s can equal 1.0
The ask that people accept infinite representations as valid is a big one.
> 0.99 is not equal to 1,
> 0.999 is not equal to 1,
> 0.9999 is not equal to 1,
> 0.99999 is not equal to 1,
> 0.999999 is not equal to 1,
> and so on, ad infinitum.
You are correct about all of these, and all finite strings of the above form.
> Saying that if you add enough "9"s it suddenly equals 1.0 makes absolutely no sense to me, and I seriously doubt that anyone will be able to convince me that it does make sense. I've read every single post in this thread and none of you have gotten me any closer at all to believing or understanding that 0.9 repeating equals 1.
I think it's because you, and a lot of other people in this thread, are turning the question on its head. The difficulty does not so much lie in figuring out whether 0.999… is equal to 1 or not, but rather in what we mean when we write 0.999….
I know I'm repeating myself from elsewhere in the thread, but I'll try again. Try to go through these step by step, and feel free to let me know where you lose the thread.
DEFINITION: A finite decimal representation of a real number is a finite string of the form `a_m a_{m-1} … a_0 . b_1 b_2 … b_n` where each `a_i` and each `b_i` is a natural number between 0 and 9 inclusive (a digit). We say that this finite decimal representation represents the real number
a_m*10^m + a_{m-1}*10^{m-1} + … + a_0 + b_1*10^{-1} + b_2*10^{-2} + … + b_n*10^{-n}.
EXAMPLE: The string `12.98` has `m=1`, `n=2` with `a_1=1`, `a_0=2`, `b_1=9` and `b_2=8`. It therefore represents the real number
1*10^1 + 2*10^0 + 9*10^{-1} + 8*10^{-2}
Within this standard framework, there is no way to ask "what is 0.999…?. It is not yet defined, because we have only defined what finite strings mean. The standard definition for what one means by 0.999… follows. (One can obviously also define these things 0.888…, 1.999…, etc., but let's stick to one case here).
DEFINITION: Let `(c_n)_{n natural}` be a sequence of real numbers (let me know if you need a definition of sequences!). We say that the sequence has the limit x as n tends to infinity (these are words, you don't have to ascribe meaning to "infinity" in that sentence – it's just a word, like "gnarf"!) if, given any real eps>0, there exists an M such that for all m > M, |c_m - x| < eps.
Definition (this is the definition you have to wrap your head around before continuing): Consider the sequence `(c_n)_{n natural}` where `c_n` is the finite sum
9*10^{-1} + 9*10^{-2} + … + 9*10^{-n}
"THEOREM": The limit defining `0.999…` does exist. It is `1`.
PROOF: You can fill this in. If you can't, I'm happy to do it.
As you can see, at no point in the above did feelings or beliefs matter :-)
That means that the "infinitely small" doesn't exist; "smallest apart from zero" doesn't exist either.
An infinitely small number is zero when projected onto the real number line. If you introduce an infinitesimal quantity to the reals, then for every number there is a unique real number to which that first number is infinitely close (that is, the difference between them is infinitesimal). You can use that real number as a (good) approximation of all the nonstandard numbers in its halo. (As long as you're comparing it to other real numbers.)
What does "infinitely small" mean?
> As in 1/∞ ?
What notion of division are we talking about here? The division most people expect is that of real numbers. ∞ is not a real number, so you'll have to specify what you mean.
Zero. Just zero. The difference is zero. 0. Because 0.999… = 1.
I think people intuitively see that infinitely zero equals zero.
If people accept the former, and that the RHS of the former is in fact 0, they've already also accepted that 0.999…=1. I don't see what the discussion is at that point
I don't have a direct computation for making the latter obvious, just indirect ones like 1 - 0.999... and 3 x 0.333...
How can they compute 1-0.999… when they clearly have no idea what 0.999… is?
Put 1.0 on top, 0.9 on the bottom. Start subtracting from left to right, and keep writing nines on the bottom as you go to the right. In no time you'll see that the answer is infinite zeros.
How do they know that that's a real number?
The problem is that if we include such a number in our formal system of math, we quickly find contradictions and the whole system falls apart. So such a number is incompatible with any formal system of math (though I guess you could start building one which does include such a number and see what properties it has).
Herein lies the problem, the people you are talking with do not use a form system. There system of math has something similar to the same flaw of their system of grouping of things, which would include the whole grouping that contains every grouping that doesn't contain itself. People rarely deal in formal systems and thus they can handle completely illogical statements fine as long they are protected from seeing the consequence of it.
Assume x is the smallest real number greater than 0. Then x/2 is also a real number and is greater than 0 but less than x. Therefore, x can't be the smallest real number greater than 0.
Well only to the extent that you don't want to throw away any of the other axioms. Sometimes you do and there are some fun systems of math, but few have any practicality and those that do are often so advanced that even someone with an undergraduate focus in math can't appreciate those systems.
It is much the same with computer science. I personally enjoyed playing around with formal concepts of computation and adding some extras to see what happens. For example, what happens to a Turing machine if part of the machine can time travel or has access to an oracle. Does this make concepts like time travel inherently contradictory to our notion of computation?
But the practicality of these exercises does not exceed their entertainment value.
If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist.
It's not a value you can meaningfully write out, but you can't write out pi, e, phi, root 2, 1 / 3 in base 10, root -1, etc. "I can't write it down" isn't a particularly unique property for numbers.
Not at all. Standard calculus uses standard real numbers, for which there is no infinitesimal. One may well speak of infinitesimals as a mental tool when building a mental model for calculus, but those infinitesimals are not actual real numbers (or a well-defined mathematical object at all - in standard calculus).
Had you actually meaningfully studied this subject, or did you just link to a Wikipedia article you half-heartedly skimmed one day?
There is in the surreal and hyperreal number systems. I got that from skimming wikipedia though....
However, basic arithmetic taught to children requires that adding trailing zeros does not change the value of a number. You'll have a hard time doing arithmetic once you change that assumption.
And what does this mean? I will remind you that for an integer d between 0 and 9, 0.ddd… means the limit of \sum_{i=1}^N d/10^i as N tends to infinity.
0.000...1 = 1/∞Fine by me. Define whatever notion you're using. You can't just throw out non-standard things and expect people to know what you mean.
0.999... + 0.000...1 = 1
0.000...1 = 1/∞
0.999... = 1 - 1/∞It's 0.999... and not 0.999...0
In the same way, it's 0.000... and not 0.000...1.
0.999... = 0.999...9
0.999...9 + 0.000...1 = 1
0.999...0 + 0.000..1 = 0.999..1
0.000...1 = 1/∞
0.999...9 = 1 - 1/∞
0.999...0 = 1 - 1/∞ - 9/∞ = 1 - 10/∞
If x/∞ = 0, then 0.999...x = 1.
If x/∞ ≠ 0, then 0.999...x ≠ 1.If Universe is finite, then finite number of elements can make only finite number of combinations, thus this discussion is repeated infinite number of times again. Why I should waste my time again?
0.000...1 = 1/10^∞ = 1/∞Here John Conway explains them: https://www.youtube.com/watch?v=1eAmxgINXrE
0.999...1 = 1 - 1/∞ - 8/∞ = 1 - 9/∞Of course it's hard because in day to day life, even for the vast majority of STEM practitioners, the nuance of the proof that 0.9999... is 1 is not of much utility.
Whenever one sees a 0.999[... to however many digits] one can safely assume it's less than one or perhaps more realistically "almost 1". To say 0.999... with the very specific detail that the 9's go on forever is actually a strange thing to say and outside of most people's experience.
There are simple enough proofs of this that normal folks who paid attention in high school can follow, but I think it has to be framed more as a clever brain-teaser than as a proof.
Oh absolutely. I'm not expecting STE(no M this time!) practitioners to necessarily be aware of why 0.999…=1 in their daily lives, but I do expect them to have encountered enough situations in their field of expertise where scraping the surface using shallow intuition and gut feeling lead them wildly astray. I'm therefore surprised that they're willing to deny this basic fact to the face of mathematicians. The ones I've interacted with also don't happen to be the types that'll start arguing Anatomy 101 facts with a heart surgeon at a bar, but somehow arguing over basic calculus with mathematicians is fine.
Indeed. I've torn my hair out trying to convince smart people with PhDs in hard sciences and had to give up in frustration.
I usually find that the most success can be had by kicking the ball to them immediately and having them define what they actually mean when they say "0.999…". If we're going to debate whether that thing equals another thing, we better make sure we know what we're talking about. Inevitably, this either causes the dead-set person to give up, or give a myriad of definitions that are either meaningless, ill-defined, or causes them to realize that they don't actually know what "0.999…" means (or what they want it to mean). It is hard to have the patience to chase down the consequences of their ill-fated definitions, though.