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Proving 0.999... Is Equal To 1
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."
It is easy to explain.
1. 1/9 = 0.111111111111111111111111111111.....
2. Multiply each side by 9
3. 9/9 = 0.999999999999999999999999999999......
4. Simplify fraction
5. 1 = 0.999999999999999999999999999999......
Monty Hall trips up even serious math enthusiasts.
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this probably isn't necessary for most of the Slashdot crowd, but...
(a+b)(a-b) = b(a-b) --> a + b = b
Required division by (a-b) on both sides. Since a = b, this is division by zero.
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Unfortunatelly, your proof is not valid. You are trying to prove something which you postulate in your first step.
How do you know 1/9 equals 0.1111111.... ?
He begged the question! For anyone confused about the term "beg the question," this is exactly what it means: assuming the proposition to be proved in the premise.
But that begs the question: is the classical meaning already dead, replaced with the much more easily understood modern usage demonstrated here?
"I zero-index my hamsters" - Willtor (147206)
Actually, a good physicist should have been able to give an answer (or something close to it) as well...
Eventually, they will come to a point where they would be required to move less than 1.616252(81)×1035 meters closer together. From the uncertainty principle, we know we cannot measure position more accurately than that. So either they will not move at all, or they will superimpose at that point.
'Sensible' is a curse word.
I am compelled to answer...
Divide both sides by (b - c - a) is dividing by zero.
See my journal for slashdot ID's by year. Mine created in 2005. http://slashdot.org/journal/289875/slashdot-ids-by-year
They're not proving "0.99999 = 1" at all. That's not true. They're proving that "0.999... = 1". One is an infinite sequence of digits, and the other isn't. The distinction is important. The proof of "0.999... = 1" has nothing to do with rounding, and to suggest so indicates a (common) gross misunderstanding of the problem.
First, you only measure things with such poor precision because you're working well above the quantum level.
Second, natural numbers are certainly important. For one, they're critical to our understanding of the rest of mathematics, which is important for fancy things like being able to take measurements and manipulate them at all. For another, we work with whole numbers of objects all the time -- two apples, ten antelope, four huts, etc. It's not "10 +/- 0.01 antelope".
It's more fun to work out why this proof fails when using non-standard analysis (in which 0.999... != 1).
Quidnam Latine loqui modo coepi?
People assuming they did something wrong when the result "doesn't make sense" isn't the problem.
People failing to distinguish between a notation and a number, creating the belief that "0.99(9)=1" doesn't make sense, is the problem.
Consider this proof, which follows simple steps to reach a conclusion that doesn't make sense:
i^2 = -1 (definition of i)
i^2 * i^2 = -1 * -1
i^4 = 1
sqrt(i^4) = sqrt(1)
i^2 = 1
-1 = 1
Then if you want you can add 1 to both sides and divide by 2, to find 0 = 1.
Now, do you know why this proof is bogus? When I was in high school, we were introduced to imaginary numbers, and I drew up a slightly more obfuscated version of the above; it had a lot of people (including a couple relatively sharp teachers) in "I know you did something wrong because the result doesn't make sense" mode for a long time.
The fault, of course, lies with the sqrt() step. For a=a to imply sqrt(a)=sqrt(a), we have to interpret sqrt(a) as the pricple square root function, so sqrt(x^y) = x^(y/2) doesn't necessary work when x isn't a real number.
Without the motivation of "this result cannot be right", I wouldn't have puzzled this out. More than that, the solution comes from understanding that rules we take for granted only apply to certain types of number. Applying that to 0.99(9), it's easy for people to convince themselves that repeating decimals are a special class of number subject to "some rule I just don't know".
But in this instance, that reasoning is flawed, because .99(9) really is just a regular real number in a weird notation.
Surely the problem is that you're assuming sqrt(1) = 1 when actually it is +- 1? You're throwing away the sign change in that step :)
Yes it is. Insofar as using infinity as an arithmetical value is valid, 10/infinity = 9/infinity = 1/infinity = 0.
Also a terminology issue, the number of ones in 0.111111... is indeed "Countable".