Ask for a number between .9 repeated and 1
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"There isn't one, 1 is the very next number right after 0.999... Checkmate atheists." (In all seriousness I don't think it's a very convincing argument for someone who doesn't buy the proofs -- it requires you to believe and have internalized the idea that there are an infinite number of reals between any two distinct reals, and therefore that any pair of reals with nothing between are the same number. Those seem like bigger logical leaps to me than the simple proofs for someone who hasn't thought about this stuff.)
How about, ask for an integer between 1 and 2. Can't think of one? Guess they're the same number then.
Apples and oranges. For any two different real numbers, there's a number between them. Integers work differently.
This is a bold assertion, and one that is not obviously true, especially in cases like 0.999... and 1.0
Those aren’t different real numbers. That’s the whole point of the conversation.
Obviously saying it is not obviously true is false if 0.999... == 1.0
I don't understand how they're the same number.
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.
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 small" number?
> 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.
I don't have a strong opinion or much mathematical knowledge, but an "infinitesimal" number is a thing that most people have heard of even if they're fuzzy on what it is. If there is such a thing, what is the difference between 0.999... and 1 - 1/∞?
Those are great questions that not every system is required to address in the same way. (In a similar vein, +0 != -0 in Java)
This is breakdown in notation and/or convention. There is no ground truth, just what's true within the system.
So does this mean that an infinitely small number is zero? As in 1/∞ ?
In real numbers, there doesn't exist such a thing as "infinitely small number" that is apart from zero. Yes, there exists infinitely many numbers between any minisculely small number and zero, but the way they are defined, every single number you can grasp, is finitely small. The "infinitely" small gap is inaccessible. In some other number systems it isn't, but in the standard reals it is.
That means that the "infinitely small" doesn't exist; "smallest apart from zero" doesn't exist either.
You can read about this in any work on nonstandard analysis. ("Nonstandard" is just the name, much like "imaginary" numbers.)
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.)
> So does this mean that an infinitely small number is zero?
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.
There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero.
> There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero.
Zero. Just zero. The difference is zero. 0. Because 0.999… = 1.
You are stating that 0.999... = 1 proves that 1 - 0.999... equals zero. I am stating that 1 - 0.999... = 0.000... proves that 0.999... = 1.
I think people intuitively see that infinitely zero equals zero.
In calculus, yes.
It's the smallest number bigger than 0.
That doesn't exist. An open interval doesn't have a smallest number.
It does exist. The other poster just clearly showed that it exists by referring to it.
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.
You are certainly correct that people arguing the opposite side probably don't have a formal system in mind, but I think the intuition that an open interval in the Reals doesn't have a smallest number is easy to grasp even without any formal training. So you can force them to see the consequences of it through fairly straightforward logical contradictions.
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.
In math, when assuming the existence of something proved a contradiction, we conclude that the thing does not exist. The description may exist "integer between 3 and 4", but there is not described object. A description names a set or a class, and that class can have 0,1, or more numbers.
Of course it does. It's called the infinitesimal. It's common definition for real number is 1 / infinity: https://en.wikipedia.org/wiki/Infinitesimal
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.
> If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist.
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).
There is no smallest positive infinitesimal either. At least in theories that manage to define those rigorously. And it’s mostly a formal trick anyway; standard epsilon-delta calculus avoids them entirely.
Had you actually meaningfully studied this subject, or did you just link to a Wikipedia article you half-heartedly skimmed one day?
At the very least, don't write "Of course it does". It does not in the real number system.
Why is 0.000... bigger than 0?
It isn't.
Or is it? Say I'm a layman and I decide that in the system of math as I understand it, 0.000... is larger than 0. Yes, if I was going to be completely form with my own system of math I would eventually have to face the problems this introduces and resolve it, but until then I can generally adopt a self contradictory system and continue to live my life unaffected. Much like many people live their whole lives using naive set theory for their understanding of sets.
You are correct. This was a rhetorical question to get traderjane to question whether "bigger than 0" really applies here.
strictly bigger than 0
Doesn't exist.
bigger or equal
0.
Ask for a letter between G and H.
You're missing the point. This would be an analogy fit for talking with someone who's looking for an integer between 1 and 2.
0.00...1
> 0.00...1
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/∞And what does the right hand side of that mean? Division is commonly defined for a real numerator and a real, non-zero denominator. You are using the common symbol, but with ∞ in the place of the denominator. Since ∞ is not a real number, you must be using a non-standard definition of division, and have to define what you mean.
No, there's no 1. 2OEH8eoCRo0 is exactly right. Subtract 0.999... from 1 and you get 0.000...
0.999... + 0.000...1 = 1
0.000...1 = 1/∞
0.999... = 1 - 1/∞You're repeating the same wrong thing you said earlier.
It's 0.999... and not 0.999...0
In the same way, it's 0.000... and not 0.000...1.