Poll: Does 0.999.. equal 1 ?

[Mr Thin]

Yes.

I'm no mathematician, so I can't give you the actual sum(s) that prove it, but if it's 0.9 recurring, that stretches on to infinity.

Thus, the numerical difference between 0.999... and 1, is infinitely small.

And... an infinitely small amount of something is equal to zero? So the difference between 0.999... and 1 is zero.

That's how I always understood it, anyways.

Some other guy posted the link to a relevant Wikipedia, check that out.

 

[Volkov]

Yes, they are equal. There are many proofs that show this. Here is a limit-based one.

The term 0.9999... (with infinitely many 9s) is an infinite sum, defined by:

limit as n -> ∞ of the sum, indexed by k, from k = 1 to k = n, of the quantity 9/(10^k).

By properties of geometric series, this is equal to:

limit as n -> ∞ of the expression 1 - 1/(10^n).

limit of 1 as n -> ∞ is 1.

limit of 1/(10^n) as n -> ∞ is 0.

By well-known properties of limits, the limit of a sum/difference is equal to the sum/difference of limits. Therefore, limit as n -> ∞ of 1-1/(10^n) is equal to 1 - 0 = 1. Therefore, 0.999... = 1.

Here is another way to look at it. For two numbers to not be equal, there has to be a finite difference between the two numbers. But no matter how small a finite number you name, it is obvious that the difference between 1 and 0.999... is smaller. Therefore, 0.999... is 1.

In general, the Wikipedia article explains everything.

http://en.wikipedia.org/wiki/0.999...

To those saying that "an infinitesimal difference is still a difference" - actually no, it is not. This is called the "Archimedian property", and it is a basic property of all real numbers (and, more generally, of all ordered algebraic structures): no ordered algebraic structure can have infinitely large or infinitely small elements. Now, if you do consider an infinitesimal difference to be a difference, then it has to be a real number as well (since difference between two real numbers is always a real number). Which means there must be an infinitesimal real number, which violates the Archimedian property. Therefore, an infinitesimal difference is NOT a difference (aka zero), which means the two numbers (0.999... and 1) are equal.

 

[Vidi Kitty]

The part where you can't express or work with how small of a difference there is with math, so you cut to .999~=1

 

[Bon_Clay]

Yes. If you don't understand the nature of infinity stay away from the field of mathematics. There are no opinions, only right and wrong answers in math.

The 9 repeats infinitely. That means the suppose value of 0.0000*infinite zeros*00001 that separates the two just doesn't exist for all intents and purposes.

 

[thenumberthirteen]

NOOOO! Not this topic again. I suggest you use the search bar as there have been threads on this already. Large, annoying threads.

 

[Volkov]

Also, this is flawed. Math gets pretty exact, but that doesn't make it universal law. They way that we work math allows for some stupid loop holes, like 1/3=.3333... but 3/3 doesn't equal .9999...

Its all a part of not being able to properly express a value with the decimal system alone, which is where fractions come in IMO.

This is incorrect. First, "Math" is not a type of science, is a set of conclusions drawn from a set of axioms. So it's not a "law", it's a set of conclusions, rules, etc. Often used as a tool to model various physical processes.

More importantly, 3/3 DOES equal 0.999... What other "stupid loopholes" are you talking about?

 

[GundamSentinel]

No it isn't. It has an asymptote at 1, so 1 is a value it will never reach, no matter how many nines you add. Even infinite nines would in principle leave it an infinitely small amount short of 1.

If it doesn't work, how can we calculate with integrals (and differentials)? Those calculations are based about exactly that.

My post represents my more feelings than mathematical truth. You're right, and that's probably the reason why I always hated differential equations. ^^

 

[Hiikuro]

This is equivalent (in that the representation 0.(3) represents 1/3 exactly):
1/3 = 0.(3)

So this must also be equivalent:
3*1/3 = 3/3 = 3*0.(3) = 0.(9)

Which means this is equivalent:
1 = 0.(9)

Infinity has some strange properties, hasn't it?

 

[veloper]

So many elegant proofs and wasted because the question is flawed. It should say recurring, not just "0,999" which is 0,9990.

 

[Vidi Kitty]

Yes. If you don't understand the nature of infinity stay away from the field of mathematics. There are no opinions, only right and wrong answers in math.

But if you do and you understand that .999~ will never quite get that last little bit to make it a whole 1, no matter how much its argued or how much people 'prove' it, then you realize that all of this is pointless.

 

[Coldie]

The part where you can't express or work with how small of a difference there is with math, so you cut to .999~=1

Math is not like Physics where you can "assume x ~0", math is absolute in its precision. If you can't express something, then it doesn't exist. Math can "express" and "work with" absolutely everything. Roughly speaking, if you can imagine it, you can formalize it in mathematical terms and use mathematical laws to work with it in nice, orderly manner.

 

[mattsipple4000]

NOOOO! Not this topic again. I suggest you use the search bar as there have been threads on this already. Large, annoying threads.

lol sorry new here and wanted to unload my brain! fun to chat tho !

 

[Bon_Clay]

Yes. If you don't understand the nature of infinity stay away from the field of mathematics. There are no opinions, only right and wrong answers in math.

But if you do and you understand that .999~ will never quite get that last little bit to make it a whole 1, no matter how much its argued or how much people 'prove' it, then you realize that all of this is pointless.

Its not pointless at all because this understand of the nature of infinity is what allows tons of calculus and other complicated mathematics to be done.

 

[Volkov]

But if you do and you understand that .999~ will never quite get that last little bit to make it a whole 1, no matter how much its argued or how much people 'prove' it, then you realize that all of this is pointless.

1. There are many reasons why the study of limits is NOT pointless. They have enormous utility in many applied fields.

2. The "if you do understand that .999~ will never quite get that last little bit..." statement is wrong, because there is no "last little bit". There is no "bit" or "number" etc. that separates the two real numbers, therefore they are the same. It's very important to realize that this is NOT a matter of opinion. This is a factual, direct conclusion from the basic axioms which define what a real number is.

Now, if you DO decide to change what a real number is, then some ambiguity may arise. But this is equivalent to saying "I won this track meet because I redefined what "running" is, and showed up in a jet plane. Doesn't my argument make sense?"

 

[mattsipple4000]

So many elegant proofs and wasted because the question is flawed. It should say recurring, not just "0,999" which is 0,9990.

the ellipsis indicates recurring

 

[Indecipherable]

x=.9999...
10x=9.9999...
10x-x=9.9999...-.9999...
9x=9
x=1
.9999...=x=1
.9999...=1

Also, this is flawed. Math gets pretty exact, but that doesn't make it universal law. They way that we work math allows for some stupid loop holes, like 1/3=.3333... but 3/3 doesn't equal .9999...

Its all a part of not being able to properly express a value with the decimal system alone, which is where fractions come in IMO.

No, this is the same answer I got when I emailed a Mathematics Professor this very question several years back.

0.9 recurring = 1.

 

[Eclectic Dreck]

No it doesn't. The gap is so infinitesimal that it hardly counts, but 0.9999 recurring does not equal one.

You are indeed mistaken. If we are talking about an infinite series of 999 (implied by the ellipsis) then the answer is indeed 1. There are a number of things that demonstrate this but this is the simplest:

1/3 = 0.333...
0.333... * 3 = 0.999...

The rules of arithmetic assert that if we perform an operation on a number and then do the inverse of that operation we get the starting value. So, even though it looks different, it is in fact equivalent to 1.

If it is not an infinite series (it doesn't matter how many 9's there are, if it is a finite number it isn't enough), then it is not equal to 1. It's just really close.

 

[godevit]

I'm really sick of this math trends.....keep in mind we made it.

 

[ZephrC]

Yes. If you don't understand the nature of infinity stay away from the field of mathematics. There are no opinions, only right and wrong answers in math.

But if you do and you understand that .999~ will never quite get that last little bit to make it a whole 1, no matter how much its argued or how much people 'prove' it, then you realize that all of this is pointless.

0.999... is not a process. It's not ever going to "get" anywhere. It's a number, and that number is also more frequently written as 1.

 

[SL33TBL1ND]

Basicly it boils down to limits:

Lim(x)=1 for x->1

That's what I was going to say.