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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."
(0.999...)st Post!
I was able to prove that with even one less "9" after the decimal point, it STILL equaled 1. I plan on doing this for a few more iteration until I can prove that . = 1
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0.999... = 1 is second place to the Monty Hall Problem on the list of things that people have difficulty understanding and accepting the proof of. It is second place because the only department where I do not see graduate students giving me a confused look is the math department; with the Monty Hall problem, I will sometimes get a confused look even from people in the math department.
The other reason I put it in second place is that most people have difficult understanding the problem at all, whereas very few people have trouble understand what the Monty Hall problem is asking.
Palm trees and 8
Now I can replace my SLA with 100% uptime.
1/3 = 0.3333...
2/3 = 0.6666...
0.3333.... + 0.6666.... = 0.9999....
1/3 + 2/3 = 1 = 0.9999.....
They damn well better, how else will I measure out all this dental floss.
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
You can tell how powerful someone is by the magnitude of the crime they can commit and be able to get away with.
just as long as no-one proves 0 = 1 we computerpeople are safe...
People, what a bunch of bastards
In the high school gym, all the girls in the class were lined up against one wall, and all the boys against the opposite wall. Then, every ten seconds, they walked toward each other until they were half the previous distance apart. A mathematician, a physicist, and an engineer were asked, "When will the girls and boys meet?"
The mathematician said: "Never."
The physicist said: "In an infinite amount of time."
The engineer said: "Well... in about two minutes, they'll be close enough for all practical purposes."
0.99999... is equal to 1, then 0.999999...8 is equal to 0.99999... and 0.9999999...7 is equal to 0.999999...6 etc etc etc until 1 = 0! Holy shit!
Or we could just admit that using a tool incorrectly produces idiotic results.
Seven puppies were harmed during the making of this post.
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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Actually, if you define 0.999... as having an infinite number of decimal points, then it is true. And that's how that ellipsis is defined! It means exactly infinite repeating decimals.
You've demonstrated the first hurdle that this problem raises in people's brains: they start thinking about adding "one more" decimal point to the expression, meaning they're thinking of a large but finite number of decimal points. And the second hurdle: people find it hard to believe that you can do mathematics with "infinity" as a meaningful quantity.
No kidding!!! What do you say at this point?
If pressed, many logicians will admit that the modern foundation of mathematics (ZFC) is probably inconsistent.
See this article:
http://www.math.princeton.edu/~nelson/papers/warn.pdf
The author discusses an informal survey he took among loogicians on page three.
If someone ever discovers a paradox, we can simply scale back to some other system and keep most of what we know, but still...
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Wrong, wrong and wrong.
First off, you're not talking about sets, but separate finite numbers.
Then, infinity is neither rational nor irrational.
Then, all numbers that have "infinite repeating decimals" are rational. See : http://en.wikipedia.org/wiki/Rational_number
So that means 0.999999..... is rational. Which rational you ask? Why! 9/9 :D
Finally, if you say 0.99999999..... is less than 1 : what is the difference between both?
We know it's less than any positive epsilon (0.1, 0.01, or 0.00000.....00001).
Which means it's nil.
There's no place for a single mosquito fart between 0.999999... and 1.
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.
My UID is prime. Hah!
Suppose you have 3 numbers, a, b and c such that c = b - a.
Multiply each side by (b - a) to get:
c(b - a) = (b - a)(b - a) => Or....
cb - ca = b^2 - 2ba + a^2 => Now add (ab - a^2 - cb) to both sides
ab - ca - a^2 = b^2 - cb - ba => Or....
a(b - c - a) = b(b - c - a) => Divide both sides by (b - c - a) and.....
a = b
There you go! Proof that any two numbers (such as 0 and 1) are equal.
(Yes, I know there's a flaw in there. Let's see who'll spot it first.)
My sci-fi novel, Ghost Thief, is now available from Amazon.com.
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
Okay, but this isn't a problem with the foundation of math being inconsistent, this is a problem with people not knowing how to write the number normally known as "1" in a different way. Most people would grasp "3/3" as being the same as 1, but this *looks* different because they're unused to seeing it.
The fact that the fractions 1/3 (known in decimal notation as .3...) and 2/3 (known in decimal as .6...) have a sum that can be written funny doesn't mean that they don't still add up to 1.
A mathematical amusement causes people confusion and consternation. It's like asking someone why they appear reversed left-to-right in a mirror, but not top-to-bottom, and saying there's an inconsistency in the foundation of physics.
The problem is that partial understanding of a subject and an associated problem in that subject makes things *appear* inconsistent when they are not.
"Murphy was an optimist" - O'Toole's commentary on Murphy's Law
In the high school gym, all the girls in the class were lined up against one wall, and all the boys against the opposite wall. Then, every ten seconds, they walked toward each other until they were half the previous distance apart. A mathematician, a physicist, and an engineer were asked, "When will the girls and boys meet?"
The mathematician said: "Never."
The physicist said: "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."
The engineer said: "Well... in about two minutes, they'll be close enough for all practical purposes."
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".
I think about this from time to time when messing with changing units. 2 cm squared is 4 cm2 and 4>2. But .02 m squared is .0004 m2 and .0004 .02. They both describe the same amounts. I obsessed over this for a while one day until I realized that the relative values of a scaler and an area meant little, but for a while I was really worried.
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second one orders a half a beer. The third orders a quarter of a beer. The bartender says, "You're all idiots," and pours two beers.
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?
The problem with the argument you present is that people who don't believe 0.999...=1 also don't believe that 0.333...=1/3. They can't quite wrap their heads around the concept of infinity, so in their minds 0.333... continually comes closer to 1/3, but never quite reaches it because they can only imagine a finite number of digits. They honestly think of infinity as being a really large finite number, so they believe that no matter how many digits you add to 0.333..., it never quite reaches 1/3.
Another part of the problem is that many people simply can't wrap their heads around is that they don't separate the idea of a number and the symbols used to represent numbers, thus they cannot grasp that some numbers can be represented in more than one way by our number system.
As soon as you get to "You know you can represent 1/3 as 0.333... right?", you hit a brick wall. 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. Taking your approach, you simply shift from arguing about whether or not 0.999... equals one to arguing about whether or not 0.333... equals 1/3. You have to get at the root of the problem of why they refuse to believe those numbers are equal before you can get anywhere.
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.
A patriot must always be ready to defend his country against his government. -edward abbey
Also a terminology issue, the number of ones in 0.111111... is indeed "Countable".