0.999... Poll

What is your stance on this Paradox(0.999... repeating decimal and its value)
1.Transformatism(1 is exactly the same number as 0.999... written in different form)
2.Strong Convergism(1 is equal to 0.999... in all aspects)
3.Weak Convergism(0.999... converges to 1 at infinity)
4.Weak dualism(1 is equal to 0.999... practically, but is different in exact values)
5.Strong Dualism(1 is different and inequal number to 0.999...)
6.Separatism(1 and 0.999... belong to different number classes and cannot be compared)

1.Transformatism(1 is exactly the same number as 0.999... written in different form): 1 is the same thing as 0.999...

2.Strong Convergism(1 is equal to 0.999... in all aspects): 1= 0.999... and  1-0.999...=0

3.Weak Convergism(0.999... converges to 1 at infinity): 1 - 0.999... = 0 at infinity point

4.Weak dualism(1 is equal to 0.999... practically, but is different in exact values): 1!=0.999... but 1-0.999...=0

5. Strong Dualism(1 is different and inequal number to 0.999...): 1!=0.999...  and 1-0.999...!=0
6.Separatism(1 and 0.999... belong to different number classes and cannot be compared) : 1 incomparable to 0.999...

1=0.999...
Eggs came first.
Plane takes off.

Is this related to infinite series?

This is NOT a paradox.
Using valid arithmetics:
1/3 = 1/3 // (a = b)
0,333... = 1/3 // (writing 1/3 in another form)
3*0,333... = 3*1/3 // (multiplying both sides by 3)
0,999... = 3/3 = 1

LOOK AT THE 0,333 = 1/3 part.
They are BOTH representating the SAME number, right?
So does 0,999 and 3/3. And, since 3/3 is just other way to write 1.

0,999 is just other way to write 1.
There is NO difference. Except notation.

>>5
You're a moderate transformatist then.

1 and 2 are the same thing.  The rest are flat-out wrong.  It's equivalent to saying there exists a real number x for which  (10x - x)/9 != x.

>>7
You're a strong convergist then.

>>8
It's not a matter of opinion at all.  In ZFC (or pretty much any functional system of axioms) .999... = 1.  A mathematical proof is not something that's up for debate.  The fact that (10x - x)/9 = x for all x is a tautology is enough proof, but there are hundreds more.

>A mathematical proof is not something that's up for debate
Ah i forgot that, your holiness. I'll repent by chanting the ZFC theorem 100 times.

>>10
The point is that if something's wrong in mathematics, you take issue with the axioms and not the theorems.  If the Banach-Tarski theorem was not enough for the average mathematician to throw out the axiom of choice, then surely a schoolchild's opinion that .999... should not be equal to one is not enough justification for the reformulation of our entire system of arithmetic.
It's pretty obvious that the person who developed this terminology 1-6 (probably OP himself) is not a mathematician, because not only do the descriptions have no rigor to their distinctions, but the question is equivalent to saying "What is your stance on this arithmetical theorem that has been proven true in standard arithmetic?"  An actual mathematician might ask "What is your stance on axiom A", or "What is your stance on unproved conjecture B", but asking for a stance on a theorem is pure lunacy.

>>11
What should we do with this lunatic schoolchild? Talking sense into such unruly kids doesn't help, those little spoiled devils need a strong hand. To dare to think that one could change our entire system of arithmetic is heretical to the highest degree, why don't they learn?

In ZFC (or pretty much any functional system of axioms) .999... = 1.
False.  ZFC is a system of axioms for set theory, not real analysis.  And there are number systems where you could define 0.999... in a way which makes it not equal to 1.

>>13
Traditionally, a real number is defined as either a Dedekind Cut or a Cauchy sequence of rationals (defined from equivalence classes of pairs of integers, defined from equivalence classes of pairs of finite ordinals), wherein the theories of real analysis take hold.  I would be very interested to see an alternate definition of a real number where you could see a distinction between the two.  Do you have a source?

>>14
Well, in the hyperreals you might define 0.999... to be \sum_{i=1}^n 9\cdot10^{-i} for some "infinitely large" hypernatural number n such that m<n for all m\in\mathbb{N}.  In this case the notation is ambiguous.

An algebraic approach is to simply define x<y on decimal expansions by looking at the first (from the left) digit in which x and y differ, so 0.\overline{9}<1.  However, if you work out the consequences of this things start to break down.  For example, if additive inverses exist then 9\cdot0.\overline{9}=9.\overline{9}-0.\overline{9}=9 which implies 0.\overline{9}=1.  We also don't have multiplicative inverses for all numbers (e.g., 3\cdot0.\overline{3}=0.\overline{9}, not 1!).  If you limit yourself to the nonnegative decimal expansions then addition and multiplication are well-defined, and you end up with an ordered semiring.  [For more detail, see "Is 0.999... = 1?", Fred Richman.]

>>15
The hyperreal approach is interesting.  (The other approach is new to me as well, but seems unnatural.)  I've never formally learned nonstandard analysis, except for how to define it from predicate calculus.  How do you go about proving the two numbers are distinct?  It would have to converge to 1 -h for some infinitesimal h if that were the case, right?  Would the number be greater if you chose a larger infinite constant for the upper bound?

>>16
That's right.  By the transfer principle (all first-order sentences which hold in the reals also hold in the hyperreals), we have \sum_{i=1}^n 9\cdot10^{-i}=1-10^{-n}, with 10^{-n}>0.  See "A strict non-standard inequality .999... < 1", http://arxiv.org/abs/0811.0164

Can't we show that they are equal with a delta-epsilon proof? Or am I assuming too much(standard algebraic axioms)?

Oooh

I see how it works

but I hate that it works

1>0.999...999...999>0.999...000...000>0.999>0

(1/x)*x=1
1=0.999...  and  1-0.999...=0
(1/(1-0.999...))*(1-0.999...)=(1/(1-0.999...))*0
1=0

>>21
0=1-1
(1/(1-1))*(1-1)=(1/(1-1))*0
1=0

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

I say it's all a canard

7. unknown(the answer cannot be discerned)
8. nigras(nigras)

>>21
WHEN X is NOT 0.
Thus, fail.

1/3 is one third of 1, not one third of 0.999...

1 multiplied by its self will always be one.
0.999 will eventually reach .000
6.Separatism

this goes way beyond the sub-sub-sub etc atomic level and in a practical situation; this would happen almost everywhere. the possibility of something for example weighing exactly 30 kg is 1/infinity. so Transformatism is logical.
in this world anyway.

peeps, if you really want to get trolled that bad head over to /b/,
you have an undying source

>>28
Well put, you've converted me from 4 to 5

>>28
Well put, You've converted me from 4 to 5.
1-1 = 0
1-.999... != 0 but an infinitesimally small quantity.

>>28
Well put, You've converted me from 4 to 5.
1-1 = 0
1-.999... != 0 but an infinitesimally small quantity.

>>28
Well put, You've converted me from 4 to 5.
1-1 = 0
1-.999... != 0 but an infinitesimally small quantity.

Ummmmm guys .9999.... is just a (dumb) way of writing something.  It's like debating whether or not knife should be spelled with a 'k' at the beginning.  It has nothing to do with mathematical truths and is simply a critique of our decimal system.

0.999999 and 1 are two different ways to denote the same mathematical object. What is the big deal?