0.99999999.......

[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.

[J. J.]

     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.

[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.

[J. J.]

     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

[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.

[muon2]

Your example is consistent with finite mathematics, but there is a definitional difference when dealing with real algebra. The root problem is in considering decimal expansions of a value using the ellipsis notation. That ellipsis really doesn't work in finite math, but allows one to take rational expressions like 1/3 and write it as a decimal value 0.333 ... . Real numbers which include rational and irrational numbers are defined in terms of infinite sequences and are incompatible with the definitions of finite math. In a sense one is converting the rational value 1/3 into a real analog 0.333 ... .

The ancient Greeks recognized that finite math and rational expressions could not explain all the numeric values they knew. Though they could prove it to be so, their number system lacked the tools to express these irrational values. The Arabic number system eventually provided those tools, which led to decimal expansions of all real numbers, rational and irrational. Yet even with this powerful tool there are cases where a finite rational expression like 1 will do fine, even though 0.999 ... is equal as a real number.

The definitional problem you cite exists, but is solved by adopting the definitions of real algebra. If you confine your math to the finite and rational you can avoid the problem. But like the ancient Greeks, you'll be in a bind if you want to express irrational numbers in a manner consistent with rational numbers, eg. to be able to express both 1/3 and the square root of 2 in decimal notation. The definition that takes care of that inconsistency creates the question raised by this thread.

No, I'm saying that 1 = 1, when one is used to represent a "threshold number."  In case where the numeral "1" (or any other numeral, including 1/3 or square root of 2) is used to represent a concept involving a minimum amount, as it can be, "1" cannot be an infinitesimal amount smaller than one.

When you talk about a threshold number, such as your fission example for PiT, you are talking about the use of numbers for measurement. Measurements are never made with infinite precision, so real numbers would not be used, a terminating decimal is just fine. Even though the theoretical ratio of a a circle's area to it's diameter is pi, we could not measure it to be exactly pi. I can't measure anything with a value of 0.999 ...  or even 0.333 ..., so I wouldn't use them for an experimental threshold.

Real numbers are a concept with their own definition. When one speaks of 1 as a real number like pi it is well-defined, but not in a way that is useful for measurement such as with a threshold.

[J. J.]

When you talk about a threshold number, such as your fission example for PiT, you are talking about the use of numbers for measurement. Measurements are never made with infinite precision, so real numbers would not be used, a terminating decimal is just fine. Even though the theoretical ratio of a a circle's area to it's diameter is pi, we could not measure it to be exactly pi. I can't measure anything with a value of 0.999 ...  or even 0.333 ..., so I wouldn't use them for an experimental threshold.

But, by saying 1 = 0.999...., you are measuring something, either 1 or 0.999..... 

I am saying 1 = 0.999 ...., except when is 1 is used as a threshold value.  And 1 can be a threshold value.

To put it this way, x is a threshold value, designated M.  For some condition to be met, the number must be equal to or greater than x.  M => x

Under your suggestion x = 0.999... x  Any number, even infinitely smaller than x cannot equal M.

So x could equal 0.999 ...x, provided that x does not equal M.  Because of the nature of numbers, and how we use them, that caveat is important.