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Crazedwraith wrote:I posted that and every one ignored me so i'll post it again.

That's because it's not a complete proof. The most it can prove is a conditional: if x = 0.999... is a real number, then x = 1. Assuring that x is actually a real number is important, and should not be ignored.

Cecil Adams says technically, yes. And the World's Smartest Man is never wrong.

FORTY-TWO!!

Sorry, it had to be said.. I say there is a difference, though. And infinite decimal, by definition, can't be the number it is rounded up to.

~ver

verilon wrote:FORTY-TWO!! :mrgreen:

Sorry, it had to be said.. I say there is a difference, though. And infinite decimal, by definition, can't be the number it is rounded up to.

What is the numerical value of the difference? Why can't we just say that every terminating decimal also has a non-terminating representation?

Kuroneko wrote:

verilon wrote:FORTY-TWO!!

Sorry, it had to be said.. I say there is a difference, though. And infinite decimal, by definition, can't be the number it is rounded up to.

What is the numerical value of the difference? Why can't we just say that every terminating decimal also has a non-terminating representation?

Because that's a contradiction. Two numbers cannot be variables that represent each other. By definition and by grammar both, that statement is paradoxical, and therefore cannot exist. .999~ only approaches 1, as Wong said, and therefore will never reach 1, in the same way a tangent curve can approach 0 on the y-axis, but will never reach it.

~ver

verilon wrote:Because that's a contradiction. Two numbers cannot be variables that represent each other. By definition and by grammar both, that statement is paradoxical, and therefore cannot exist. .999~ only approaches 1, as Wong said, and therefore will never reach 1, in the same way a tangent curve can approach 0 on the y-axis, but will never reach it.

There are no variables here. Whether I refer to one as 1 or 2/2 or whatnot makes no difference. They are all different representations of the same number, and it does not matter whether I refer it as 0.999 either (I'll use an underline to denote repetition from now on--an overline is usually used, but I cannot type that). Explain to me why decimal representations absolutely must be unique.

verilon, it's a bit futile to argue against the rules of math. Writting 0.9(9) or 1 is precisely the same thing. Just check a Calculus book (ask for Apostol in a library, for a good one).

edit: Fuck me, it seems this entire thread is about the subject. Clearly, I'm not paying enough attention. Why is this even a matter of discussion?

Colonel Olrik wrote:verilon, it's a bit futile to argue against the rules of math. Writting 0.9(9) or 1 is precisely the same thing. Just check a Calculus book (ask for Apostol in a library, for a good one).

The RULES OF MATH state that a DECIMAL and a WHOLE NUMBER cannot be the same.

~ver

verilon wrote:

Colonel Olrik wrote:verilon, it's a bit futile to argue against the rules of math. Writting 0.9(9) or 1 is precisely the same thing. Just check a Calculus book (ask for Apostol in a library, for a good one).

The RULES OF MATH state that a DECIMAL and a WHOLE NUMBER cannot be the same.

Whole numbers are a subset of Fractionay numbers.

verilon wrote:The RULES OF MATH state that a DECIMAL and a WHOLE NUMBER cannot be the same.

They state no such thing. Did you make this up?

Given the accuracy of instruments like a speedometer measuring a car's speed to the closest mile an hour and so forth, it can never be infinitely accurate, so the decimal is simply rounded off when it reaches a certain arbitrary limit. Or so I'm lead to believe.

Admiral Valdemar wrote:Given the accuracy of instruments like a speedometer measuring a car's speed to the closest mile an hour and so forth, it can never be infinitely accurate, so the decimal is simply rounded off when it reaches a certain arbitrary limit. Or so I'm lead to believe.

Physical possibility is irrelevant to mathematical questions.

Kuroneko wrote:

Admiral Valdemar wrote:Given the accuracy of instruments like a speedometer measuring a car's speed to the closest mile an hour and so forth, it can never be infinitely accurate, so the decimal is simply rounded off when it reaches a certain arbitrary limit. Or so I'm lead to believe.

Physical possibility is irrelevant to mathematical questions.

That dawned on me, but I just wanted to say something.

So given in mathematical terms infinity does exist and in fact there are many types of infinity (we don't even truly know the biggest one) then does this not seem folly?

Kuroneko wrote:

verilon wrote:The RULES OF MATH state that a DECIMAL and a WHOLE NUMBER cannot be the same.

They state no such thing. Did you make this up?

Whole numbers: 0,1,2,3,4,5,....
Integers: -3,-2,-1,0,1,2,3...

I meant to say that a decimal number that is not a whole number or integer cannot be a whole number or integer and a finite decimal cannot be the same as an infinite decimal by all laws of reasoning and reality. A PARADOX CANNOT EXIST, THEREFORE .999 CANNOT EQUAL 1.

And in terms of paradox, I do not mean PHYSICAL possibility - I mean by possibility IN GENERAL.

~ver

Colonel Olrik wrote:

verilon wrote:

Colonel Olrik wrote:verilon, it's a bit futile to argue against the rules of math. Writting 0.9(9) or 1 is precisely the same thing. Just check a Calculus book (ask for Apostol in a library, for a good one).

The RULES OF MATH state that a DECIMAL and a WHOLE NUMBER cannot be the same.

Whole numbers are a subset of Fractionay numbers.

My mistake on that.. see above post; I worded what I was saying wrong.

[EDIT] Also, .999 < 1, so .999 CANNOT equal 1.

~ver

verilon wrote:

Colonel Olrik wrote:

verilon wrote: The RULES OF MATH state that a DECIMAL and a WHOLE NUMBER cannot be the same.

Whole numbers are a subset of Fractionay numbers.

My mistake on that.. see above post; I worded what I was saying wrong.

[EDIT] Also, .999 < 1, so .999 CANNOT equal 1.

~ver

But if it goes on for an infinite number of significant figures, does that not equal one given we are talking theoretical maths?

Admiral Valdemar wrote:But if it goes on for an infinite number of significant figures, does that not equal one given we are talking theoretical maths?

How can it, if it only approaches 1?

~ver

verilon wrote:How can it, if it only approaches 1?

~ver

Someone explained it before. Infinity is the theoretical "number" that is larger than all other numbers. 1/infinity is the number smaller then all other numbers. If 0.999~ goes on for infinity, so that its infinitely close to 1, and no number is greater then it, then it MUST equal one, because its the number that is greater then all other numbers under one. The only number that fits this definition is one itself.

kojikun wrote:

verilon wrote:How can it, if it only approaches 1?

~ver

Someone explained it before. Infinity is the theoretical "number" that is larger than all other numbers. 1/infinity is the number smaller then all other numbers. If 0.999~ goes on for infinity, so that its infinitely close to 1, and no number is greater then it, then it MUST equal one, because its the number that is greater then all other numbers under one. The only number that fits this definition is one itself.

But if it's infinite, it can't equal 1, because 1 is finite.

~ver

verilon wrote:But if it's infinite, it can't equal 1, because 1 is finite.

~ver

Not true at all. The number CONTINUES to infinity, and can be analysed as such, but that does not mean it does not have an actual finite value. Because the more you add to it, the closer it gets to 1, if you continue adding INFINITELY, it becomes one AT infinity. It also becomes anything else, depending on the limit of the figure.

Let me put it another way, if you have 1.000~ that has an infinite number of zeros after it, is that in any way different than 1? No. It continues forever, Just the same, I can write 360/7 and that is a finite number, but in decimal form it repeats indefinitely.

Colonel Olrik wrote:Why is this even a matter of discussion?

Because I'm an evil, evil person.

verilon wrote:

Kuroneko wrote:

verilon wrote:The RULES OF MATH state that a DECIMAL and a WHOLE NUMBER cannot be the same.

They state no such thing. Did you make this up?

Whole numbers: 0,1,2,3,4,5,....
Integers: -3,-2,-1,0,1,2,3...

I meant to say that a decimal number that is not a whole number or integer cannot be a whole number or integer and a finite decimal cannot be the same as an infinite decimal by all laws of reasoning and reality.

Wrong. A non-terminating (`infinite') decimal can represent a whole number. You're simply begging the question here. Prove it cannot do so.

verilon wrote:Also, .999 < 1, so .999 CANNOT equal 1.

If x = 0.999 < 1, then 1 - x > 0. Then, I can pick some y such that x < y < 1 (for example, the halfway point y = (x+1)/2). But, if so, then y does not have any decimal expansion--if y = 0.a1a2a3... (an's being digits of y), then if there is a digit ak < 9, then y < x, a contradiction. But if every digit is 9, then y = x, also a contradiction.

Therefore, we are forced into to conclude one of the three:
1. x = 0.999 = 1, (there is no y between x and 1; the reals are continuous), or
2. There are some real numbers that have no midpoint between them. (there is no y; the reals are not continuous)
3. There exist real numbers that have no decimal representation whatsoever. (There is some y between x and 1.)

Which conclusion do you pick?


Edit: overlooked possibility added.

verilon wrote:

Kuroneko wrote:

verilon wrote:FORTY-TWO!!

Sorry, it had to be said.. I say there is a difference, though. And infinite decimal, by definition, can't be the number it is rounded up to.

What is the numerical value of the difference? Why can't we just say that every terminating decimal also has a non-terminating representation?

Because that's a contradiction. Two numbers cannot be variables that represent each other. By definition and by grammar both, that statement is paradoxical, and therefore cannot exist. .999~ only approaches 1, as Wong said, and therefore will never reach 1, in the same way a tangent curve can approach 0 on the y-axis, but will never reach it.

~ver

The problem, ver, is that 1 and .9999... are not different numbers. They are the same number. Once again I would point you back to the definition of a unique (real) number. Basically if two numbers are unique then there exists a number which is also unique that can be placed inbetween the first two (i.e. if x=1 and y=.9999... then they are not qual if a number z exists such that y,z<x. However no such number z exists and thus x and y are the same number).

One and point nine repeated infinitely are non unique numbers, they are the same thing.

Mathematicians do not agree with your illogic, verilon.

0.999~ = 1, due to a technicallity.

Not to put a fine point on it, verilon, but, put simply, 0.999 is a whole number.