Proving 0.999... Is Equal To 1 1260
eldavojohn writes "Some of the juiciest parts of mathematics are the really simple statements that cause one to immediately pause and exclaim 'that can't be right!' But a recent 28 page paper in The Montana Mathematics Enthusiast (PDF) spends a great deal of time fielding questions by researchers who have explored this in depth and this seemingly impossibility is further explored in a brief history by Dev Gualtieri who presents the digit manipulation proof: Let a = 0.999... then we can multiply both sides by ten yielding 10a = 9.999... then subtracting a (which is 0.999...) from both sides we get 10a — a = 9.999... — 0.999... which reduces to 9a = 9 and thus a = 1. Mathematicians as far back as Euler have used various means to prove 0.999... = 1."
Re:I went one further (Score:1, Interesting)
It must be great to have an infinte amount of time.
This is so old... (Score:4, Interesting)
This is so old...
Even Blizzard issues a press release about it years ago because people kept arguing about it on the Blizzard forums.
http://www.mbdguild.com/index.php?topic=14915.0 [mbdguild.com]
Re:This is second place (Score:4, Interesting)
The trip-up is that it's repeating...since we have no concept for infinity, and, that there's no method of resolving a fraction w/ repeating decimal...it's not an accurate representation of the fraction - that's the flaw.
Therefore, Fractions are Good. Decimals are Evil!
Good thing our banks, credit card companies, and governments don't use repeating fractions.
Re:This is second place (Score:2, Interesting)
People don't really know what numbers are (Score:4, Interesting)
This just goes to show that people don't really know what numbers are, at least when they are infinite decimal numbers. A finite decimal number corresponds to a rational number, e.g. 9.99 corresponds to 9 + 9/10 + 9/100. The way you describe infinite decimal numbers of by denoting a sequence of finite decimal numbers that goes towards this infinite decimal, in our case: 0.9, 0.99, 0.999, etc. This, by the way, is how you construct the real numbers (pi is described in such a way).
In doing so, however, there are multiply ways of describing the same number; the sequences 0.9, 0.99, 0.999, etc. and 1, 1, 1, etc. describe the same number, and this apparent non-uniqueness is probably what bugs people.
Re:Or (Score:1, Interesting)
As a mathematician, I have always hated people who claim that 0.999... = 1 can't be true. Especially because they (almost) always gladly accept that fx. 0.333... = 1/3, which, as you show, yields the equality. I cannot comprehend that one accepts 0.333... = 1/3 but not 0.999... = 1.
However, strictly, 1/3 = 0.3333... needs to be proved as well.
Re:This is just faulty math (Score:1, Interesting)
Re:(0.999...)st Post! (Score:4, Interesting)
What I'm saying, is I don't think anyone would accept such a proof, because while recognized, floor and ceiling are rarely considered useful due to inclusion of error, and most people don't even think of ceiling (only floor).
So, sorry, 2+2=6 is a stretch compared to 2+2=5. You might as well say 2+2=1000000, ceiling-rounded to the nearest million
Re:I went one further (Score:3, Interesting)
Only in a few cases (and the notable case of "infinity is not a number"). Anyone familiar with the derivation of limits, derivatives, and integrals should be familiar with finite numbers that are the result of an infinite-step process.
Re:Finally (Score:3, Interesting)
since 0.999 can also be expressed as 1 - 1/infinity,
1 = 1 - 1/infinity
0 = - 1/infinity
0 * infinity = -1 / infinity * infinity
0 = -1
1 = 0
Re:This is just faulty math (Score:3, Interesting)
Re:This is second place (Score:3, Interesting)
Re:I've tried what you suggest, and it DOESN'T WOR (Score:5, Interesting)
People who believe that 0.999... does not equal one also believe that 0.333... does not equal 1/3, and for many of the same reasons.
For once in my life I can claim someone is underestimating the average person!
I don't believe .999... = 1. Let me qualify that a bit, I intellectually and academically know it, but on a softer, more psychological level, I don't actually believe it. When presented with it, my first reaction would be "Hell no! Stupid.", even though I know it is true.
Why? Because your mapping two concepts that we all were taught as a kid isn't true. Does .9 = 1? Or .99? Or .999? or ... Or .999999999999(a ridiculous but non-infinite number of times)? Most grade school kids would say "no", and be correct. Then you hit the infinite jump, and suddenly it becomes true. So you run into two problems, the problem of it not being immediately obvious (common sense), and the problem of conceptualizing infinity.
On a lower level, its like saying A = ~A. You have a proof saying basically that ~A was A all along, so the actual preposition was wrong, which makes sense, but on a surface level all you can see is A =~A.
I have no problem whatsoever with 1/3 = 0.3333... This makes sense, its like stating A = A. 1/3 being 0.3333 is obvious. I would even get in trouble in lower level math classes for not mucking with fractions, and going straight for the decimals, since I never say fractions outside of cookbooks and socket sizes. 1/3 = 0.33333... makes sense, it is clear and obvious, and can be explained with a single phrase (not a proof); "the "/" means division". .999999... doesn't have this.
No, I'm not stupid, or at least for this reason. I know damn well that 0.9999... = 1, and if I ever find myself in a situation where that bit of knowledge can be applied (usefully, not just for building my ego on the internet), I will do it properly. My first reaction is still "bullshit!" on a visceral level, though. I don't perceive it as true, even if I know it is.
I suppose I can map this experience to most of the "social knowledge vs. science" debates in our culture currently. I won't.
Re:(0.999...)st Post! (Score:3, Interesting)
It isn't "my" semicolon notation, it's called lightstone's notation. That said, I probably bastardized it. I believe, but could be wrong, it's discussed here:
* Infinitesimals and Integration
* A. H. Lightstone
* Mathematics Magazine
Vol. 46, No. 1 (Jan., 1973), pp. 20-30
(article consists of 11 pages)
* Published by: Mathematical Association of America
* Stable URL: http://www.jstor.org/stable/2688575 [jstor.org]
Re:This is second place (Score:3, Interesting)
I am not a mathematician, but I have always considered the Monty Hall trick to be more of a word trick than any basis in mathematics. Look at it this way:
If you pick one door out of a million and Monty Hall opens 999,998 others and it's between yours and the other door, there's a good chance Monty Hall knew where the car was since the chances of him doing that at random are so small, so of course your chance is better if you switch to the other door since there is a strong probability he didn't miss that one door just by chance.
On the other hand, if Monty opens 999,998 doors at random and still hasn't revealed the car, despite the unliklihood of that happening, then the odds are still 50/50 that you have the right door. The odds at first might have been 1 in a million, but now they are 1 in 2 since the other 999,998 have been eliminated without a biased factor (Monty's choice).
It's the human element that always seems to get lost here. The real question is whether the other 999,998 doors are eliminated by someone who knows where the car is (Monty) or by chance.