O.9999999... = 1?

[Spartaculous]

0.00000...1
Meaning an infinite number of zeros followed by one one.

This is not a real number (in the usual mathematical sense of the phrase).

technically, this is a real number.

No. This is not a real number.

[sweetandcool]

This is not a real number (in the usual mathematical sense of the phrase).

technically, this is a real number.

No. This is not a real number.

It's in the same vein as declaring that you have a triangle with 4 sides.

As soon as you tack the 1 at the end of that number, you now have a finite number of '0's followed by a 1.

So the number as described by Diplomacy&Warfare does not exist.

That all being said, I am 99.9999.......% certain that he was just trolling.

[Diplomacy&Warfare]

technically, this is a real number.

No. This is not a real number.

It's in the same vein as declaring that you have a triangle with 4 sides.

As soon as you tack the 1 at the end of that number, you now have a finite number of '0's followed by a 1.

So the number as described by Diplomacy&Warfare does not exist.

That all being said, I am 99.9999.......% certain that he was just trolling.

It seems like at least one of you missed the fact that I was responding to a post asking for an explanation of the smallest positive number. This number is infinitely small, and therefore there is an infinitely small difference between it and zero. Of course, the 0.000...1 explanation doesn't neatly fit onto a piece of paper. The two main potential problems are:
(a) the number has a finite number of 0s, like 0.00001, and therefore is larger than, say, 0.000001
(b) there are an infinite number of zeroes, and therefore the one never functionally appears in the sequence

I believe sweet was saying my explanation runs into solution (a). It doesn't, because the one is only functionally "tacked on" after an infinite number of zeroes. This doesn't make sense, but few things involving infinity make sense, and this number is essentially:

1
---------------------------
∞

Also, arguing that there's no difference between it and zero is pointless, because the difference between any real number x and 0 is always |x|.
|(1/infinity)| = 1/infinity and therefore there is an infinitely small difference between (1/infinity) and zero.

In fact, this is not a real number, any more than (i/2) is a real number. For a proof, positive, real numbers such as 3, pi, (179/7), etc can always represent the length of, say, a piece of paper. A piece of paper can be 3 meters long, pi meters long, or (179/7) meters long. On the other hand, numbers that are either not positive or not real can't be the length of a piece of paper: a piece of paper cannot be 0 meters, -18 meters, i meters, infinite meters long. A piece of paper also cannot have infinitely small length, and therefore (1/infinity) is not a real number. I didn't say it was a real number. What it is is the smallest positive number, so I believe I have answered the question.

Now it's time to answer the title of this thread. 10 - 9.999... = (1/infinity). Therefore, there is an infinitely small difference between 9.999... and 10. This difference may or may not exist, as the difference between the number and zero is the number itself. This means that:
(let x=(1/infinity))
x-0=y
y=x
therefore,
x-0=x
and x=x
This has proven to be thoroughly pointless. My conclusion is that while I can give a basic explanation of the smallest positive number, you need a math major to explain whether or not 9.999...=10 with any certainty.

[Spartaculous]

It seems like at least one of you missed the fact that I was responding to a post asking for an explanation of the smallest positive number.

To be clear here: assuming you are working with the real numbers, there is no smallest positive number.

Theorem. There is no smallest positive real number.

Proof. Suppose for the sake of contradiction that there is a smallest positive real number. Call this number x. Then, because x is positive, 0 < x/2 < x. (Since the real numbers are a field and fields are closed under division by non-zero elements, if x is a real number, x/2 must also be a real number.) Hence, x/2 is a smaller real number than x, which is a contradiction. QED.

[sweetandcool]

It seems like at least one of you missed the fact that I was responding to a post asking for an explanation of the smallest positive number.

To be clear here: assuming you are working with the real numbers, there is no smallest positive number.

Theorem. There is no smallest positive real number.

Proof. Suppose for the sake of contradiction that there is a smallest positive real number. Call this number x. Then, because x is positive, 0 < x/2 < x. (Since the real numbers are a field and fields are closed under division by non-zero elements, if x is a real number, x/2 must also be a real number.) Hence, x/2 is a smaller real number than x, which is a contradiction. QED.

I was about to whip out the same exact proof (:

Your proof should make clear what most non-math people probably take for granted: You can always find a real number between two real numbers. Infinitely many real numbers in fact.

I think that's why most non-math people don't understand why the argument that no number exists between 0.9999...... and 1.0 proves that 0.9999....=1.0

[sweetandcool]

No. This is not a real number.

It's in the same vein as declaring that you have a triangle with 4 sides.

As soon as you tack the 1 at the end of that number, you now have a finite number of '0's followed by a 1.

So the number as described by Diplomacy&Warfare does not exist.

That all being said, I am 99.9999.......% certain that he was just trolling.

It seems like at least one of you missed the fact that I was responding to a post asking for an explanation of the smallest positive number. This number is infinitely small, and therefore there is an infinitely small difference between it and zero. Of course, the 0.000...1 explanation doesn't neatly fit onto a piece of paper. The two main potential problems are:
(a) the number has a finite number of 0s, like 0.00001, and therefore is larger than, say, 0.000001
(b) there are an infinite number of zeroes, and therefore the one never functionally appears in the sequence

I believe sweet was saying my explanation runs into solution (a). It doesn't, because the one is only functionally "tacked on" after an infinite number of zeroes. This doesn't make sense, but few things involving infinity make sense, and this number is essentially:

1
---------------------------
∞

Also, arguing that there's no difference between it and zero is pointless, because the difference between any real number x and 0 is always |x|.
|(1/infinity)| = 1/infinity and therefore there is an infinitely small difference between (1/infinity) and zero.

In fact, this is not a real number, any more than (i/2) is a real number. For a proof, positive, real numbers such as 3, pi, (179/7), etc can always represent the length of, say, a piece of paper. A piece of paper can be 3 meters long, pi meters long, or (179/7) meters long. On the other hand, numbers that are either not positive or not real can't be the length of a piece of paper: a piece of paper cannot be 0 meters, -18 meters, i meters, infinite meters long. A piece of paper also cannot have infinitely small length, and therefore (1/infinity) is not a real number. I didn't say it was a real number. What it is is the smallest positive number, so I believe I have answered the question.

Now it's time to answer the title of this thread. 10 - 9.999... = (1/infinity). Therefore, there is an infinitely small difference between 9.999... and 10. This difference may or may not exist, as the difference between the number and zero is the number itself. This means that:
(let x=(1/infinity))
x-0=y
y=x
therefore,
x-0=x
and x=x
This has proven to be thoroughly pointless. My conclusion is that while I can give a basic explanation of the smallest positive number, you need a math major to explain whether or not 9.999...=10 with any certainty.

I have a Master's in Mathematics. Your proof just shows that 0=0.

If you let x=1/(infinity) then x=0
(well really this should be stated as a limit, but I digress)

So then according to your proof:

0-0=y
0=0
therefore,
0-0=0
and 0=0

------------------------------------------------------------------------------------------

Spartaculous has covered why there is no smallest positive real number.

But I would also like to mention that in Mathematics once something is proven (yes, in very high level Math sometimes mistakes in proofs occur, but then someone will notice and it will eventually be disproven), then it is ironclad true. It's not a matter of opinion. Either something is true or it is false.

It is 100% true that 0.9999.....=1.0.

And this is not even a wacky result.

Another math result that truly is incomprehensible, but true (but I won't blame anyone for disputing): The infinite sum, 1 + 2 + 3 + 4+ ...... = -1/12.

That is, the sum of all Natural numbers is equivalent to -1/12.

[JustAGuyNamedWill]

Another math result that truly is incomprehensible, but true (but I won't blame anyone for disputing): The infinite sum, 1 + 2 + 3 + 4+ ...... = -1/12.

That is, the sum of all Natural numbers is equivalent to -1/12.

? Explain to us simpletons

[sweetandcool]

Another math result that truly is incomprehensible, but true (but I won't blame anyone for disputing): The infinite sum, 1 + 2 + 3 + 4+ ...... = -1/12.

That is, the sum of all Natural numbers is equivalent to -1/12.

? Explain to us simpletons

Heh, I don't understand the proof either, really.

I think the lesson to be learned here is that you ever encounter anything related to infinity, just mark your mental map with "Here be dragons" and just accept you might be facing results that don't make sense.

[JustAGuyNamedWill]

I think i found it

Ill let the smarter people figure this out lmao

[sweetandcool]

IMG_0679.jpeg

I think i found it

Ill let the smarter people figure this out lmao

Oh, okay, I guess there is an elementary proof. Very nice!

I'll explain it for you Will.

Line 1: Let S be the sum of all Natural numbers.

Line 2: Let Y be the alternating sum of all Natural numbers. We are going to use this to calculate S.

Line 3: Consider S - Y.

Line 4: Distribute the negative (or just consider S + (-Y)) and rearrange the terms of the sum S - Y. In doing so, a pattern becomes clear.

Line 5: S -Y = 4 + 8 + 12 + 16 + .....

Line 6: Since all terms of S - Y are divisible by 4, we can factor it out. So
S-Y=4(1+2+3+4+....)

Line 7: But wait! That's just 4 times the sum of all natural numbers! So really
S-Y=4S.

Line 8: Basic algebra yields S=-Y/3. Since it is known that Y=1/4 (an entirely separate proof), we get S=-(1/4)(1/3). We evaluate and arrive at our conclusion:

S= -1/12

[sweetandcool]

So really that proof is a bit of a cheat, because now you need to prove that Y=1/4, which I assume is an advanced proof. Or it cheats by utilizing the result of a different advanced proof, etcetera.

[JustAGuyNamedWill]

I think?? I understand? At least the concept.

Still is a weird answer tho lmao

[kingofthepirates]

-1/12 is really wacky. I believe the process to get it is called Ramanujan summation. I'm learning calc rn, so I don't really know how/why it works (my teacher has expressed distaste towards the subject when a friend of mine brought it up), though the logic of the proof seems consistent.

[JustAGuyNamedWill]

It just feels counterintuitive, even though the math (presumably) checks out.

Like, it should be impossible to add a positive number to another number and get less than what you started with, which is one.

[kingofthepirates]

yeah. numbers to wacky things. like you can uniquely map every number between 0 and 1 to the entire number line. crazy things lol.

[JustAGuyNamedWill]

Even funnier is the numbers are the most stable concept we’ve come across. Yet we still don’t understand them

[sweetandcool]

-1/12 is really wacky. I believe the process to get it is called Ramanujan summation. I'm learning calc rn, so I don't really know how/why it works (my teacher has expressed distaste towards the subject when a friend of mine brought it up), though the logic of the proof seems consistent.

Probably because the common sense answer is that it equals infinity. Off the top of my head I don't actually know much about the -1/12 result. I think it has some relation to zeta functions. It really is less of a matter of literally taking the sum of the Natural numbers and getting -1/12 and rather a matter of doing some funky things with some zeta function and the function spitting out -1/12 as an answer.

That is, it's essentially assigning the value of -1/12 to this divergent infinite sum, and it's a result that makes sense in the context of the function.

However, it doesn't really make sense from a layman's perspective.

Your teacher probably dislikes it because they either don't understand the math that led to that result (likely) or they hate that the result is being taken out of context (also likely).

[sweetandcool]

It just feels counterintuitive, even though the math (presumably) checks out.

Like, it should be impossible to add a positive number to another number and get less than what you started with, which is one.

Agreed, it makes sense to just use common sense. A positive integer plus a positive integer is always going to be a positive integer, so the result doesn't really make sense except for in the proper context.

The answer is infinity, but infinity is cool and likes to break the rules, so the answer is really -1/12 in the context of the function they were using to yield the result.

[Diplomacy&Warfare]

[snip]
I have a Master's in Mathematics. Your proof just shows that 0=0.

If you let x=1/(infinity) then x=0
(well really this should be stated as a limit, but I digress)

So then according to your proof:

0-0=y
0=0
therefore,
0-0=0
and 0=0

------------------------------------------------------------------------------------------

Spartaculous has covered why there is no smallest positive real number.

But I would also like to mention that in Mathematics once something is proven (yes, in very high level Math sometimes mistakes in proofs occur, but then someone will notice and it will eventually be disproven), then it is ironclad true. It's not a matter of opinion. Either something is true or it is false.

It is 100% true that 0.9999.....=1.0.

And this is not even a wacky result.

Another math result that truly is incomprehensible, but true (but I won't blame anyone for disputing): The infinite sum, 1 + 2 + 3 + 4+ ...... = -1/12.

That is, the sum of all Natural numbers is equivalent to -1/12.

I have no objection to you flexing your superior mathematical skills. However, can you at least take the time to read my post? I, myself, stated that the smallest positive number was not real. (1/infinity) is not a real number, in the same way that infinity, i, and (i/2) are not real numbers.

Additionally, you also clearly have not read the question I was answering:
Can someone explain that smallest positive number?
Nothing about this question indicated the number had to be a real number.

Yes, my proof ends up being something like x=x, and if x = 0 then it's just 0=0. However, the point of that particular proof was pointing out how useless a particular line of reasoning was.

Finally, you made the claim that 0.999999999... = 1 while implying I had disagreed. Which I had not. I am happy to defer that problem to you.

In conclusion, you failed to read the claim or reasoning you were disagreeing with, failed to read the question I was providing an answer to, failed to read what I was proving with my proofs, and "debunked" me by answering an unrelated question.

My claim, if you feel like actually debunking it, is:
(1/infinity) > 0

[sweetandcool]

Dip, I read what you wrote.

I'm not sure how to reply to you because you have contradict yourself in a few places and you say things that unfortunately just don't make sense.

Okay, so in your first paragraph here you claim that 1/infinity is not a real number. You give examples of i and I/2 not being real numbers. So are you claiming 1/infinity is complex number, and not a real number? Regardless, as I said before, 1/indinity is 0. So it is a real number, and also a complex number.

Paragraph two, I did answer the question you were answering. Spartaculous and I just showed you why there is no "smallest positive number". We answered the question.

Paragraph three I don't understand.

Paragraph four, is confusing because by claiming that there is a smallest positive number, particularly in the context of the discussion, implies that you don't believe that 0.9999..... =1.

Your conclusion is not incorrect, because I read everything. I'm sorry if your feelings are hurt. Just as I assume you read what I write, I would like you to believe that I read what you write

Your final claim is incorrect. 1/infinity=0. This is something you would learn in beginning calculus, though the concept could easily be taught in Algebra.