0.99999999.......

[Associate Justice PiT]

     Yes. As BRTD said, the math behind it is pretty clear-cut.

[Associate Justice PiT]

It's certainly very close to 1, and if I was using it in some mathematical equation and didn't need to be too precise, I'd put it down as 1, but in the end though 0.9999999... ≈ 1, at the same time 0.9999999... ≠ 1 either.

It's infinitely close to 1, however, making it mathematically equal.

Quite frankly I fail to see how it's equal to 1, indeed it's very close, but just by looking at it you can see that it's not exactly equal.

Fun isn't it?

f(x) = 1 / x², a lot of people would say, never touches the x-axis, but only gets closer and closer to it.

It does, though, in theory. Under the same principle as the original question here, the function is considered to be infinitely long, and thus it also becomes infinitely close to the x-axis

     It all goes back to Zeno's paradox, really. Achilles races with a tortoise. Achilles starts at point A while the tortoise starts at a point B further ahead. By the time he reaches point B, the tortoise has reached point C. By the time Achilles reaches point C, the tortoise has reached point D, & so forth. The implication is that Achilles never reaches the tortoise, but we know that would be false. As such, we realize that at x=infinity, the graph of f(x) = 1/x² does in fact equal 0. I find limits to be rather fascinating, really.

[Associate Justice PiT]

I do not dispute that 0.9999999... is indeed extremely close to one, but so say that it equals one make absolutely no sense whatsoever (but when has mathematics ever done that).  The difference will indeed be very (understatement) small, incalculable in fact, but no matter how much I look at it and think about it, it still strikes me as less than 1.  That said, if I don't need to be too precise when displaying the results of a calculation, I'll just treat it as one all the same (considering that when rounded you get 1), but I'll never be able to view it as equal to one though, that is, exactly equal.  I will look into this further.

OK, despite Libertas extremely nasty behavior....his math isn't wrong.

We agree that any number divided by itself equals 1, right? That can't be disputed! So 3/3=1.

Alright, now if you had to write 1/3 as a decimal number, what would it be? Is it 0.33? or 0.(lots of 3s?). It's clear that it's an infinite number of 3s behind the decimal point, correct?

But now if you multiply that number by 3, what do you get? wouldn't that be 0.333.... x 3 = 0.999...?


To argue that 0.999... is not equal to 1, you would have to dispute that 3 x 1/3 is equal to 1

Even though it has moments like this that just don't make any sense on the outside, I actually quite enjoy reading about mathematics.


On the outside, 0.999... and 1 look completely different to each other, and would be seen as not exactly but certainly nigh on equal to each other.  When I first go into the subject, I knew it wouldn't be consistent, but there are something that really do take the cake.  Makes me wonder what it'd be like if we used base 6 numbering instead, for one thing.

I can indeed see where you're coming from here, I'd put most certainly 3 multiplied by 1/3 as equal to 1, such as when you take a circle, slice it into three equal slices and put them together again you get 1 circle.  So, 3 by 1/3 = 1, no argument from me there.  Tough one.  Anyway, you do indeed have a point there, unlike Libertas, who just trolls.  0.999..., weird number, cosmetically unequal to 1, yet for the purposes of most equations the former's treated just like the latter.

     Well what happens is that the difference is infinitely small. As we know, 1/infinity=0. Because of that, any two numbers where the distance in between them is infinitely small are equal, because an infinitely small number is just equal to zero.

[Associate Justice PiT]

To your comment to BRTD, there cannot not be a "last" 9. The nature of an infinite series is that it never ends, therefore it is meaningless to say there is a last value in the sequence. It would be just as meaningless to ask for the largest integer. There is always another one higher.

That's my point really. If you have an infinite number of 9s, you can't have an 8 after them.

     As muon suggests, if you could have that then you could have a largest integer after an infinite number of smaller ones. Maybe it's not totally counterintuitive, but formally defining it would be impossible.

[Associate Justice PiT]

If you needed ten billion dollars, and had $9,999,999,999.00 do you have ten billion dollars?

I think the answer "infinitely close to one in the base ten system" might be the best.

One problem I could see is looking at a situation where X has to be a number equal to or greater than one, X >/= 1.  0.9999999999... would not be greater than or equal to x in that case.

     It would be more like if you had $9,999,999,999.99999.... The problem with your example is that you are short by an amount that is defined as $1, a non-zero value. In the topic's question, 0.99999999... is short by an infinitely small amount, which is mathematically defined as 0. You see what I mean?

[Associate Justice PiT]

If you needed ten billion dollars, and had $9,999,999,999.00 do you have ten billion dollars?

I think the answer "infinitely close to one in the base ten system" might be the best.

One problem I could see is looking at a situation where X has to be a number equal to or greater than one, X >/= 1.  0.9999999999... would not be greater than or equal to x in that case.

     It would be more like if you had $9,999,999,999.99999.... The problem with your example is that you are short by an amount that is defined as $1, a non-zero value. In the topic's question, 0.99999999... is short by an infinitely small amount, which is mathematically defined as 0. You see what I mean?

I question if yo can really define this as zero, anymore than can define it as 42. 

For example: X is defined the amount of molecules of a substance needed for a reaction, and the amount of X needed is 1 unit.  0.999999999... units would not be enough.

So long as X isn't a minimum amount, the definition would work.  As soon as it becomes a minimum value, there is a definitional problem.

     If you figure that any number can be subdivided into infinitely small units, then you run into the problem of Xeno's Paradoxes, in that the runner can never catch the tortoise in that he must catch up to the tortoise at an infinite number of points before he can actually reach the tortoise; in other words, the graph of the distance between the runner & the tortoise approaches the x-axis, but never reaches it.

     As such we have the concept of limits. Experience dictates to use that the runner will eventually pass the tortoise, but for him to do so requires that the graph of the distance reaches the x-axis, or otherwise stated that he catches up to the tortoise. The idea is that x != 0 within a finite number of terms, but it does equal zero in an infinite number of terms. In application to this problem, 0.9999999... is defined by the series 9/10 + 99/100 + 999/1000..., which converges to 1 at inifinity.

     That's my shot at it, though muon2 can explain it better than I can.

[Associate Justice PiT]

     If you figure that any number can be subdivided into infinitely small units, then you run into the problem of Xeno's Paradoxes, in that the runner can never catch the tortoise in that he must catch up to the tortoise at an infinite number of points before he can actually reach the tortoise; in other words, the graph of the distance between the runner & the tortoise approaches the x-axis, but never reaches it.

     As such we have the concept of limits. Experience dictates to use that the runner will eventually pass the tortoise, but for him to do so requires that the graph of the distance reaches the x-axis, or otherwise stated that he catches up to the tortoise. The idea is that x != 0 within a finite number of terms, but it does equal zero in an infinite number of terms. In application to this problem, 0.9999999... is defined by the series 9/10 + 99/100 + 999/1000..., which converges to 1 at inifinity.

     That's my shot at it, though muon2 can explain it better than I can.

Actually, this is not a Zeno's Paradox situation.  It is how we use numbers.

We can say that for something to be X, it must have a greater than or equal to value than a specific number.

To oversimplify this, and reveal my lack of understand nuclear fission, at a specific temperature, pressure, shape and I'm sure some other things, let's say that substance Z has a critical mass of 1kg for a chain reaction.  Exactly one kg.  We could say that  the critical mass of Z is k.

You can say Zk >= 1.

Under your definition, you are saying, 0.999... equals 1.  1 = 0.999... 

If Zk >= 1, and 1 = 0.999 then we see that Zk does not equal 1 but an infinitesimal amount smaller than one.

Logically, Zk can be equal to 1.  This statement is then true:  Zk = 1.

You claim 1 can equal  an infinitesimal amount smaller than one.  1 = 0.999...

Therefor Z, which has a critical mass of one can have a critical mass of an infinitesimal amount smaller than one.  Zk = 0.999...  That statement does not follow Zk = 1 or Zk > = 1.

In other words, when 1 is a "threshold number," i.e. you need at least 1, 1 cannot equal an infinitesimal amount smaller than one.

The problem is, we can use a number as a threshold number.  In physics you use threshold numbers with things like critical mass.  In terms of legislative procedure, you use it with things like a 2/3 vote.

I think it is more of a definitional problem.

     A fair point, though I'm not sure you would ever come up an infinitesimal short in an instance where a threshold number is needed.

[Associate Justice PiT]

     A fair point, though I'm not sure you would ever come up an infinitesimal short in an instance where a threshold number is needed.

I've actually seen some things like this is calculating 2/3 votes and quorums. 

You are basically saying N is any number, and that P is an infinitesimally small amount.  You are saying that N = N-P.  Is that correct?

If so, I'm saying that N cannot be a number that is a threshold.

For example, some parliamentary manuals define a majority vote (M) as any number that is more than half of the legitimate votes cast (V).  I could define that as M => V/2 + P.  M is a threshold number (and it can be repeating in some unusual circumstances).   A bare majority (M₁), in theory, could in theory, equal half of the legitimate votes cast (V) plus an infinitesimally small (P) amount.  M₁ = V/2 + P

M₁ is a number. 

If M₁ is any number (N), then M₁ = N. 

So M₁ = M₁ - P. 

If that would be the case M₁ - P = V/2 + P or M₁ = V/2.  V/2 is not a majority. 

Therefor, M₁ cannot equal N, any number.  Of course M₁ is a number (N).

I submit that N cannot be a threshold number.  If you want to describe a threshold or minimum number M₀.  N = N - P, provided N is greater than M₀.

     Ah, I see what you mean. I hadn't thought of that, since I'm accustomed to a majority being defined as 50%+1 vote, though 2/3rds or 3/5ths of votes are examples that are not uncommon in the United States. So yes, it makes sense that 0.999999... equalling 1 is not a proper definition in certain contexts.

[Associate Justice PiT]

     Ah, I see what you mean. I hadn't thought of that, since I'm accustomed to a majority being defined as 50%+1 vote, though 2/3rds or 3/5ths of votes are examples that are not uncommon in the United States. So yes, it makes sense that 0.999999... equalling 1 is not a proper definition in certain contexts.

Even with some of those supermajorities, you have the same type of problem (I think the 2/3 vote threshold would be one where this could commonly occur).

If M₀ represents a minimum value I could see this definition:

N = N - P > M₀

     My point exactly. Votes of those types are examples where defining 0.999999... as 1 falls flat.