then i say that you didn't pay attention. without () your thesis is wrong. But in general yeah, that would be it.

0.333.... = 1/3
3*.3333= 3*(1/3)= (3*1)/3
0.999.. = 1

Or

0.111... = 1/9
9*0.1111..= 9*(1/9)= (9*1)/9
0.999.... = 1

Or

x= 0.999...
10x = 9.999...
10x - x = 9.999... - 9.999...
9x = 9
x=1
1 = 0.999...

Or

0.999... = 9(1/10) +9(1/10)² + 9(1/10)³ + .... = (9(1/10))/(1-(1/10))=1

I still insist, that in any practical application of numbers in the real world:

First Situation (Used in Economics, Most Physics, Statistics)

.(3) ≠ 1/3 Correct

.(3) ≈ 1/3 Correct

.(3) = 1/3 Incorrect

.(9) ≈ 1 Correct

.(9) = 1 Incorrect

OR Second Situation (Used in education, some physics, some theoretical graphing)

.(3) = 1/3 Correct

.(3) ≠ .(3) Correct

There's a big difference between '=' and '≈'.

The only time '=' is an acceptable substitute when simplifying for the sake of education, for theoretical graphing, or when clearly marking .(3) as = to 1/3 and not to the value associated with the real number .(3).

In that last situation, .(3)*3 = 1 ≠ .(9) because .(3) is defined as 1/3.

This ambiguity has been discussed at length and only some extreme theorists and smart ass students pretend not to know the difference between .(3) as a real number and .(3) as a representation of 1/3.

If anything this draiks proofs just prove the flaws of ignoring the recognized ambiguity and the problem with using "close values" instead of exact values.

Lastly, using scientific notation [Σ(3*(1/10)^n)] is the same as saying .(3).  Proofing with that notation doesn't make .(9) any closer to 1.

I don't see what you're basing your argument on Steven.

0.(3) is a single number in the set of real numbers and 0.(3)=0.(3) always, there are no exceptions.

The idea that 0.(3) is only close to 1/3 is because of a lack of understanding of the concept of infinity - the idea that if you multiply 0.(3) by 10 that at the end of all the threes there is now a zero. This ignores the fact that there is no end of all those threes, the number of threes was infinite and is still infinite.

Here is a source with some fairly well qualified people reinforcing this - http://www.newton.dep.anl.gov/askasci/math99/math99167.htm

Couldn't find a better source in the time I could be bothered looking it up.

The idea that .(3) is only close to 1/3 is because of an understanding in the usefulness of computational viability.

The reason why .(3) ≠ 1/3 is because that  1/3*1/10^∞ that is missing.   1/3 IS NOT .(3), it's approximately .(3).  Because of infinity.  Infinite repetition defines the decimal representation as inaccurate.

.(3) is not always .(3), why? Because .(3) is ambiguous.  Some times it is used as an exact representation of .(3), othertimes it is an approximation of the decimal value of 1/3.

Theoretical Mathematicians tend to support the idea that .(3) is close enough.  Applied Mathematicians (the ones who count - get it. Ha!) actually use numbers to do things in the real world do not accept .(3) as an accurate measure of 1/3.

The idea that .(3) + .(3) + .(3) = .(9) AND 1 is ridiculous.  It's basically saying, "Oh well, we can't figure out a better way to express it, so it must be right."  It's not.

Proofing it does not prove the real world values .(9) and 1 are the same.  It just proves that the methods are flawed.

But it's the best system we have, so we don't throw it out.  Pretending it does not have problems is counter productive.  This is an attempt to give quantitative identity to the universe, not a religion taken on faith.

My background is in economics (my Minor), so I am biased.  My position is that all things can be represented numerically, just because we haven't figured out how does not mean we should quit or make shit up.

Edit: I forgot to mention that the concept of infinity isn't proven to exist.  The universe is not infinite, and once we leave our universe there's either a totally different set of values or a different set of rules.  Likewise, when decending in scale, infinity isn't proven either; I do not buy infinitely divisible particle theories.  So sooner or later those 1/3's are going to end in something other than 3.

I think I see what you're saying but I'm not certain.

In theoretical mathematics "0.(3) + 0.(3) + 0.(3) = 0.(9) AND 1" is true by definition because 0.(3) does equal 1/3 and 0.(9) does equal one. That has been proven multiple times in this thread, it is not just close to (as you seem to suggest with your 'close enough' comment) but it is identical to. If you're denying this then you are just plain wrong.

However if you're trying to state that in application (as opposed to purely in theory) 0.(3) as expressed properly (0.3333...) is never actually 1/3 as there is no way to have actual infinite recurrence. Thus in applied mathematics 0.(3) != 1/3.

This debate is very very old, and yet still breeds misgivings among many =P  I've encountered a number of people who have a problem with the proofs and I've found that once someone is entrenched in the position you cannot convince them.

So I will ask those who don't buy it a question out of genuine curiosity that relates to the question.  I've asked others in the past to find one person who has a Masters or Doctorate in mathematics who supports the position that 0.999...~=1 rather than 0.999...=1.

Of course it really doesn't change my point even if there is. The point is that the debate boils down to one of definitions of the mathematical kind and to most(all?) with the highest credentials in these matters they have one single interpretation of every proof that has been put forward (5 of them iirc).  And that is that 0.999...=1. 

I'll be the first to say that a consensus of experts a fact is not, but this is a case where it seems to be more than consensus ...its unanimous. And with 5 different proofs using different approaches from different parts of mathematics most people are more than sufficiently convinced by that.  But I admire the spirit of questioning authority so I havn't rule it out, I'm just infinitely leaning that way. 

Jokes aside, I wanted to post this to put the debate into perspective for others and encourage them to talk to their professors and instructors.  I'll avoid the heated side of the debate since I've had it many times before and the purpose has more to do with people working it out for themselves than actually convincing someone else.

PS - I wasn't joking about the question BTW, I'm actually pretty curious if there is one.

I asked my dad the questoin back in the day. He doesn't know of a single friend/colleague that thinks 0.9999 is not 1. It's the accepted consensus that it is 1. I find it funny that some people that don't have any credentials whatsoever are still arguing against it considering the PhDs are, from what I've heard, unanimous about it. Do you all think you know math or are smarter and know math than PhDs in math or what?

Yes. My rationale is as follows.


The number .999... goes on infinitely.

At every decimal place you add a 9, the difference between the decimal and the whole number becomes smaller by a factor of 10. (The difference between .9 and 1 is .1, The difference between .99 and 1 is .01, etc.)

Because of this, the difference .999... and 1 is infinitely small (since the difference shrinks by a factor of ten at every 9 after the decimal place, and this continues infinitely).

The only number that is infinitely small is 0, therefore the difference between .999... and 1 is 0.

If the difference between .999... and 1 is 0, then they have to be the same number.

 No. It equals .999, silly.