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If we say that infinitesimals exist, it still happens that 1 = 0.999…. It just happens that 0.999 ≠ 1 - 𝛚. 0.999… = 1 is a property of the way we write some rational numbers, not of the number system itself.
wouldn't 0.999.. be equal to 1 - 10^w since it's only a countably finite series of nines.
0.999... with infinite nines is equal to 1.
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black magic! I wonder if there are any programming languages that are able to handle this properly?
https://www.wolframalpha.com/input/?i=.9...+%3D+1
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I like this one better: 0.9 = 1 - 0.1 0.99 = 1 - 0.01 0.999 = 1 - 0.001 0.9999 = 1 - 0.0001 0.99999... = 1 - 0.00000... with a 1 at the end of the infinite series of 0
>[...] at the end of the infinite series of [...] ...uhh
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What is an "infinitely small" number? 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 small" number? 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.
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Yes. How much faster than 99.999..% the speed of light would you need to go to get as fast as the speed of light? If your answer is 0.00..1% the speed of light, this answer is nonsensical because infinity never ends, and you never ever have the ability to add that 1 at the end. So, the only answer can be 0.00.., and 0.00.. = 0, so 99.99.. has to = 1.
Hmm, This reminds me a lot like Zenos paradox. Also, I’d like to understand how if we can say 99.999... is = 100, wouldn’t we also be able to say 99.9999888999... = 99.9999..., and therefore also just 100? And so on?
You've just introduced a novel symbol into the discussion. What is the definition of 99.9999888999...? Before anyone can answer what it equals, you must define it. Recall that if the digits repeat, we already have a definition, so that case is fine. Your case is not à priori well-defined.
Why do we accept .999... as a valid notation. Why not only allow 1 to denote this concept?
That would be adding a special rule for purely cosmetic reasons. It's typically not done in mathematics, where concise rules are usually more cherished than special-casing things.
I think this is a notation and definition problem. To me, it behaves differently in `Y = 1 / X`, which distinguishes quite strongly between `X = 1 - 0.9999` and `X = 0.9999 - 1`! If 0.9999 ought be exactly equivalent to 1.0, there ought to be no difference between `1 / (1 - 0.9999)` and `1 / (0.9999 - 1)`. To me, 0.9999 indicates a directional limit, which can't necessarily be evaluated and substituted separately fro…
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Infinities are difficult and your explanation is wrong. This notation .666...7 is meaningless. The notation .666... indicates a decimal representation that does not end. The representation has infinitely many sixes and does not end. If there is a 7 in the representation then it must be at a specific decimal. If it was at the end of the representation then it would be a finite decimal representation. The notation is m…
This reasoning is understandable, but also incorrect, and we can point to where it breaks down: the idea that the 7 is "at a specific decimal" doesn't hold true, due to that pesky ellipsis. The crazy bit about the infinite repetition is that the 7 in 0.666...7 is not at a specific decimal. It's not even at "the last decimal" because there is no last. So, let's show this via a proof by contradiction: --- 1. we assert…
A sequence in the mathematical sense is a function whose domain is the natural numbers. Please define that function for the creature you're working with here. Otherwise you're trying to prove things about an object with no definition. You will end up in trouble.
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Infinities are difficult and your explanation is wrong. This notation .666...7 is meaningless. The notation .666... indicates a decimal representation that does not end. The representation has infinitely many sixes and does not end. If there is a 7 in the representation then it must be at a specific decimal. If it was at the end of the representation then it would be a finite decimal representation. The notation is m…
Yes, exactly, .666...7 means a number that has 6 at each decimal position and 7 for index k, where k is greater than any natural number. This exactly is .666... (just as parent commenter explained).
Ill-defined. Try again.
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See my comment here since it's related to this: https://news.ycombinator.com/item?id=23008366 Assuming that all my suspicions in that comment are correct and these proofs actually are invalid proofs (not the results which are true), then the question might become: does it matter if the proof of a fact is incorrect if the fact itself is correct? That is a philosophical question and I'm honestly not sure how I'd answer…
I don't necessarily understand your use of the word "invalid", when what it seems you mean is incomplete and/or too informal for your taste. > does it matter if the proof of a fact is incorrect if the fact itself is correct? Your language isn't allowing for a notion of precision, or for multiple forms of correctness, and it's not considering audience, communication or level of expertise either. I don't think it's a q…
> I don't necessarily understand your use of the word "invalid", when what it seems you mean is incomplete and/or too informal for your taste.
> [...]
> It does matter if a proof is wrong, if there is a step in the proof that can be shown to be false. But that's not the case here, what you want is additional definition.
I'm going to try to be more formal, but not entirely formal since (1) the details are almost never-ending and require a lot of formal logic and (2) I'm not 100% sure about the reasoning myself.
What I mean by a "proof" is a sequence of logical steps that start with axioms of your logical system. We are obviously not looking at things that formally, but I actually think it is still an important point. The rational numbers can be thought of as being defined by certain axioms of arithmetic. E.g. you have the natural numbers as well as the minimal extra values so that you can add, subtract, divide, multiply, etc.--in other words you have a field. Let's call these the arithmetic axioms. Then when you go to the real numbers you basically extend the rational numbers in such a way so that you have completeness and retain all the previous properties. So basically for real numbers you have the prior arithmetic axioms and you have the completeness axiom.
Next comes the question of what is actually to be proved. For that we _must_ make some sort of definition of what we mean by "0.9...". Let us define it as the limit of the sequence of partial sums (all of which are rational numbers) _if_ it exists. So to prove "0.9... = 1" means to prove that the limit exists and equals 1.
So lets say we believe we have a proof that the limit equals 1 using an argument like this:
0.9... = x => 9.9... = 10x => 9 = 9x => 1 = x => 0.9... = 1
This is not a formal symbolic proof, but there is one thing we can see immediately: this proof does not make use of the completeness axiom. Therefore it should logically be the result of a sequence of logical steps starting from what I earlier referred to as the arithmetic axioms. Now here is the point where the surreals come in. The point with the surreals is that they contain rational numbers and the arithmetic axioms still apply. (To be clear, I haven't thought this through 100%, but I am almost certain this is true modulo my hand-wavy reference to "arithmetic axioms".) That means that that same proof should work inside the surreal numbers to also prove that the limit of the partial sums is 1. But here is the key important point. Within the surreal numbers, the limit of the partial sums is _not_ 1. So what does this tell us? The original supposition that there exists a proof only making use of the arithmetic axioms cannot be true. So the proof must make use of the completeness axiom. Well the surreal numbers is _not_ complete so that axiom doesn't exist there and therefore the fact that the proof exists and works within the real numbers and not the surreal numbers is not a contradiction.
Okay this is a bunch of logic mumbo jumbo and no grade student should be expected to worry about things at this level, but we can still give a more simplified version of the proof that actually is in essence correct. Some people in this thread use the argument: "Well since we know the number must be less than or equal to 1 and we also know it must be bigger than any number smaller than 1, then it must be 1." That argument (while a priori assuming convergence) is at least implicitly using an intuitive idea of completeness. Is it logically formal and 100% rigorous? Of course not. But it is explicitly making use of a property of real numbers while the first proof is not. I think if students are to be taught anything about the real numbers, then they should get some general intuition for this property. The algebraic version of the proof is invalid (my claim, which hopefully is at least a little more well-supported given this comment here) and it doesn't give the intuition they should (hopefully) get anyway.
In any case, hopefully this does some to help clear up what I've meant throughout these posts.
P.S. Finally, in response to this:
> By the way, that blog post claims "The set of real numbers contains no infinitesimals." Wikipedia claims: "the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers..."
So what? Are you saying those statements are contradictory? If so, how? And if not, what are you saying?