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## Does 0.9999(repeating) = 1?

Yes 705 56.9%
No 578 46.65%
Voters 1239 .

### Does 0.9999(repeating) = 1?

Senior Member

Using the math displayed in this thread thus far I've found some interesting things.

Going to say technically it doesn't actually equal 1, but it's one of those loopholes in math in which you can indeed proof it to equal 1.

KirbyCake

Senior Member

Quote:
fmwyso:
Why would it be dx? There is no dx whatsoever Statiqhock...

x = .999...

10x = 9.999...

10x = 9 + x

9x = 9

x = 1

Does that help you understand? It is definitively 1, stop trying to act like you can conceptually misconstrue definitions; we simply don't need to in this case.

Where's your justification for subtracting 2 infinite decimal numbers
can an operation subtract even be defined for 2 infinite decimal numbers? How do we know these numbers are in the real number field.

Magni

Senior Member

Quote:
imaloony:
No, no it does not. What you are doing is called rounding, and rounding kills.

Another person joins the "0.999... is a finite number" club.

The Convict

Recruiter

Here's another way to put it...
1 - .999... = 0.000...

Since there are an infinite amount of zeros, due to the mechanics of subtracting
(see: 1-.09 = .01, 1 -.009 = .001)
there is no valid way to express an infinite amount of zeros between the . and the 1, at least not in the conventional math that is used even in calculus (there may very well be theoretical methods for expressing this.) Because there will always be one more required step to solve the equation (1 more zero in the chain), the number repeats, infinitely. This logic does not mesh with our current understanding of reality, that is, that there are "building blocks" of the "building blocks" of protons, neutrons, and electrons. These are the smallest pieces of existence, and nothing can hold a smaller value than these things other than a concept referred to as "0", nothingness.
However, since the limit of x approaching 1 gets close to 1, yet never actually touches it, we wind up with what seems like a logical paradox. However, there must be a threshold - when does something become small enough to be zero? The answer would be, for our current understanding of the world, when something would be smaller than the smallest pieces of existence, we must either redefine the lower border of existence as we know it, or submit that it is no longer a part of existence, it is no longer something, it is nothing, it has a value of zero. This would also be exhibited in numbers, something infinitely small enough would be zero.
This leads us back to the number shown as 1-.999...
Since the result would be infinitely small, using the conclusion from the previous paragraph, 1-.999... = 0.

I should probably go back to work.

Should.

Raisttlin

Senior Member

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

Wikipedia shows several proofs that disagree with you Mr. Red!

Edit: Does no one use the internet to look something up while they are on the internet?

SplendidSorrow

Senior Member

Quote:
Chyrsippus:
Here's the thing: Decimals do not provide a unique expression of all real numbers. Certain real numbers can be expressed in multiple, equally valid, decimal expressions. For instance, 0 and -0 are two ways of expressing the same value. Similarly, 1 and 0.999... are equivalent. The proof of this has already been posted on the thread.

Something similar is true with fractions. 1/2 = 2/4, even though the representations are of course different. This is actually all a bit of a problem, in that it shows how numbers themselves are a very different animal from our representations of them.

1 is also used to represent 100%

.9 repeating is almost never represented as 100%

1 is also used to represent .9 repeating

.9 repeating is not used to represent 1

While you can mathematically prove they are equal, representationally they are not the equivalent in all cases.

BedderDanu

Recruiter

Quote:
sudhirking:
Hello Young Mathematicians!
The answer to this question is.... YES! The proof is quite simple and exploits the well known formula for geometric series.

.9999999999 (continually) may be regarded as the sum among .9 + .09 +.009 etc. The reason why I choose to write it in this fashion is as to force this .99999 to like something like an infinite sum. Clearly each decimal can be written in a fraction form.

9/10 + 9/100 + 9/1000 + ....

After we realize that each division of 10 or by 100, or 1000, is merely 10 raised by incrementing integers, we can rewrite this as a sum:

.999999999= 9(1/10)^1 + 9(1/10)^2 + 9(1/10)^3 + .........

While one may think that adding infinitely many amount of numbers together yields infinity,this object is well realized in mathematics to be a convergent geometric series. The value (which we can easily prove separately) is given by,

9(1/10)^1 + 9(1/10)^2 + 9(1/10)^3 + ......... = 9/(1-(1/10)) - 9 = 9/(9/10) - 9= 1

Thus, it is from this conclusion that we claim .9999999 is in fact 1, where we place enough trust in our understanding for the geometric series as to revolutionize our understanding of .99999999!

There is one case where the above doesn't apply, but you need to redefine the traditional definition of 1 to do it:

Imagine the number 1 as a length equal to the collection of all points greater than 0 and less than or equal to 1 on a number line. This is more commonly represented as the range (0,1]

In only this case, the number 0.999... is not equal to the number 1. It is instead equal to (0,1). AKA all numbers greater than 0 and less than 1.

But this artificial construct is definitely not traditional numbers.

2pudge1cup

Senior Member

Quote:
Tainted Moose:
Why are Reds commenting on this while it's still in GD yet has NOTHING to do with League?

no, it has a lot to do with league.

the no votes are people you should always queue dodge when they on your team or they'll be a BR/feeder/moron. it explains matchmaking so perfectly.

Kyryck

Senior Member

So what you're saying is that when you have 0.99999blah blah of a computer program, it'll function, because you actually have the whole 1.0 of the program.

Really?

So if you have a system that can only function when it's completely whole, if you only have 0.9999blah blah of that system, it'll work, because it's just like having the whole 1.0 of that system?

I'm sorry, but mathematically proving something like this seems to me to be entirely different than having it actually operate like that in literal reality. While I'm not a mathematician, I'm sure that there are some accomplished ones that can produce things that make mathematical sense but don't make work out when applied in reality.

Perhaps I'm simply literal minded over mathematically minded and feel the concept of a whole thing invalidates a mathematical proof that shows that having less than a whole is actually the same thing as having that whole.

Hurricaneo0oo

Senior Member

Quote:
RiotDerivative:
That's fine. I am human. I am allowed to be incorrect.

If it disappoints you that someone who studied Math can still be wrong about something in Math, you will disappointed in life a lot. :-(

You can still fix your mistake! Edit your original post! End this! Come on Mr Frodo, throw it in the fire!