The good things about maths is that you can argue about anything and maths progress through debates. Great mathematicians in the past also made many mistakes and their theories either refuted or become theorems. Euler's conjecture was refuted when someone found the counter-examples which was not possible to be calculated during his time. Lame made a mistake in his attempt to proof Fermat's Last Theorem. We know for sure the summation of 1-1+1-1.. is either one or zero for odd and even terms respectively but this Grandi's series could be 1/2 and it depends on how people interpret it. Of course, 0.99999..recurring could be 1 but I beg to differ even when 0.999999..recurring=3x0.3333333..recurring or 3 x 1/3=1. By the way God particle has being found and scientists are 99.99995% sure about finding it at CERN. Maybe they need to be 99.9999%..recurring sure before it could be accepted:) http://www.dailymail.co.uk/sciencetech/article-2167188/God-particle-Scientists-Cern-expected-announce-Higgs-boson-particle-discovered-Wednesday.html

Last edited by Stangerzv (2012-07-02 20:54:15)

Hi;

I like being able to use the formula for geometric series.

This proof is originally from Euler although anyone else could do it.

Hi MrButterman;

Interesting how this thread got so long.

These type threads are on every forum. Mostly they are so long because the opponents of .9999... = 1 can not be convinced by any of those proofs or any others.

Welcome to the forum.

Hi y'all

You might say that those on both sides of the argument tend to take it to the limit!

1/2 a > good day!

Well, I guess I can try to be a little bit more clear...

jdgall wrote:

Calligar wrote:

I myself do not...personally believe this as a mathematical "fact," but also realize how futile it is to argue against it.  So like those many, it is impossible to convince me as well, after all, there is a reason this idea is so highly controversial.

The term mathematical fact might be a little vague.  First, not every number is rational -- not every number can be represented as the quotient of integers.  For example the width of a square whose area is 2 is not a rational number.  That is, we need the full blown set of real numbers.  Figuring out what the real numbers (really) look like is a hard challenge, and providing a description of them in set theory was a major challenge.  There are two main approaches: Dedekind's cuts and Cauchy sequences.  They produce the same set.  Essentially, take a bounded sequence of rational numbers, and identify a "number" L with this sequence.  The real numbers are the rational numbers with all these Ls.  Thus in the construction of real numbers, we see that every real number is the limit of a sequence of rationals.

In other words, the real number "1" is by definition the limit of the sequence
    <0,9,0.99,0.999,0.9999,etc

Firstly, when I say mathematical fact, I am really just referring to what is currently accepted and arguably "known" in mathematics.  Secondly, the limit of sequence is just another way to represent it being infinitely close, but still not exactly equal to the number (unless I'm mistaken).  Just like for it representing 1/3 with <0.3,0.33,0.333,etc. (I might not have put everything in proper terms, sorry if that causes any confusion, wasn't sure how to say it simply off the top of my head).  Also would like to make a note, you messed up slightly when you posted; it should be 0.9, not 0,9 for the first one unless I'm mistaken (but doesn't really have any relevance to anything).  In other words, it is just more rules that exist that otherwise, as I was saying, prove it's a mathematically fact.  Remember when I said this...

Calligar wrote:

To be fair, the proofs they offer can be argued from a logical standpoint (so long as you understand everything that is going on), but there is nothing in mathematics that can prove how they are different otherwise.

In mathematics, there is no way to represent the difference between 0.¯9 and 1.  All proofs (including false ones) either assume things (for specifically this), or simply define it as one only because of the infinitely close distance (there might be a few other reasons, but those are the 2 major I see).  Even though some people will argue things like it is 0.0...1 away, or 1/10¯0 away, which might arguably seem right, one can argue about the infinite distance, therefore making it an impossible argument to win.  So this argument doesn't carry on (with me) over confusion, I'll explain in more detail.

Rules in math already say this is true (even though it is very controversial) for many reasons.  However, it largely also comes down to one major thing: saying that one number equals a number that isn't exactly the same.  In other peoples eyes, that would be like saying 2 = 1; the only reason for it not being like this is because of the rules for it (plus all of math as we know it would collapse).  Also, the whole reason for this controversy in the first place, is because of using decimals in a way that don't accurately represent what it is.  Decimals can't accurately represent everything, unfortunately, no system can (at least that I know of).  However, unfortunately in this case, things like hyper real numbers define this, which is where the controversy comes in (I am not saying hyper real numbers are a bad thing though).  There are other similar cases of this.  1/3 = 0.¯3, right?  But wait, there is a difference, isn't there (asking rhetorically)?  Why?  Because 1/3 can not be accurately represented in decimal form.  Changing 1/3 into a decimal will give you 0.3...3...3...forever.  However, there isn't that number at the "end" to...well end it.  It is just like pi in that sense; pi = 3.14159265..., however there is no end to it.  There is no way to accurate represent the number in decimal form, or in any form for that matter (that I know of), besides for just calling it pi.  It is that very reason I've seen people argue pi ≈ 3.14159265...instead of =; because no matter how long the decimals go, it doesn't exactly equal pi because there is no end to it (rather pi is mathematically = or ≈ to 3.14159265... in this case is irrelevant to what I'm trying to say, plus I never said which one it is).

Please just keep in mind I'm not exactly arguing for or against.  I'm trying to say overall that it is futile to argue.  I might be personally against the current idea in math, but I have learned to accept that this is currently mathematical fact.  I just wanted to make all that clear.

Last edited by Calligar (2012-11-21 23:57:16)

0.999... doesn't exist. Recurring 9's aren't allowed.

But if 1/3 = 0.333... Then Why does 0.333... Not become = to 0.4 Because that would then be the Same Infinite Calculation as...

0.999... becoming = to 1

But approaching 0 and seems to be approaching Are both not Actually ever going to get there ?

But the number 0.9999... itself doesn't exist...

.9 + .1 = 1
.99 + .01 = 1
.999 + .001 = 1
.9999 + .0001 = 1
.99999 + .00001 = 1

But the a Two differences remain constantly the same! just another 9 and 0 ...further down the line does not make them nearer to being equal to 1

Hi Bob

I didn't mean it in that sense. Such a number isn't allowed in mathematics.

Stefy wrote:

I didn't mean it in that sense. Such a number isn't allowed in mathematics.

What number is that? 3 ?

I think you'll find it is used in quite a few areas of mathematics. 

And so is 0.999999999 recurring.

To save me some time, I refer you back to my post # 999.

Bob

Why not?

Bob