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Ross Stratman

Euclidean geometry also suggests tha .999(repeating) = 1. Here goes.

1) For every distinct point on a line (number line included) There MUST exist at least one point between them.

2) If .999 repeating is actually LESS than 1 then there MUST exist a distinct point on the number line separate from 1.

3) Since there must exist a number between the two (see Theorum 1) then... WHAT, pray tell, is that number?

Flavio

.99999... = 1 ONLY in standard analysis.

In non-standard analysis that IS NOT true.

YOU should know that axioms matter, and since 1900s mathematicians don't believe there is anything are the "right" axioms!!!! Haven't you ever read about the parallel postulate debacle...

So learn something yourself, and not necessarily .999... = 1. That depends on axioms.

BTW for computable reals, that is also not true, 0.9999 != 1.

Polymath

Yes, it's true...I do understand that axioms matter, and that with other axiom sets, there are mathematically consistent ways of proving that the numbers aren't equal. But I do make explicit (not in the first post, but in subsequent ones) that I'm talking about the standard real number set. Furthermore, the people who disbelieve can't even argue logically in the standard real numbers, let alone the hyperreals or other sets.

craig


Proof 1/3=0.33333rec

a=0.99999rec
10a=9.999999rec
9a=9
a=1
a/3=1/3
a/3=0.3333333rec
as a/3 is 1/3
1/3=0.333333rec
QED

no name

So I am just going to put my two cents in.

I like you algebraic proof. It was nice to terminate the repeating decimal that way. However, let's first look at 1/3
10x = 3.3333333...
- x = .3333333...
9x = 3
x = 1/3

It works. Yay. If we were to stick 1/3 back in for either equation, it would work. Let's do pi next. I will just do 3.14
10x = 31.4
- x = 3.14
9x = 28.26
x = 28.26/9 = 3.14
That worked just as well, but only up to the two digits we used. If x = 3.14, then 10x = 31.4. Hey look, it checked out. For the final test.

10x = 9.9999999...
- x = .9999999...
9x = 9
x = 1
what? .999999... = 1? But that's impossible? .999999...<1. Well, yes that's correct. if you plug in .999999... back, you get 10x = 9.999999... and x = .999999...

Wow, it worked for an irrational and repeating number, so it doesn't really matter what type of number it is, the algebraic proof works. I also don't want to hear about how 3.14 is not irrational. Do it to however many digits you want, it still works.) Surprisingly, but yes, it works.

Guilherme

What about this? .9999... is a fraction (since it periodically repeats 9). So it is a racional. A racional arbitrally close to 1 is 1.

Bradley

Very sorry, and I regret doing this but after lots of reading and thought on the issue, I question the validity of the basis of the whole argument that .9r = 1.

Time and time again in these comments multiple posters on both sides of the argument have acknowledged that the fundamental issue of the conflict is that the mathematics aren't flawed - rather that the symbolic representation of the idea is flawed.

I'm inclined to believe that the importance of the entire argument is lacking if this factor cannot be eliminated. I believe it is impossible to exhaustively convey the existence of any entity, object, or concept through notation (which also makes me question my ability to convey my very thoughts to you all now through language).

Our very idea of symbolic notation and mathematics is specific to our existence. Likewise, multiple mathematical concepts and operations utilized in the 'proofs' that have been thrown around in this article are also limited to the bounds of our system of symbolic representation. Who is to say we are allowed to define all these items through our various systems of notation when even the entirety our own existence throughout history is as minute in the grand scale of the known universe as the infinitesimally small amounts this argument seeks to explain.

We can prove conclusively that, based on our conceptual representation of these unquantifiable amounts that the numbers do make sense and lead us to the conclusion that .9r really does equal 1. However, we cannot hope prove this exhaustively. There are so many possibilities in the universe (which, by the way, we don't even know the true expanse of) for explaining the argument against .9r=1 that it would be impossible to fathom them all.

I'm not dismissing the blood, sweat, and tears of generations upon generations of renowned mathematicians - I'm merely encouraging us all to look ahead with an open mind free of the shackles of our traditional schools of thought because we may soon find ourselves on the brink of an entirely different dimension - one in which the rules and symbology we've used so conclusively to describe processes and concepts in the world as we know now will not apply, and will have their foundations shaken from under them.

Rich

I have been thinking about this for quite some time myself, and I find this discussion quite interesting to say the least.

I think the major issue in the line of reasoning that 0.999... with the 9's repeating off to infinity is equal to 1 is that, each of your proofs add or subtract "infinitesimals," and so some information is lost in these results. Let me explain, however. I do not disagree in any way that 0.9999... is another real-number representation for the real number 1. This is because to treat 0.999... as a real number we come to the conclusion that 1 - 0.999... is equal to some number closer to zero than any other real number, and thus we might then say that 1 - 0.999... = 0 since they differ by an infinitesimal quantity.

First, consider your proof using simple algebra. I personally like this proof, it is quite neat. Anyway, what you do is something like:

0.999... = x; thus clearly
9.999... = 10x; hence by subtraction of these
9.999.... - 0.999... = 10x - x = 9x, thus x = 1 = 0.999....

What I want to analyse is when you say:
9.999... - 0.999... = 9.
Here you are subtracting two numbers which are indeed infinite geometric series. So it might be suitable to turn to that discussion to see the crux of my problem so far:

When you consider the geometric series, say 0.999... = 0.9 + (1/10)9 + (1/100)9 + ..., mathematicians have discovered through some simple multiplication and subtraction of finite geometric series that the sum of the first n-terms is:

Sn = a + ar + ... + ar^(n-1)
= (a - ar^(n))/(1-r)

Which can be simply obtained. Now, when we say that the sum of an infinite series equals a finite number, what we are saying is that
lim[n->inf] Sn = S in the reals

Provided that 0inf] Sn = lim[n->inf] (a - ar^(n))/(1-r)

And the right-hand side obviously *equals* a/(1-r) since we say that r^(n) = 0 as n -> infinity.

OR DOES IT? This is the crux of the matter, as this EQUALITY has to hold if your subtraction is to be true. I mean, if there is some ambiguous undefined component in these results, then we won't be able to say even that 0.999... = 0.999..., since first we would have to establish that this component is the same for both infinite geometric series. But consider:

lim[n->inf]r^n, with 0 0 as n -> inf, but we cannot say that r^(n) = 0 as n -> inf, because it doesn't!

So this limit introduces a term in each geometric series that is infinitesimal--it is closer to zero than any other number. And the important part is that we have no clue how to add and subtract infinitesimals, just like infinity - infinity makes little sense to us. Which means that 9.999... - 0.999 = a number closer to 9 than any other real number, maybe we might write 9 +/- I, for some extremely small I. Now of course if we just want to deal with real numbers, we can ignore infinitely small numbers.

This topic is quite vital, in my opinion at least, and I like your explanations. But one might witness that in Calc II, for example, we do stuff like:

dy/dx = h(x) =>
dy = h(x) * dx

Thus we can substitute into an integral and change the limits of integration, and ultimately solve this integral. But we must recognise that dy and dx are numbers closer to zero than any other real numbers! We got them via a limiting process; therefore, infinitely small numbers are not "pointless" things to be thrown away all the time.

For the record, however, the argument that 0.999... = 1 is quite appealing in that it demonstrates how infinitely small values are handled typically in mathematics--namely, we cast them away as "really" zeros when we take the limits.

Supernova426

It seems that all of the arguments against ".9999... = 1" are stating that representations of numerical values are different from the numerical values themselves. You must use your brain to truly understand the true numerical values of repeating decimals, as Garthnak has so wisely described.

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