A very interesting Mathematical Paradox

[Dopefish]

I'm amazed and somewhat pleased to se that people who believed they weren't equal have been successfully converted thanks to the discussion on this thread.

Everyone I've tried to explain it to in person has always insisted that it's a flaw in math, since "any reasonably intelligent person with any amount of common sense knows they're different numbers!". Attempts to figure out what aspect makes them different, and also pointing out that every other number has multiple representations haven't lead anywhere unfortunately. At best I've apparently raised "an interesting philisophical point", but they continue to believe they're not equal.

[Mike_Bson]

This is my favorite proof:

0.999...=\lim_{x\to\infty}\frac{10^n-1}{10^n}=\lim_{x\to\infty}\frac{log(10)10^n}{log(10)10^n}=1

[Wnderer]

I've always seen parallels with this duality with the concept of +0 and -0.

1 - 0.999 = +0
0.9999 - 1 = -0

1/-0 = -infinity and 1/+0 = +infinity. In a sense it indicates that it matters whether a point is being approached from above or below. That there are two sides to all points and that +infinity and -infinity are the two sides of the same point and the curves in the tangent function are continuous. But I guess the utility is kind of limited since it only works at the integers and there are much more advanced concepts about infinity in mathematics.

[Ideas sleep furiously.]

Maths <> Logic.

Though they are closely related, maths is a human made concept and is subject to the same flaws as a human is.
Though there is no denying that this is brilliant.

Well... Either that, or more likely, there is no such thing as infinity.

[Goldstein]

1/-0 = -infinity and 1/+0 = +infinity. In a sense it indicates that it matters whether a point is being approached from above or below.

While it is meaningful in limit theory to talk about a value being approached, that's not what's going on here, as explained in Phlip's nice post on the previous page.

Also note that 0.999... is equal to 1, not 'slightly below', so the +0 and -0 you've defined are also equal to each other.

[Wnderer]

1/-0 = -infinity and 1/+0 = +infinity. In a sense it indicates that it matters whether a point is being approached from above or below.

While it is meaningful in limit theory to talk about a value being approached, that's not what's going on here, as explained in Phlip's nice post on the previous page.

Also note that 0.999... is equal to 1, not 'slightly below', so the +0 and -0 you've defined are also equal to each other.

Yes they are equal to each other because there can be no points in between, but the concept of a two sided point gets me the nice result that the tan(89.999...) = infinity and the tan(90) = -infinity and makes the tan curve continuous because infinity = -infinity. By which I mean, if we say the number line goes from -infinity to +infinity, then there are no points before -infinity and after +infinity and therefore no points in between +infinity and -infinity, so +infinity and -infinity are the same point. The tangent is looping through infinity, which is what the graph looks like to me. I know its unrigorous crackpot mathematics, buts its fun.

[mike-l]

But that makes perfect sense even without making 89.9999.. a different number than 90... tan of both of them is 'infinity', and you can make this rigorous by working on the projective plane.

[Wnderer]

But that makes perfect sense even without making 89.9999.. a different number than 90... tan of both of them is 'infinity', and you can make this rigorous by working on the projective plane.

I'm not trying to say that 89.999... is a different number than 90. I'm trying to say that all points in a two dimensional Cartesian graph have two sides that shows up in properties of how the point is approached. But I admit that using the x.999... to represent that concept is a bad idea. So I'll drop the suggestion. I'm not sure I understand the projective plane. I'll have to look into it.

[phlip]

But even if you did want to set up some kind of extended number system where each point had two "sides"... this wouldn't be the way to do it. For one, you can't do it to every number, only the rational numbers with terminating expansions (that is, the decadic rationals - ones where the denominator only has prime factors of 2 and 5). Non-terminating rationals and irrational numbers only have one decimal expansion. Which means that the set of points that's doubled up changes, depending on what base you're using (1/3 has one expansion in decimal, but two expansions in, say, base 12). A good number system behaves independently of its representation.

But say you did try to set up a more complete version, where every number has two sides... R x {+,−}, say... so for every real number, you now have two: eg (2, +) and (2, −). But then how do you define arithmetic on that? Presumably (2,+) + (2,+) = (4,+) and (2,−) + (2,−) = (4,−), but what's (2,+) + (2,−)? If it's either of (4,+) or (4,−), then you can't subtract... same way you can't divide by 0. If (2,+) + (2,+) = (4,+) and (2,+) + (2,−) = (4,+), for example, then (4,+) − (2,+) isn't well defined, because both (2,+) and (2,−) would fit.

One more complete treatment of the "infinitesimals" idea is the dual numbers, which instead of just having a + and a − for two different sides, it uses a whole set of real numbers for the infinitesimal part. It still has some problems compared to normal real numbers (eg you can't divide by an infinitesimal, even though it's non-zero) but it's still interesting to work with.

[Wnderer]

But even if you did want to set up some kind of extended number system where each point had two "sides"... this wouldn't be the way to do it. For one, you can't do it to every number, only the rational numbers with terminating expansions (that is, the decadic rationals - ones where the denominator only has prime factors of 2 and 5). Non-terminating rationals and irrational numbers only have one decimal expansion. Which means that the set of points that's doubled up changes, depending on what base you're using (1/3 has one expansion in decimal, but two expansions in, say, base 12). A good number system behaves independently of its representation.

Yes, you're right.

But say you did try to set up a more complete version, where every number has two sides... R x {+,−}, say... so for every real number, you now have two: eg (2, +) and (2, −). But then how do you define arithmetic on that? Presumably (2,+) + (2,+) = (4,+) and (2,−) + (2,−) = (4,−), but what's (2,+) + (2,−)? If it's either of (4,+) or (4,−), then you can't subtract... same way you can't divide by 0. If (2,+) + (2,+) = (4,+) and (2,+) + (2,−) = (4,+), for example, then (4,+) − (2,+) isn't well defined, because both (2,+) and (2,−) would fit.

For signed zero, they handle the case of sign cancellation through order of operations.
http://en.wikipedia.org/wiki/Signed_zero
(-0)-(+0)=-0
(+0)-(-0)=+0
I could do something similar. Do the math normally for the numbers and the side part would be determined by
(+)+(-) = (+)-(+) = (+)
(-)-(+) = (-)+(-) = (-)
so (2,+) + (2,-) = (4,+)

Basically I'm extending signed zero concepts to all numbers. I'm trying to replace tan(90) = NaN with tan((90,+)) = -inf, tan((90,-)) =+inf.
I don't like NaNs. Maybe some new NaNs will pop up.

[krucifi]

someone explained this concept to me when i was a kid. i think i was around 12 and i got the proof quite easily.

It was explained that 0.999... =1 because if they WEREN'T equal then
there must be a number between 0.999... and 1.

0.999...+1 = 1.999....

divide it by two and you get 0.999.... which is clearly not any different than our first number.
but that means 0.999...+ 1= 0.999.... x 2

so they MUST be equal.
QED.

[math]

Yeah... It's called ROUNDING. In reality, .999≈1 because you rounded in the 10x-x=9 because 10x would end its infinite string of 9's 1 earlier than plain x.

[phlip]

Sigh...

end its infinite string

No. You fail forever. Read the thread again.

[krucifi]

where in my argument did i round?
i think i proved it pretty straight forward.
and yes if by this point you cannot see how maths work,
then you fail at maths.

[Qaanol]

Keep having 9’s forever, and they never end. There is absolutely no end to the 9’s. Then, after the 9’s end, which they don't, add a 0. No, that’s silly.

[t1mm01994]

Sigh...

end its infinite string

No. You fail forever. Read the thread again.

In my sig this goes. I just love it <3

[krucifi]

Sigh...

end its infinite string

No. You fail forever. Read the thread again.

... the irony... his name is math...

[2 tanners]

Would it perhaps make it clearer to make this point:

Marrow is confused that the 1 "at the end" (just wait, guys) seems to vanish or be rounded away.

That 1 is an infinite number of places to the right of the decimal point, yes?

Therefore its value is 1/(10*infinity) = 1/infinity = 0, as was earlier conceded. It does not approximate 0, it IS 0.

Therefore this term (call it 'a') can be used as follows

Marrow dictum 1=.999... + a
=.999... + 0
=.999...

Did that help? It's still pretty cool.

[kansasdave]

Two numbers are equal if their difference is zero.

How does this apply when one of the numbers is something weird with infinite non-zero digits, like 0.99999....?

The way math guys usually deal with things like this is to say that 0.99999... is equal to 1 if, no matter how small a separation you give 'em, they can write enough digits of the 0.99999... number to get the difference between that and 1 inside your separation.

The idea is that, if these two numbers were really different, there would be some separation between them. Sure, it would be small, but it would be there, and it wouldn't change.

Of course, someone out there will say, "what if we make the separation an infinite number of zeroes followed by a 1?"

Well, that's zero.

;-P

[RebeccaRGB]

Of course, someone out there will say, "what if we make the separation an infinite number of zeroes followed by a 1?"

Keep having zeroes forever, and they never end. There is absolutely no end to the zeroes. Then, after the zeroes end, which they don't, add a 1. No, that's silly.

[lightvector]

Of course, someone out there will say, "what if we make the separation an infinite number of zeroes followed by a 1?"

Keep having zeroes forever, and they never end. There is absolutely no end to the zeroes. Then, after the zeroes end, which they don't, add a 1. No, that's silly.

It's not necessarily silly. One could imagine trying to define arithmetic on decimal sequences where the sequence is indexed, say, by the ordinals less than \varepsilon_0 rather than by the natural numbers.

http://en.wikipedia.org/wiki/Ordinal_number

Then, we can easily have infinite sequences, followed by values that come after that, followed by more infinite sequences, and more values afterwards, and so on. For instance, if we want an infinite number of zeros, followed by a 1, this is just the sequence where all terms are 0 except the \omegath term, which is 1. Of course, it might be hard to make the arithmetic have nice properties, but it *can* be done sensibly, if we are willing to step outside the real numbers.

[xkcdfan]

Then, we can easily have infinite sequences, followed by values that come after that, followed by more infinite sequences, and more values afterwards, and so on. For instance, if we want an infinite number of zeros, followed by a 1, this is just the sequence where all terms are 0 except the \omegath term, which is 1. Of course, it might be hard to make the arithmetic have nice properties, but it *can* be done sensibly, if we are willing to step outside the real numbers.

No, that actually doesn't make sense.

[Macbi]

Then, we can easily have infinite sequences, followed by values that come after that, followed by more infinite sequences, and more values afterwards, and so on. For instance, if we want an infinite number of zeros, followed by a 1, this is just the sequence where all terms are 0 except the \omegath term, which is 1. Of course, it might be hard to make the arithmetic have nice properties, but it *can* be done sensibly, if we are willing to step outside the real numbers.

No, that actually doesn't make sense.

Prove it.

[t1mm01994]

Then, we can easily have infinite sequences, followed by values that come after that, followed by more infinite sequences, and more values afterwards, and so on. For instance, if we want an infinite number of zeros, followed by a 1, this is just the sequence where all terms are 0 except the \omegath term, which is 1. Of course, it might be hard to make the arithmetic have nice properties, but it *can* be done sensibly, if we are willing to step outside the real numbers.

No, that actually doesn't make sense.

Prove it.

Only if you prove, without using the definition of 2, that 1+1=2.

[Turtlewing]

Then, we can easily have infinite sequences, followed by values that come after that, followed by more infinite sequences, and more values afterwards, and so on. For instance, if we want an infinite number of zeros, followed by a 1, this is just the sequence where all terms are 0 except the \omegath term, which is 1. Of course, it might be hard to make the arithmetic have nice properties, but it *can* be done sensibly, if we are willing to step outside the real numbers.

No, that actually doesn't make sense.

Prove it.

Only if you prove, without using the definition of 2, that 1+1=2.

Oh boy a redicilous clallange, let me:

from the deffenition of addition
1+1=2
Wow, that was easy.

Seriously though, the above number system doesn't make any lese sence than say: complex numbers. However it doesn't on first glance have much usefulness outside allowing people to talk about a set of infinite series seperated by zero or more finite series as a single number, which outside making "the 1 at the end of the ininate 9's" make some form of sematic sence probably isn't overly beneficial (as evidenced by the fact that it isn't used even though it would be easy enough for a compotent mathamatician to think it up if they needed to).

[phlip]

I think it would be hard to define operations on it, though... the traditional addition algorithm, for instance, relies on the fact that every digit has a predecessor, for the "carry" step... and not every ordinal has a predecessor. So what do you do if a digit in a limit ordinal position needs to carry? Do you have an integer part at every limit ordinal, as well as the fraction part that's expanded digit-by-digit over the successor ordinals... in which case what you've basically got is a sequence of real numbers (indexed by the ordinals) which are added/subtracted individually (a-la Rⁿ, except with "n" replaced with "all of the Ordinals"). Such a system might well be useful, but I don't see the reason to go one step further and index the decimal expansion of each of the reals...

[krucifi]

For all those who still don't believe the proof i hereby pose the ultimate question.
If 0.999... and 1 aren't equal then what number lies between them?

[jestingrabbit]

For all those who still don't believe the proof i hereby pose the ultimate question.
If 0.999... and 1 aren't equal then what number lies between them?

Its not in the reals, but people have been coming up with some stuff that could be made to work.

the traditional addition algorithm, for instance, relies on the fact that every digit has a predecessor, for the "carry" step

Yes. This makes me think that the smallest, natural place to take this idea would be the nonstandard natural numbers. They are totally ordered, with an order type ω + (ω* + ω) · η according to WP. So, you could talk about the (p/q, i)th decimal place, where p and q are positive, and i is an integer. They're ordered so that (p/q,i) is more significant than (r/s,j) if p/q > r/s or p/q=r/s and i>j.

But I would have no idea where to put the carried 1 in this sort of situation.

[mike-l]

You also need to make sure that whatever operations you define actually have 1 - 0.999.... = 0.000... 1. Also, what is 0.999... in this system? Are the digits beyond the naturals equal to 9? 0? Do you not just have the same problem when every digit in every (fractional) location is 9?

Has anyone actually shown that there is a reasonable definition of addition, multiplication, etc that has the usual properties and agrees with the usual notion in most cases (ie all 'ordinal' places are 0 and the 'real' part does not end in infinite 9s?)

[t1mm01994]

^This. So, you want to introduce a new "layer" of numbers, which comes right after the infinite layer of "normal" numbers. Well, we'll make the layer behind it all 9's too.. I guess this either has to be made impossible, which poses other problems, or we have to keep on adding layers, which I'll make all 9's too, until we have come to the infini'est layer, and we have defined a very sloppy way of doing mathematics.
I see infinitesimals having more potential than this system.

[snowyowl]

I still think it's convincing to just draw a line of length 1 and a line of length 0.999... and then ask people which one is which.

[mike-l]

I still think it's convincing to just draw a line of length 1 and a line of length 0.999... and then ask people which one is which.

That muddies the point... you're just convincing them that they are indistinguishable, but so are 1 and 0.999999999999999 (not repeating). These are different numbers, while 1 and 0.(9) are not.

[krucifi]

For all those who still don't believe the proof i hereby pose the ultimate question.
If 0.999... and 1 aren't equal then what number lies between them?

Its not in the reals, but people have been coming up with some stuff that could be made to work.

so are we thinking imaginary numbers? not the same as root negative one obviously.... or maybe... no it's not.
although that being said some concepts are hard to grasp like the concept of multiple infinities ( infinite actually)
but even though my knowledge of math is actually quite basic (bout to do a degree WOO HOO but no) i do enjoy
maths and get the concepts really easily. ordinals i managed to tackle but only scratched the surface of the implications.

but the idea of two numbers which seem different being the same doesnt really strike me as being mathematically unsound.
infact i like the idea of maths having these quirks and strange proofs. and this one isnt all that strange. it's completely valid but i do wish i could be explained better for those who genuinely dont get it.
especially the trailing zero's ending in a 1. if there is room for a 1 then there is room after it for more zero's completely defeating the purpose of the infinitesimal.

i like the idea of Hilbert's hotel. maybe that could help explain this. the idea of adding 1 to infinity makes no difference to its value if doing so was even possible... so why would adding a one to the end of an infinitesimal? bearing in mind one can't?

however a number (imaginary) inbetween 0.999... and 1 wouldn't disprove they were equal.

[xkcdfan]

Oh boy a redicilous clallange, let me:

from the deffenition of addition
1+1=2
Wow, that was easy.

Seriously though, the above number system doesn't make any lese sence than say: complex numbers. However it doesn't on first glance have much usefulness outside allowing people to talk about a set of infinite series seperated by zero or more finite series as a single number, which outside making "the 1 at the end of the ininate 9's" make some form of sematic sence probably isn't overly beneficial (as evidenced by the fact that it isn't used even though it would be easy enough for a compotent mathamatician to think it up if they needed to).

Oh boy a ridiculous challenge, let me:

from the definition of addition
1+1=2
Wow, that was easy. But wasn't actually a proof!

Seriously though, the above number system doesn't make any less sense than, say, complex numbers. However, it doesn't on first glance it doesn't have much usefulness outside of allowing people to talk about a set of infinite series separated by zero or more finite series as a single number, which, outside of making "the 1 at the end of the infinite 9's" make some form of semantic sense, probably isn't overly beneficial (as evidenced by the fact that it isn't used, even though it would be easy enough for a competent mathematician to think it up if they needed to).

[krucifi]

Oh boy a redicilous clallange, let me:

from the deffenition of addition
1+1=2
Wow, that was easy.

Seriously though, the above number system doesn't make any lese sence than say: complex numbers. However it doesn't on first glance have much usefulness outside allowing people to talk about a set of infinite series seperated by zero or more finite series as a single number, which outside making "the 1 at the end of the ininate 9's" make some form of sematic sence probably isn't overly beneficial (as evidenced by the fact that it isn't used even though it would be easy enough for a compotent mathamatician to think it up if they needed to).

Oh boy a ridiculous challenge, let me:

from the definition of addition
1+1=2
Wow, that was easy. But wasn't actually a proof!

Seriously though, the above number system doesn't make any less sense than, say, complex numbers. However, it doesn't on first glance it doesn't have much usefulness outside of allowing people to talk about a set of infinite series separated by zero or more finite series as a single number, which, outside of making "the 1 at the end of the infinite 9's" make some form of semantic sense, probably isn't overly beneficial (as evidenced by the fact that it isn't used, even though it would be easy enough for a competent mathematician to think it up if they needed to).

Lmao harsh but naughty naughty no red text you rebel

i suggest a nice magenta or perhaps king fisher blue for that retro look.

[Turtlewing]

Oh boy a redicilous clallange, let me:

from the deffenition of addition
1+1=2
Wow, that was easy.

Seriously though, the above number system doesn't make any lese sence than say: complex numbers. However it doesn't on first glance have much usefulness outside allowing people to talk about a set of infinite series seperated by zero or more finite series as a single number, which outside making "the 1 at the end of the ininate 9's" make some form of sematic sence probably isn't overly beneficial (as evidenced by the fact that it isn't used even though it would be easy enough for a compotent mathamatician to think it up if they needed to).

Oh boy a ridiculous challenge, let me:

from the definition of addition
1+1=2
Wow, that was easy. But wasn't actually a proof!

Seriously though, the above number system doesn't make any less sense than, say, complex numbers. However, it doesn't on first glance it doesn't have much usefulness outside of allowing people to talk about a set of infinite series separated by zero or more finite series as a single number, which, outside of making "the 1 at the end of the infinite 9's" make some form of semantic sense, probably isn't overly beneficial (as evidenced by the fact that it isn't used, even though it would be easy enough for a competent mathematician to think it up if they needed to).

Thanks for the editing, as long as people like you exist I never have to do it myself

[xkcdfan]

retarded comment

spell-checked version of quoted comment. fixing the spelling didn't make it any less retarded.

Thanks for the editing, as long as people like you exist I never have to do it myself

So do you like not understand what "prove" means or something

[krucifi]

retarded comment

spell-checked version of quoted comment. fixing the spelling didn't make it any less retarded.

Thanks for the editing, as long as people like you exist I never have to do it myself

So do you like not understand what "prove" means or something

Ooo me oo me me oo me sir please sir pick me i will prove it
but i will need the proof that 1=0.999.... first is that ok?

[HarvesteR]

It seems to me that most of the people who are having trouble with this concept are essentially having a problem understanding infinity itself.
Let me try to make it more clear (this is by no means an attempt to prove anything... I'm just trying to word it differently to explain it from a different angle that might make more sense to some people)...

0.999... is not the same as 0.999

So 9.999... - 9 equals 1, and 9.999 - 9 equals 0.999

At some point, whoever is trying to refute this concept is inevitably trying to put an end to the infinite series (hence the very comical nested quote a few posts above) and say that .999... at some point will have an end.

Infinity is surprisingly hard to explain... most people will get stuck trying to visualize it, and inevitably fail because visualising a number implies seeing it's extent, and that in itself goes against the concept of infinity. Just try to think of infinity as a convention, instead of a number.

The .999... = 1 concept works and ONLY works if there is an infinite number of trailing 9's. In any other case the result will be some very long finite sequence of decimal places

For those also wondering about if 0.333... = 0.4 is true by the same principle, that's not the case.
The correct form would be 0.3999... = 0.4

I hope this helps a little...

Cheers

[Turtlewing]

retarded comment

spell-checked version of quoted comment. fixing the spelling didn't make it any less retarded.

Thanks for the editing, as long as people like you exist I never have to do it myself

So do you like not understand what "prove" means or something

No, actually my not proof was intended to be a stupid, pointless not proof (I figured this was implied, however you seemd to feel it needed to be made explicit and i deffered to your aparent expeience as an amature editer). In short it was a stupid answer to a stupid question.

1+1=2 is part of the defenition of addition (when applied to most common number systems). Asking to prove 1+1=2 without using the deffenition of 2 is asking for something like:

let there be a thing
let there be another thing
by the defenition of addition there are now more than 1 things, and less than 3 things. let's call this case "there are 2 things".
(prety stupid and useless, but to drive home the stupid and useless I made it seems even less formal and well thought out in the original post)

Asking for a proof of 1+1=2 without using the defenition of _addition_ on the other hand is actually "prove addition for the natural numbers". which is still pretty pointless since the proof of addition is basicly just a restatment of the defenition of counting, and counting is generally held to be a fundamental postulate that is well verified by observation. So the proof will likley involve a practical demonstration of counting, but at least this question is driving that something that is worthwile to do every now and then (questioning your assumptions).

I had hoped my stupid not-proof would draw attention to the distinction between questioning fundamental assumptions (which I had hoped was the point of the original challange to prove 1+1=2) and issuing challanges that are poorly formulated and thus meaningless to complete.

However back on topic:
The problem people generally have with the .99...=1 issue seems largely to stem from people being used to rounding numbers and treating them as equivilent even if they know that not to be the case. That's why most people first comment about rounding and the numbers being "effectively" the same. Then when they're told that's not true the are actually the same, just written differently they get defencive and try to imagin a difference.

I know in may imaginings I tend to think of "infinate 9s" as an ever growing list of 9's. Thus it's easy to imagin an ever growing list of 9's followed by a 9. ie the set is expanding in the middel not being added to on the end. Once I've made that mistake the "infinate 0's followed by a 1" doesn't seem nearly so stupid. However it's based on the fallacy that Infinity is a "growing" set.

The demonstration which convinced me (many years ago) was the one where you demonstarte:
1/3 = .333...
1/3+1/3+1/3 = 3/3 = 1
so by substitution:
.333... + .333... + .333... = 1
simplify using unrounded addition on every decimal place to:
.999... = 1
(remember you aren't rounding so every one of those infinate decimal places is a 3+3+3=9)

It's not a proof so much as a way of explaining it that can work with my (ultimately innacurate, but intuatively useful) visualization of infinity.