Is 0.999... Truly Equal to 1?

[Integral]

You are absolutly correct, the extended Reals are not a field because of the defined properties of infinity.

 

[Hurkyl]

nd this "unique positive integer" your talking about is exactly what shows up in the number itself on paper. 0.9999~ shows the 9999~ corresponding to a ZERO. If it was truly 1, then that number before the decimal place would NOT be zero.

You entirely misunderstood.

The correspondense is that there is a 1st digit, a 2nd digit, a 3rd digit, and so on; there is an n-th digit for each positive integer n.

Furthermore, every digit can be labelled in this way; for every digit to the right of the decimal place, there is a positive integer n for which you can say that it's the n-th digit.

0.9~ is a sequence such that for each integer n, the n-th digit to the right of the decimal point is a 9.


And, as a sequence of digits, 0.9~ is, indeed, inequal to 1. Furthermore, 0.9~ < 1. according to the lexical ordering of sequences of digits. But 0.9~, as a decimal number, is equal to 1..

 

[e(ho0n3]

I don't know if this has been said already, but the reason I think of why a lot of people don't 'buy' that 0.999... = 1 is simply because these two numbers are syntactically different. In other words, if the numbers don't look the same then they are not the same. I think people should just realize that numbers aren't as straightforward as they think they are. So far nobody has given a convincing logical argument that 0.999... and 1 are different numbers so why is this thread getting so large.

 

[Hurkyl]

Yet, oddly, nobody seems to have any trouble accepting that $\frac{4}{3} = \frac{8}{6} = 1\frac{1}{3}$, and most people can understand that $1 = 13 \mod 12$. I still haven't understood why so many have trouble with $0.\bar{9} = 1$.

 

[Erazman]

alright I've changed sides...
makes more sense now

 

[steersman]

Yet, oddly, nobody seems to have any trouble accepting that , and most people can understand that . I still haven't understood why so many have trouble with

1 and 1/3 are different forms of representation. The example you gave is correct, 1/3 is predicated on there is a 1/3 that can be found and definitely measured. It is a potential representation. This is a predicate in mathematics not reality. The case is different with 0.9~=1. The number 1 is a reality representation.

See, the thing is 0.9 maybe the same as 0.9~. It just depends on how far you are willing to measure. So in a sense 0.92 is more than 0.9 in the way you are justifying your case, despite this being just a matter of measurement.

It is not possible to travel at the speed of light, why is this? because you must expend an infinite amount of energy - which is not possible, because actual infinities do not exist. You would need an actual infinity in 0.9~=1.

This is a philosophical problem not a mathematical problem. It is to do with the philosophical meaning of infinity not any mathematical definition of it.

It's like saying: well if this were possible then this would equal 1. But its not possible.

 

[Hurkyl]

Measurement? Reality? Where did you get any of this stuff?


0.9~ does not depend on any choices whatsoever. It has a 9 in the n-th place for ALL positive integers n, and there exist no other places to the right of the decimal places.

It is not formed by starting with 0. and adding 9's one by one.

 

[steersman]

It is not formed by starting with 0. and adding 9's one by one.

Who said it was?

 

[Hurkyl]

You sort of implied it with "See, the thing is 0.9 maybe the same as 0.9~. It just depends on how far you are willing to measure."

 

[steersman]

You sort of implied it with "See, the thing is 0.9 maybe the same as 0.9~. It just depends on how far you are willing to measure."

No no. The point I was making was that every number that isn't whole is already (potentially) infinite in terms of the infinite regress involved when trying to quantize something.

 

[Integral]

1 and 1/3 are different forms of representation. The example you gave is correct, 1/3 is predicated on there is a 1/3 that can be found and definitely measured. It is a potential representation. This is a predicate in mathematics not reality.

The sentence I have bolded makes no sense to me, could you provide a translation into relevant language?

The case is different with 0.9~=1. The number 1 is a reality representation.

Just what is a reality representation? Is there a mathematical definition for that?

See, the thing is 0.9 maybe the same as 0.9~. It just depends on how far you are willing to measure. [\quote]
I don't care how far you measure, as long as we are talking bout Real numbers 0.9 in NEVER = .9~

So in a sense 0.92 is more than 0.9 in the way you are justifying your case, despite this being just a matter of measurement.

Could you please demonstrate a "sense" were 0.92 is NOT greater then 0.9?

It is not possible to travel at the speed of light, why is this? because you must expend an infinite amount of energy - which is not possible, because actual infinities do not exist.

Wrong thread, discuss Relativity and the speed of light in the correct forum. (BTW you are not exactly on the mark with that statement either!) This is logic applied to Math, no Physics needed or wanted.

You would need an actual infinity in 0.9~=1.

Just so happens that Math HAS a definition of infinity and it means that 0.9~ =1. Perhaps if you knew even the basics of Real Analysis you would have known this.

This is a philosophical problem not a mathematical problem. It is to do with the philosophical meaning of infinity not any mathematical definition of it.

You could not be more wrong. This is a property of the Real Number system, This system has been very carefully constructed on well known and understood axioms followed by careful and through proofs of each and every theorem.

It's like saying: well if this were possible then this would equal 1. But its not possible.

News to me, and every mathematician in the world, perhaps you know something we don't or... Just maybe...

You don't know a LOT that Mathematicians DO KNOW.

 

[steersman]

You could not be more wrong. This is a property of the Real Number system, This system has been very carefully constructed on well known and understood axioms followed by careful and through proofs of each and every theorem.

Maths has limited utility in answering this problem because it's answer does not correspond with reality. That's why I call it a philosophical problem. I do concede though that in maths 0.9~=1. But if that's all you care about then you'll never learn anything.

 

[Integral]

The problem has meaning ONLY in Math. Out side of the definitions of math the the string of symbols .999... has no meaning what so ever.

 

[steersman]

Noone said this was a math problem. It could easily be a philosophical problem. Indeed, many philosophers have pondered this very question in different words and symbols

 

[Hurkyl]

Well, the fact that 0.9~ = 1 (and everything about the decimals!) is chosen so that they're a model of the real numbers. (*sigh* "real" was a poor choice of name, but anyways...)

The real numbers are defined to be a complete ordered field.

To put it loosely, "ordered field" simply means that +, -, *, /, and < all work "properly". The definition is "complete" is more difficult, and is not needed for what follows.


The first thing to notice is that there cannot be any numbers between 0.9~ and 1; if I change any of the digits of 0.9~ into something other than a 9, I'll get a smaller number.

The next thing to notice is that, if we assume $0.\bar{9} \neq 1$:

$$0.\bar{9} < \frac{0.\bar{9} + 1}{2} < 1$$

which contradicts the fact that there are no numbers between 0.9~ and 1.


Thus, in order for the decimal numbers to fulfill the purpose for which they are created... that is, to be a model of the real numbers... it cannot be the case that $0.\bar{9} \neq 1$. In other words, 0.9~ = 1 must be true.



There are, indeed, a lot of interesting philosophical questions that relate to mathematics, but they aren't about whether 0.9~ = 1 in the decimal numbers.

 

[wtfc]

I'm sure most of you already know this. The real point of the thread is finding different ways of approaching it. You see, I have a friend who refuses to accept what seems to me to be so obvious: .9 repeating (infinite 9s after the decimal) is exactly equal to the whole number 1.

Here is how I look at it.
You start of with .9999... and continue until you either run out of time or get tired. Now you are at a crossroads. At this stage you decide if it is still .9999... or 1. Conclusion is what one reaches when one is out of time. Just because you are tired of traveling through the infinite, .9999 does not equal 1 irrespective of all the mathematical equations.

Rounding off is only to solve equations. To further extend this, you either use whole numbers or fractions. You should not criss-cross domains like this.

 

[Hurkyl]

You start of with .9999... and continue until you either run out of time or get tired.

What is there to "continue"? .9999... is a decimal with an infinite number of nines; it is not a suggestion that you start with .9, then go to .99, then to .999, and continue in this fashion.

 

[balkan]

i can't wait till scientists figure out the smallest quantization of all matter and then use it to construct a new mathematical system without infinities
that'll rid the world of all this nonsense... or at least, i'll be a very happy man...

 

[Prometheus]

What is there to "continue"? .9999... is a decimal with an infinite number of nines; it is not a suggestion that you start with .9, then go to .99, then to .999, and continue in this fashion.

I sympathize with people who have trouble understanding infinity in mathematics.

Mathematical infinites exist by definition. It is not that there is any actual correlate to these infinities in the real world. For example, the series of numbers, 1, 2, 3, ..., is not infinite in the real world, where real people count them. Numbers are only infinite in the world of mathemeatics, where they can be defined as infinite, and thereby given attributes of this infinity.

When a person attempts to add actual instances of the number 9 to the end of this number, the world of mathematics is being substituted by the real world, and this is a mistake.

 

[Hurkyl]

i can't wait till scientists figure out the smallest quantization of all matter and then use it to construct a new mathematical system without infinities

(a) The real numbers are a mathematical "system" without "infinities".
(b) There are lots of other "system"s, each with their own pros and cons.
(c) You say it like the use of "infinities" is bad.
(d) What does science and quantization of matter have to do with anything?

 

[hello3719]

i can't wait till scientists figure out the smallest quantization of all matter and then use it to construct a new mathematical system without infinities
that'll rid the world of all this nonsense... or at least, i'll be a very happy man...

I'll take that as a joke.

 

[Prometheus]

I'll take that as a joke.

Are you laughing at his phraseology, or at the deeper meaning that he is trying to convey?

Or, do you understand the difference?

 

[loseyourname]

He's laughing because there is nothing nonsensical about infinity in mathematics. In fact, infinity, and particularly infinite sequences and series, can be quite useful.

 

[HallsofIvy]

He's laughing because there is nothing nonsensical about infinity in mathematics. In fact, infinity, and particularly infinite sequences and series, can be quite useful.

and has nothing to do with physical discoveries.

 

[balkan]

well, to elaborate:
i'm perfectly fine with the concept of infinities within contemporary mathematics, it works fine for now and is quite usefull...
but i suspect that a full understanding of physics can only be made through a parallel evolvment of a new mathematical system... and i suspect this system not to involve infinities due to the simple fact that nothing in this universe is infinite, as far as we know... most likely not infinitely small either... only time will tell though...

the old mathematical system will still be appropriate for most tasks, just like classical physics compared to quantum mechanics, and it will have coherence, but it will not be usefull to describe the mechanisms of the universe to a sufficiently precise degree...

 

[hello3719]

well, to elaborate:
i'm perfectly fine with the concept of infinities within contemporary mathematics, it works fine for now and is quite usefull...
but i suspect that a full understanding of physics can only be made through a parallel evolvment of a new mathematical system... and i suspect this system not to involve infinities due to the simple fact that nothing in this universe is infinite, as far as we know... most likely not infinitely small either... only time will tell though...

the old mathematical system will still be appropriate for most tasks, just like classical physics compared to quantum mechanics, and it will have coherence, but it will not be usefull to describe the mechanisms of the universe to a sufficiently precise degree...

hahaha, This would mean that our "old"(includes finite and infinite concepts) mathematical system would be more complete than the "new"(includes only finite concepts) one.

 

[Didd]

use your calculator to calculate 1/9.
It is 0.1111... and then 9/9=9*1/9=9*0.1111... but this means adding 0.1111...
itself 9 times. You will get 0.9999...
so, 9/9=9*1/9=0.9999...
But 9/9=1. So 1=0.999...

 

[Hurkyl]

And I'll reiterate; the theory of the real numbers has no infinities, and the only infinitessimal is 0.

 

[balkan]

hahaha, This would mean that our "old"(includes finite and infinite concepts) mathematical system would be more complete than the "new"(includes only finite concepts) one.

it would be more "complete", yes, but not more precise. I'm not just talking about a new array of functions, I'm talking about an entirely new system...
just like quantum mechanics are more precise dispite the fact that it deals with quantization and probability, where classical physics claims to predict precise locations and have fluent change of energy levels...
ironically, the "imperfect"/"unprecise" theory is more correct...

hurkyl:
i know, but that's not what I'm talking about... I'm talking about a new way of calculating...

 

[loseyourname]

Kind of like the way cellular automata have been used recently to model biological systems?