@Bolded: exactly, .(3) or .33333333333333333333333333333333333333333... is a representation of of 1/3, it's a symbol for 1/3.  The digits .(3)... do not equal 1/3, the symbol .(3) represents 1/3.  The digits .(3) is approximately 1/3, it's the closest we can get in decimal form, but they are not equal.

Which is why those proofs don't mean anything.

Generally it does.

Put it this way: 0.9999.....does not equal 1, BUT

0.(9) is equal one. So one someone aska you what 1/3 is in numbers you give the answer 0.(3) . Multiply 0.(3) by three and that is equal to 1/3 times 3. And that equals one.

If someone doesn't get what i'm saying, when you write some number like 0.9999999........... ==> that does not mean that the number is infinate. It ends at some distant point that you can't accurately pinpoint, but it still ends.

In this case the () with a number inside means that the number goes on for infinity.

And because 0.(9) goes on for infinity, it is so close to the actual 1 that it is percieved to be that number.

I was rather shocked about this the first time I read it on Wikipedia.

This is a geometric series (a * (r^n)) of the form .9(.1 ^ n) power. Because we can prove the convergence of geometric series (in certain instances) using the formula a / (1 - r), after plugging in the terms, we get .9 / (1 - .1) = .9/.9 = 1. Series and sequences are fun! (But screw Taylor series and polynomails, those were horrible...)

Thank you.  But I don't like the wording of the last sentence. Perception may matter when measuring a circle for a drawing or building.  But when working sets of data, a lot of precision is lost.

1/3+1/3+1/3 = 1 and .(3) + .(3) + .(3) = .(9) no big deal.

but when you have 1/3 + 2/7 + 6/11 + 4/13 + 9/17 + 3/23 = .... that would lose a lot of accuracy if you convert each component to a decimal before adding them up.

Of course, but i was speaking of this one particular subject. The one you brought up is a completely different matter and should be solved accordingly.

As to the wording: i had maths in Polish, not English, excuse my lack of accurate wording on the subject. I cannot agree with the first part of your post though.

If anything, with this set of data, this kind of simplifying is a must, or else you would have gone haywire trying to  calculate precisely. When your making measurements of some kind of mathemetical object, there is no way to do it "on a hunch of perception" , you have to do it by the established equations.

I agree, but it depends on the idea of simplification.  Using numerical symbols (to avoid fractions), rounding, etc is easier for some, and keeping original forms and having exact answers is easier to others.

The problem is, in the OP, there is a flaw.  .(3) ~= 1/3

That's my story and I'm sticking to it.

Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:

• Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.^[33]
• Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".^[34]
• Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.^[35]
• Some students regard 0.999… as having a fixed value which is less than 1 by an infinitesimal but non-zero amount.
• Some students believe that the value of a convergent series is at best an approximation, that .

These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999….

It IS 1.  It's not a ton of nines, it's not a bagillion bazillion, it's an infinite amount of them.  The number of nines is NOT approaching inifity, it has REACHED infinity.

I take it that you mean my OP about 0.(3)=1/3.

Of course you are right that 0.(3) is more like ~=1/3 but the same thing goes with 0.(9) which is actually ~1.

The thing is, that this has been decided by people smarter than us or two ioi's put together. It's mathemetical law, and you have to abide by it. It's not debatable i'm afraid. Trust me, i don't like the idea either, but whenever i would raise such a topic in math class in high school, i would get this answer: It's not debatable.

Edit: point of my rant about math law==> if 0.(9)=1 then 0.(3) will be equal to 1/3.

So here it goes, I hope this is not a joke thread because I'm about to waste 3 minutes of my life writing this up. From here on out S is sigma

0.999999 can be represented as the infinite sum of S(9 * (1/10) ^n) where n = 1 to infinity.This is basically sayying that 0.99999... = 9/10 +9/100 +9/1000 +....

Meanwhile the sum of a geometric series is PROVEN to be equal to :

a/(1-(1/r))

Where a is the first term, 0.9 in this case, and r is the rate of growth, 1/10 in this case. So where does that leave us?

0.9/(1-(1/10) =

= 0.9/( 9/10) =

= 0.9/0.9 =

= 1

And that is the easiest and way to prove this. The whole 0.3333 = 1/3rd is crap and not a real proof by any stretch of the imagination.