## 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."*
## (0.999...)st Post! (Score:5, Funny)

(0.999...)st Post!

## Re: (Score:3, Funny)

2+2=5 for sufficiently big values of 2.

## Re:(0.999...)st Post! (Score:5, Funny)

2+2=5 for sufficiently big values of 2.

or for sufficiently small values of 5.

## Re: (Score:3, Funny)

The Who were right in

Mobile- One and one don't make two, one and one make one.One and three is one, too.

One and two is zero.

## Re:(0.999...)st Post! (Score:5, Funny)

But, if you choose the rounding method known as "floor", then 0.999... is 0, right? So for sufficiently bad rounding methods, 1 = 0.

## Re: (Score:3, Funny)

## Re: (Score:3, Insightful)

The floor operation arguably doesn't make much sense for infinite decimal places. I don't know if it's ever used for anything in mathematics except finite-precision numeric methods.

(also, it was a joke. Laugh.)

## Re: (Score:3, Funny)

There is no such thing as beyond infinite...

Unfamiliar with Pixar physics, I see.

## Re: (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)

* Publishe

## 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:(0.999...)st Post! (Score:5, Funny)

Geez these first posters. Like spammers, always looking for a new attack vector. I'm sure he's been sitting on this particular exploit for a long time, just waiting for his opportunity to strike. You've won today, but we're all onto your trick when you try to (0.999...)st post the next story...

## Re: (Score:3, Funny)

## Re:(0.999...)st Post! (Score:5, Funny)

Shoot, I just spent my last 0.999... mod points.

## I went one further (Score:5, Funny)

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

## Re:I went one further (Score:4, Insightful)

And seriously... is this really front page material? The simplest proof is to say "express 1/9" as a decimal. Now multiply both sides by 9. I remember this in elementary school algebra.

## Re:I went one further (Score:5, Insightful)

## Re:I went one further (Score:5, Insightful)

a lot of people have this implicit assumption that if a few simple steps bring them to a result that doesn't look like it makes sense, then they did something wrong.

It should be noted that this is

nota bad thing. Indeed, it is one of the first things that a good math teacher will teach to the class - all answers should go through a 'does this make sense?' filter before you consider the problem done. It is only very rarely that it causes problems, and it is exceedingly common that it prevents them.## Re:I went one further (Score:5, Insightful)

The problem with that, though, is that people have trouble accepting that there was nothing wrong with what they did -- a lot of people have this implicit assumption that if a few simple steps bring them to a result that doesn't look like it makes sense, then they did something wrong.

Nope, the problem is that the people who discuss this question are lousy teachers. They set it up deliberately to create a block in other people's minds that makes it unnecessarily difficult for them to understand what is being claimed and why it is true.

If instead they said, "It is possible to represent numbers in different ways. We all know this, and it's completely uninteresting, but I'm going to bore you with it anyway. You know you can represent 1/3 as 0.3333... right? No big deal. Now curiously that also means you can represent 1/1 = (3*1/3) as 3*0.3333... or 0.99999... It's just a different representation of exactly the same value. You can of course also represent 1 as 5*1/5 1/2+1/2 and all kinds of other awkward and unintersting ways, too."

I'm not sure why people insist on presenting this result in the most counter-intuitive way possible and then wasting vast amounts of time trying to undo the damage they've inflicted with their incompetent introduction of the problem. My guess is that they are simply not very smart, as anyone who isn't fairly dumb would see that there is an obvious pedagogical problem at play here, and correct their presentation accordingly, rather than blindly and stupidly repeating the rote "0.9999... = 1" introduction to the remarkably dull fact that you can represent the same value in different ways.

Of course, in an insanely strictly typed language with infinite precision 0.999... would not quite be the same as 1, as the former is a real and the latter is an integer, so despite having the same value their different types would mean they could not be used identically in all circumstances.

## I've tried what you suggest, and it DOESN'T WORK (Score:4, Insightful)

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

alsobelieve 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.## 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:I went one further (Score:4, Informative)

## Re:I went one further (Score:4, Informative)

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.

## Re:I went one further (Score:5, Informative)

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 :)

## Re: (Score:3, Informative)

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 :)

Yeah, that's the problem, but for those interested...

The base problem with this is that unlike the logarithm for real numbers, the logarithm for complex values is not a function (or, if you like, it's a "multi-valued function"). This comes from the interesting fact that

x^1has 1 solution,x^2has two solutions,x^3has 3 solutions, and so on. We kind of fudge around it in reals, becausex^nwill only ever have one or two solutions, but in the complex plane it hasnsolutions, and things are much more compl## Re: (Score:3, Informative)

I'm probably too late to get modded up, but since none of the existing responses gave the exactly correct explanation, I'll have to post rather than moderated.

sqrt(1) is 1. It's not -1. By definition.

A list of transformations of an equality like the one given in the grandparent's "proof" is shorthand for a list of "implies" statements. For example, a proof like this:

is actually shorthand for:

## I've had this argument more times than I'd like (Score:4, Insightful)

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.

## Re: (Score:3, Informative)

No, you're missing the whole point. 1/3 is

exactlyequal to 0.333... with an infinite number of trailing digits. It's not an approximation or an estimate, it is two ways of representing the exact same real number.Here's how you convince yourself: If 1/3 was really close but not quite 0.333..., then we could split the difference between those numbers and find another real number between them. But we can't, which means we were wrong to assume that 1/3 and 0.333... were distinct.

## Re:I went one further (Score:4, Insightful)

Conceptually, 0.999... keeps getting closer and closer to 1, as you add more decimal places. It approaches 1. This limit is how all calculus works. Any series that approaches another number as you flesh out the series further and further, will be that number once you have taken the series to infinity.

## Re: (Score:3)

## Re: (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: (Score:3, Insightful)

And how do you do multiplication of an infinite series of digits? I'm guessing that you don't start from the right-hand side.... but beyond that your approach seems to be simple because it is incomplete. Kind of the point of the article really.

## Re: (Score:3, Insightful)

There is no point in trying to convince me: I am well aware that 0.999... = 1. It is a very basic fact about decimals that I was taught in school at a young age.

I said nothing about multiplication by 10. I know that is trivial, but if you read the GP you will see that he claimed it is easy to multiply an infinite series of digits by a number that was not the base. It's not trivial and it requires some definitions far deeper than what he was alluding to. This is why his "simple" proof can only be proven cor

## Re:I went one further (Score:5, Funny)

And seriously... is this really front page material?

You'd rather argue about smartphones?

## Re: (Score:3, Funny)

Excellent point. I will submit a story about 1/2 = 0.4999999....

## Re:I went one further (Score:4, Funny)

Absolutely.

Did I tell you that the N899.999... is the the bee's knees?

## Re: (Score:3, Funny)

## Finally (Score:2, Funny)

## Re:Finally (Score:5, Funny)

just as long as no-one proves 0 = 1 we computerpeople are safe...

## Re: (Score:3)

But b00000000 = b11111111 in one's complement [wikipedia.org] system...

So 0 == 1 as long as you're using 1 bit wide one's complement integers...

## Re:Finally (Score:4, Insightful)

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.)

## Re:Finally (Score:5, Informative)

I am compelled to answer...

Divide both sides by (b - c - a) is dividing by zero.

## Re: (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:Finally (Score:4, Funny)

## Re:Finally (Score:4, Insightful)

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 [princeton.edu]

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...

## Re:Finally (Score:4, Insightful)

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.

## This is second place (Score:5, Insightful)

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.

## Re: (Score:3, Insightful)

## Re:This is second place (Score:4, Informative)

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.

## Re: (Score:2)

## Re: (Score:3)

## Re: (Score:3)

## Re: (Score:3, Interesting)

## Re: (Score:3, Informative)

0.999... is not a "decimal expansion" of some number, but rather it denotes a sequence of numbers: 0.999, 0.9999, 0.99999, ....

Talk about having difficulty understanding the concept. 0.9999... is not a sequence, it is a series:

...

9/10 + 9/100 + 9/1000 + 9/10000 +

Which anyone who paid attention in middle school will recognize as a geometric series (well, actually, it is a multiple of a geometric series).

The fact that "the real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers"...

## 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: (Score:3, Insightful)

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!

So, what's the exact value of PI represented as a fraction?

## Re:This is second place (Score:5, Funny)

Typical engineer. Here's the operations perspective: .999... means "sometime fail".

a reliability of 1.0 equates to never fail.

a reliability of

The sales guy will sell 1.0, and when failure happens, explain that what was really meant was .999...

Good luck with that.

## Re:This is second place (Score:5, Informative)

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?

## Re: (Score:3, Insightful)

## Re: (Score:3)

Better yet, how does he know that 0.1111.... x 9 = 0.9999.... ?

All these simple proofs leave something to be desired, and take certain assumptions into account that everyone is taught, but they often don't know the justification. For example, one proof had:

10 x 0.999.... = 9.999....

Again, why? Shifting the decimal point is a trick we're all taught, but they never proved it to us in the general case.

Ultimately, the proof is that

anynon-terminating decimal isdefinedto equal to the limit of the partial sums## In base 10 it is 0.14285714285711. (Score:4, Informative)

Also a terminology issue, the number of ones in 0.111111... is indeed "Countable" [wikipedia.org].

## Re: (Score:3, Insightful)

You are confusing a symbolic representation for a number because the symbol contains numbers in it. It is physically impossible to represent certain numbers using base 10. Pi for example. Is is less obvious, but still a fact that 1/3 and 1/9 are in fact impossible to accurately represent using base 10. The .1111... .33333... and .9999... are all of rather limited accuracy symbols, not numbers, just as if I were to say pi = 3.14159+ The 3.14159+ is a symbol representing Pi, not a number, similarly .9999999... is NOT a number, but is instead a symbolic representation of a number.

.1111... is understood to stand for the supremum [wikipedia.org] of the set {0,1/10,11/100,111/1000...}. See Rudin, "Principles of Mathematical Analysis", page 11. Likewise for .3333..., .999999...., and 3.14159+... where the sets are defined accordingly.

The fact that long division or electronic calculators come up with those results is an indication of human accounting for the limitations of our mathematical symptoms.

Calculators produce such results because they are useful approximations of the supremum.

In base 8, .11111111 = 1/8 + 1/80 + 1/800 + .... That number, multiplied by 7 becomes .77777777777... or 7/8 + 7/80 + 7/800 +... You can use the same bad math you used earlier to prove that 1 = .7777777... base 8 that you used to claim that 1 = .99999 in base 10

Here you are in error. 7*.111111...= 7*(1/8+1/8^2+1/8^3+...) = 7/8 + 7/8^2 + 7/8^3 + ... = (7/8)/(1-1/8) = 1; the reduction from an infinite geometric series to 7/8/(1-1/8) is a commo

## Re: (Score:3)

Are you really that clueless?

## Re:This is second place (Score:5, Insightful)

The Monty Hall problem and its delinquent cousin the Tuesday Boy problem are genuinely difficult because the answer is highly dependent on the way that the question is posed.

0.9999...=1 is not genuinely difficult because at the end of the day it's a very informal statement about adding an infinite number of decimals, and the only real controversy about the statement exists among 4chan trolls and Wikipedia users. Most who don't understand don't care and most who do understand also don't care.

The only people with a problem are the people who don't understand but still care, but then that's the problem with most topics these days.

## Re:This is second place (Score:5, Insightful)

The Monty Hall problem and its delinquent cousin the Tuesday Boy problem are genuinely difficult because the answer is highly dependent on the way that the question is posed.

I would argue that the Monty Hall problem is difficult because people don't take into account the fact that the result is

NOTpath independent.It would be much easier (I think) to understand intuitively if people realized that it was highly likely that they picked the wrong door to start. A more intuitive way of explaining the problem to somebody would be to increase the number of doors - to say, infinity. If there are infinity minus one closed doors with goats behind them, and a single door with a car behind it, the odds are obviously very high that you picked a goat. The probability that you picked the car is vanishingly small. Therefore, when the host opens

everydoor except yours and one other, and they all reveal goats, the odds are very, very high that the other door hides a car, and yours hides a goat.Now, reduce that to 3 doors. The same logic applies.

## Re: (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 on

## Re: (Score:3, Insightful)

Tuesday Boy is a difficult problem not because of the math behind it but because of the grammar in the question.

## Re:This is second place (Score:5, Insightful)

## Re:This is second place (Score:5, Funny)

I remember being told this in highschool. There was much objection, but the teacher shut us up by simply saying "give me a number in between them."

Duh. 0.9999... and a half!

## Re: (Score:3, Informative)

## Re: (Score:3, Informative)

This is Slashdot. Better explained this way:

When the decimal repeats, you enter a loop. The ...9991 is unreachable code.

## they have wikipedia in Montana? (Score:2)

"When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator the result is 9 9, which is 0. The final step uses algebra:"## metacafe (Score:2)

## Re: (Score:3, Insightful)

0.999.... might be equal to 1, but 0.999 isnotequal to 1:x=0.999

10x = 9.99

10x-x = 9.99 - 0.999

9x = 8.991

x = 0.999

This is true whether there's three 9s or a hundred 9s, so I can see the confusion.

## Time to Update my SLA (Score:5, Funny)

Now I can replace my SLA with 100% uptime.

## Or (Score:3, Insightful)

1/3 = 0.3333...

2/3 = 0.6666...

0.3333.... + 0.6666.... = 0.9999....

1/3 + 2/3 = 1 = 0.9999.....

## Re:Or (Score:4, Funny)

As a mathematician, I have always hated people who claim that 0.999... = 1 can't be true.

As a nerd, I have always hated people that hate others for trivial reasons. You're just a math bully.

## Oh yeah? Well... (Score:2, Funny)

a = b

a^2 = ab

a^2 - b^2 = ab - b^2

(a+b)(a-b) = b(a-b)

a + b = b

2b = b

2 = 1

## Re:Oh yeah? Well... (Score:4, Informative)

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.

## 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]

## Humans are just biased towards natural numbers (Score:3, Insightful)

Humans are used to natural numbers because they're simple. But do natural numbers even exist in the real world? For the vast majority of practical purposes, 0.99999 can be thought of as one. But "one" itself is usually just a construct in the real world. There is no such thing as the perfect one of anything. The more precise we get, the more "one" becomes more of a mathematical ideal than a reality. So we spend our entire lives rounding off, because that's practical. We teach kids to count 1, 2, 3, 4... We can't very well teach them to count 0.000001, 0.00001, 0.0001, 0.001... (or any of the infinite variations of "counting" without resorting to natural numbers).

Proving that 0.99999 = 1 is an interesting intellectual exercise. But in the real world, we do it every minute of every day.

In other words--eh, close enough.

## Re: (Score:3, Insightful)

It's not about rounding, or counting, or any real-world interpretation of any math concepts. It's about the trouble that people have understanding that 0.999... is simply notation that refers to the limit of an infinite sequence of numbers.

## Re:Humans are just biased towards natural numbers (Score:5, Informative)

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".

## Vindication (Score:2)

So after all these years, has Intel been vindicated [wikipedia.org]?

## Cribbed, Since My Memory for Jokes Sucks (Score:5, Funny)

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."

## Re:Cribbed, Since My Memory for Jokes Sucks (Score:5, Informative)

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.

## Corrected, Since My Memory for Jokes Sucks (Score:5, Funny)

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."

## And if (Score:4, Funny)

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.

## 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.

## More fun... (Score:4, Informative)

## Re: (Score:3, Funny)

## Re: (Score:3, Funny)

Oh... They aren't empty. The aliens live in them now. They think the high radiation is good for their complexion.

No, Ziaxia, I wasn't telling them anything on slashdot, GET OUT OF MY HEAD! GET OUT OF MY HEAD! AHHHHHH!!!! Don't make me explode! ^h^h^h^h^h^hcarrier lost

## Re: (Score:2)

## Re: (Score:3, Funny)

## Re:This is just faulty math (Score:5, Insightful)

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

exactlyinfinite 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.

## Re: (Score:3, Insightful)

The series is infinite, you don't lose one.

Just because you can not show the number as a whole does not mean

you can not perform operations using it.

i.e. Think of pi.

## Re: (Score:3, Insightful)

How can you multiply

## Re: (Score:3, Interesting)

## Re: (Score:3, Informative)

Far from true. A rational number is a number you could get by expressing as a ratio (real number divided by real number). Any infinite repeating decimal is easily shown as a ratio (and often of simple integers to boot), i.e., a rational number. 0.22222... is 2/9. 0.456456456456456... is 456/999. And so on.

## Re:When you add/subtract/multiply/divide infinite (Score:5, Funny)

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 [wikipedia.org]

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.

## Re:Cat and Mouse (Score:5, Funny)

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.