*(note: this post has been closed to comments; comments about it on other pages will be deleted!)*

**UPDATE!!:** The saga continues at this post.

**MORE UPDATES, WITH REFUTATIONS!**

Every year I get a few kids in my classes who argue with me on this. And there are arguers all over the web. And I just know I'm going to get contentious "but it just *can't* be true" whiners in my comments. But I feel obliged to step into this fray.

.9 repeating equals one. In other words, .9999999... is the same number as 1. They're 2 different ways of writing the same number. Kind of like 1.5, 1 1/2, 3/2, and 99/66. All the same. I know some of you still don't believe me, so let me say it loudly:

Do you believe it yet? Well, I do have a couple of arguments besides mere size. Let's look at some reasons why it's true. Then we'll look at some reasons why it's not false, which is something different entirely. The standard algebra proof (which, if you modify it a little, works to convert any repeating decimal into a fraction) runs something like this. Let x = .9999999..., and then multiply both sides by 10, so you get 10x = 9.9999999... because multiplying by 10 just moves the decimal point to the right. Then stack those two equations and subtract them (this is a legal move because you're subtracting the same quantity from the left side, where it's called x, as from the right, where it's called .9999999..., but they're the same because they're equal. We said so, remember?):

Surely if 9x = 9, then x = 1. But since x also equals .9999999... we get that .9999999... = 1. The algebra is impeccable.

But I know that this is unconvincing to many people. So here's another argument. Most people who have trouble with this fact oddly *don't* have trouble with the fact that 1/3 = .3333333... . Well, consider the following addition of equations then:

This seems simplistic, but it's very, very convincing, isn't it? Or try it with some other denominator:

Which works out very nicely. Or even:

It will work for any two fractions that have a repeating decimal representation and that add up to 1.

Those are my first two demonstrations that our fact is true (the last one is at the end). But then the whiners start in about all the reasons they think it's false. So here's why it's not false:

- ".9 repeating doesn't equal 1, it gets closer and closer to 1."

May I remind you that .9 repeating is a *number*. That means it has it's place on the number line somewhere. Which means that it's not "getting" anywhere. It doesn't move. It either equals 1 or it doesn't (it does of course), but it doesn't "get" closer to 1.

- ".9 repeating is obviously less than 1."

Hmmmm...it might be obvious to you, but it's not obvious to me. Is it really less than 1? How much less than 1? No, seriously...tell me how much less? What is 1 minus .99999999...?

Really???? *Infinitely many* zeros and then after the *infinite* list that *never ends*, there's a 1???? Surely that's stranger than the possibility that .9 repeating simply does equal 1. Or for something even stranger, consider this: if .9 repeating is less than 1, then we ought to be able to do something very simple with those two numbers: find their average. What's the number directly between the two? Or for that matter, name *any* number between the two. Let me guess: the average is .99999...05? So after this *infinite* list of 9s, there's the possibility of starting up multiple-digit extensions? Doesn't that just raise the obvious question: What about .9999999...9999999...? Namely, infinitely many 9s, and then after that infinite list, there's *another* infinite list of 9s? How, exactly is that different from the original infinite list of 9s? If you saw it written out, where would the break between the lists be?

I'm afraid that if you apply the "huh??" test of strangeness, you get a much higher strangeness factor if you say that .9999999... is *not* 1 than you do if you say it *is* 1.

- "Uhhhhh, I'm sorry, but I still don't believe you. .99999... just can't equal 1."

Well, let's look a little more carefully at what we really mean by .999999...:

This equation isn't really up for debate, right? It's simply the meaning of our place value system made explicit. That thing on the right hand side is called an infinite geometic series. They have been studied extensively in math. The word "geometric" means that each term of the series is the identical multiple (in this case 1/10) of the previous term. The **definition** of the sum of an infinite geometric series (and other series, too, but we won't get into those) goes something this:

- Start making a list of partial sums: the sum of the first one number, then the sum of the first two numbers, then the sum of the first three, etc.
- Examine your list closely. In this case the list is: .9, .99, .999, .9999, .... (Note that the actual number .99999.... is not on the list, since every number on the list has finitely many 9s.)
- Find some numbers that are bigger than every single number on your list. Like 53, 3.14, and a million.
- Of all the numbers that are bigger than every number on your list, find the smallest possible such number. I think we can all agree that this smallest number is 1.
- That smallest number that can't be exceeded by anything on the list is the
**definition**of the sum of the geometric series.

Notice that I keep putting the word **definition** in bold face. (See, I did it again!) That's because it's a **definition**, which isn't really up for debate. It is the nature of a mathematical definition that once you acccept it, you have to agree to its consequences. In other words, .99999... = 1 by the **definition** of the sum of a geometric series. It's also true if you use the popular formula

a/(1 - r) with a = 9/10, and r = 1/10.

We're left with this: merely *saying* ".99999... doesn't equal 1" admits the fact that this number .99999... exists. And if it exists, it equals 1 by definition. The only way out for you now, if you still don't believe it, is to have a different working definition of the sum of an infinite series (go talk to some math professors, and see how far you get) or to deny the very existence of the number .9999.... I have seen a lot of people doubt that the number equals 1, but very few of them are willing to deny the very existence of that number. If you want to play "there's no such thing as infinitely long decimal representations," I'm afraid you won't get very far, because there's always the number pi to worry about, too, you know.

Okay, so there's my rant. .9 repeating equals one. No, I'm sorry, it does.

A couple of questions from a non-math-geek.

Perhaps I'm wrong, but it seems to me that if .99999... is the sum of a geometric series (as I agree it is), then it should be resolvable as any other converging geometric series: that is, it should be possible to find the limit of the series as n approaches infinity.

The formula for that is given at http://en.wikipedia.org/wiki/Geometric_series#Infinite_geometric_series.

Pardon me, but I don't know how to reproduce the formula directly here, so this is a bit cumbersome, but if you can follow my substitution, it seems that in the series you've chosen a=9/10 and r=1/10. This is good, since |r|is less than 1 and therefore this series will converge. In that case the sum of the geometric series is given by...

S = a/(1-r)

a=9/10, r=1/10, (1-r)=9/10

S = (9/10)/(9/10) = 1

Yet this only establishes that 1 is the limit of the sum of the series as n approaches infinity. As I understand limits, that means that it's possible to get that sum arbitrarily close to 1 by setting n as close as one wishes to infinity.

Have I got that right so far? If so, then I guess the fundamental philosophical question is, as n approaches infinity, can it ever actually reach infinity? If the answer is yes, then the sum of the series equals 1. If no, then the sum only converges on 1.

Not so?

Posted by: Paul S | June 19, 2006 at 01:37 PM

"Yet this only establishes that 1 is the limit of the sum of the series as n approaches infinity. As I understand limits, that means that it's possible to get that sum arbitrarily close to 1 by setting n as close as one wishes to infinity."

Yes. You can get arbitrarily close to 1 by stringing a large FINITE number (n) of 9's after the decimal point. But 0.9999... (I can't do the bar here in the comments) is by definition, an infinite number of nines, which means it's equal to the limit.

Posted by: The Science Pundit | June 19, 2006 at 05:39 PM

So, what is the number both closest to 1 and smaller than 1? If not .999 repeating then what number?

Posted by: A simple question | June 20, 2006 at 02:13 AM

when are two numbers different from each other? if you can put a (real) number between them!

compare 1 and 2: there's for example 1.5 between them, so they are different.

compare 1.2345 and 1.2346: there's 1.23455 between them - so they differ.

as u cannot give a number, which lies between 1 and 0.999999..., those two numbers have to equal!

Posted by: r0bert | June 20, 2006 at 02:23 AM

This problem is from a whole class of problems people call "counter-intuitive problems" for the simple reason that what your intuition tells you doesn't match up with reality, thereby leaving you puzzled, trolling or just simply denying every proof shown to you.

Our intuition plays a much larger part in our everyday reasoning than we normally notice - and that's not a bad thing. But it mostly deals with "normal" stuff. Our intuition can deal with information quite nicely that will navigate you through a supermarket, will make you spill the right liquids into a pot in your kitchen and will make you do valid decisions.

However, it was never designed for reasoning on infinity or other weird mathematical phenomena.

So, everyone who feels a certain violent reaction after reading this article, sit back, relax, breath. And then, slowly, very, very slowly, reread the PROOFS Polymath pasted. It is hard to digest, but that is just because our brains tend to equate "infinity" with "an insanely huge number" - which isn't true. Infinity is NOT a number, it is a concept. And an infinite number of 9s can have effects that may appear strange to us but are nonetheless true.

Mmmh. But you gave me the idea to post some other counter-intuitive problems on my blog soonish :)

Should attract some trolls too :)

Posted by: Jabberwockey | June 20, 2006 at 02:40 AM

>as u cannot give a number, which lies between 1 and 0.999999..., those two numbers have to equal!

wow! one comment to end all the discussion

Posted by: KaanEngin | June 20, 2006 at 02:48 AM

It's an interesting post, and I have, I believe, heard this before from other mathy types... but I am curious how this relates to Feynman's apparently succesful use of the mathematical manipulation of infinities.

I am certainly not an expert of the subject, but it has always seemed to me that Feynman's success in this matter would call into question the previously held stipulations regarding the limits of infinite sums.

Posted by: Palantar | June 20, 2006 at 02:55 AM

Dear old chimaera2005 raised the idea of mixing philosophical and mathematical infinities, but he missed the chance to quote William Blake, who seems to have grasped exactly the kind of infinity that's being discussed here:

To see a World in a grain of sand,

And Heaven in a wild flower,

Hold Infinity in the palm of your hand

And Eternity in an hour.

- William Blake, Auguries of Innocence

(And thanks to Google for helping me find that. I'm only a poet in my spare time.)

This is a good simple explanation that works. I like it; the idea that a counter-intuitive fact can be arrived at by pure thought is one of the real beauties of mathematics. I mean, figuring out that two is one less than three, that's easy; your intuition is helping you there, so you hardly even need to think. But proving that there are an infinite number of primes, or that point-nine-recurring equals one: that's gorgeous. That's mathematics.

(And I loved that line about "trying to convince an atheist that God exists", as if that was an indication of a flaw in YOUR logic... he really doesn't get it, does he? Heh heh heh...)

Posted by: Eric TF Bat | June 20, 2006 at 03:08 AM

I don't understand why people can't understand this thought. Our number system is not perfect.

If someone went to any source of reference for mathemeatics, they will see that there are either 3 or 4 rules for mathematics (one of the rules is being debated). This concept of ".9 repeating = 1" follows the rules of our number system; there is no violation. Also, there are "weaknesses" in our number system, as exhibited by this example.

You just need to remember that the number system that we use isn't archetypically perfect, it is "flawed".

Posted by: John | June 20, 2006 at 03:42 AM

It all does make sense. My only comment would be that surely this proves that there is a slight flaw in the concept of algebra? Does this mean a rethink is required? As a race we can only get better at creating new ways of thinking and new mathematical techiques etc...

I just think, we all know 7/7 is one whole, so saying it is equal to .99999999* and rounding up is in my opinion accepting there is a flaw in the logic and then choosing to be lazy and saying it is equal to 1 because using the algebraic law we can prove so!? Almost like saying, "if it isn't broke, don't fix it" philosophy, although this clearly demonstrates there is something not quite right. thats my say anyway, im not a math person, yes i studied it, as did everyone else, i'm just a black and white person and question things when i either A] don't understand or B] am not entirely convinced.

You logic works, no doubting, hell, I have been shown an equation that proves 1 is equal to 2!!! Im just saying that maybe the logic in question is flawed and its time to examine that and create a better solution. We need to progress, and in the future things will change, so why not get the ball rolling now.

I hope you see my point, just because we have been studying it for ever and take it as gospel, doesn't always mean its right in the first place...Questions : Is the world flat, will we fall off the edge if we travel too far? Does the sun revolve around the Earth?

Crude, I know, for which I am sorry, but I think it highlights my point.

Posted by: Skuff | June 20, 2006 at 04:46 AM

FWIW, the first bit of algebra is junk and 'works' with any decimal.

10x = 9.5

- x = 0.5

-----------

9x = 9

ZOMG! 0.5 = 1

Posted by: RandomPoster | June 20, 2006 at 04:48 AM

Er, 19/20 also = 1

Posted by: RandomPoster | June 20, 2006 at 04:52 AM

You're confusing people with sloppy math.

Let's take another example, to show what I mean.

Assume you have infinity, represented by oo. Assume that adding a number to infinity doesn't make it larger, and assume that substracting a number from infinity, doesn't make it smaller. (Note the similarity between this and multiplying 0.999 by 10 to get an extra digit).

Now, let's take the following equation:

oo = x + 1.

Two possiblities: either x is infinity, or it isn't.

Assume x is infinity, in that case, the equation becomes:

x = x + 1

=>

0 = 1. Contradiction. Therefore, x isn't infinity.

Assume x is NOT infinity.

oo = x + 1

=>

oo - 1 = x + 1 - 1

=>

oo = x. x IS infinity. Once again, contradiction.

This example clearly shows that as soon as you try to define addition (or any other mathematical operator) to infinity you get contradictions. Reasoning with infinity outside of the realm of limits is meaningless (there are a few exceptions).

The simple math trick above should be used to prove the opposite, that you shouldn't use the cute 0.9999|9 notation because it doesn't make sense.

*IF* you assume that 1/3 == 0.3333|3, THEN you can prove that 0.99999|9 == 1. Because the second scenario is undesirable, you do not want to accept the premise, that 1/3 == 0.3333|3.

The problem 0.9999|9 is that it is really a mathematical limit of n to infinity, of the sum of 0.9 * 10^-n.

If you would use THAT mathematical notation for 0.9999|9 people would balk and run away scared. Reasoning with infinity isn't easy, and that's why tricks like these are harmful.

Don't talk about rubbish like an infinite amount of 9s without introducing infinity properly. Otherwise, people learn the trick, but don't gain UNDERSTANDING. That's why every time people will tell you that "It's not true because...".

[Before you disagree with me, make sure you know what the point is I'm making. It's not that 0.999|9 isn't 1. ]

Posted by: Pedantic | June 20, 2006 at 04:55 AM

You're confusing a limit with a number.

1 is the smallest number which 0.999... will never equal. 1 is thus the limit of 0.999...

Posted by: Knut Arne Vedaa | June 20, 2006 at 05:02 AM

"

FWIW, the first bit of algebra is junk and 'works' with any decimal.

10x = 9.5

- x = 0.5

-----------

9x = 9

ZOMG! 0.5 = 1"

That made no sense considering the fact that if you assigned the variable x the value .5, then 10x would equal 5, not 9.5. I think you missed something.

But I digress. Every time I explain this to anyone I always start out with 1/3 + 2/3. Gets them every time. Very well written article. Thanks for the good read.

Posted by: Daniel | June 20, 2006 at 05:25 AM

Oh, and I missed the most crucial part. 0.5 is not a repeating decimal.

0.49... = 0.5, .999... = 1, 2.999... = 3, etc.

Posted by: Daniel | June 20, 2006 at 05:27 AM

People. There is no flaw.

The math this perception is based on is absolutely ok.

The only thing that makes us think that 0,9999|9 and 1 are distinct is our intuition.

It is like an optical illusion: it SEEMS as if the man in some picture is taller than the other man, but that is just an illusion, the picture plays a trick on our mind. And it is the same here.

Pedantic, you are right to say that we can't introduce INFINITY into our equations. The reason was given by Chimaera - he was right in one point, infinity is NOT a number, it is a concept. BUT 0.999|9 IS a number. 0.999|9 is just one representation for it, we could choose numerous others. But 0.999|9 is a number and a perfectly valid one.

And because it is a number, we can introduce it into our equations. And because it is a number, we can multiply it by 10. With the effect, that every 9 is shifting to the left. Every of the infinite number of 9s.

For anyone who has trouble in thinking how this could possibly work, read up on Hilbert's Hotel which is a neat and fun thought experiment to help you to realise how countable infinity works.

Again: it is your own intuition that says "that can't be right!" - the world is strange but it is like that!

Posted by: Jabberwockey | June 20, 2006 at 05:57 AM

I'm not a maths geek so be gentle...

So does a converging infinite series equal what it converges to? Or is it still converging? (or am I talking mathamatical diarrhea?)

If you can show me that then I'll be convinced. Using the algebraic formalae as proof doesn't convince me as each number symbol "represents" a number as opposed to "being" the number.

Another thing, can't you just define a number that is in between .9999... and 1, that cannot be written as a "normal" number on the conventional number lines?

Posted by: Nasum | June 20, 2006 at 05:59 AM

Oh, just remembered a nice quote from Terry Pratchett. A group of druids called "computer scientists" use stones to compute different things - therefore called computers. It works by laying out these stones in different patterns. At one point all the calculations seem flawed. They push all the stones around, only to find out:

"Oh, there's nothing wrong with the stones. It's the world that is going nuts!"

(or something in those lines)

Posted by: Jabberwockey | June 20, 2006 at 06:01 AM

If you take one cookie and divide it into three pieces of the exact same size, each piece represents 1/3 of the cookie which could numerically be represented by the number 0.33333...

If you put the pieces together, you will logically have exactly one whole cookie, which could numerically be represented by the number 1.

There's your mathematical proof that 1/3*3 != 0.99999...

0.9999... = 0.9999... and 1 = 1. Rounding a number is always a way to compromise with the truth, in the same sense that 1.4 => 1 or 1.6 => 2.

Posted by: patrick | June 20, 2006 at 06:14 AM

Jabberwockey,

You missed my point. I'm not saying that 0.999|9 isn't a number. I'm saying that the notation 0.999|9 is a bad idea because you hide complexity.

Why do you teach 8 year olds that 0.999|9 is an infinite series of nines, but wait till they're 18 and in college before you explain what a limit is and why those concepts are related.

You can use 0.999|9 in an equation just like you can use a formal limit in an equation. If a limit becomes infinity, you can't use it in an equation anymore, for the reason I described earlier - addition and substraction are not defined for infinity. But if a limit converges to an 'ordinary' number, you *can* use it in an equation. That's what happens here with 0.999|9

Adding and subtracting limits is weird. But it's well defined. Talking about 0.999|9 without talking about limits doesn't work.

Posted by: Pedantic | June 20, 2006 at 06:14 AM

I will never understand why the mathematically incompetent bother to debate mathematics. Actually, that's a lie. I do understand. It's the same reason why Christian fundamentalists try to argue against Evolution.

0.999... is obviously equal to one. There is nothing magical or profound about it. It is just the consequence of our base 10 number system.

Regards,

Michael Tam

Posted by: Michael Tam | June 20, 2006 at 06:44 AM

thanks for the post, was very interesting.

someone further up raised what I think is an interesting question, given that 0.999... = 1

what is the number that is closest to, but less than, 1? does that question have any meaning?

by the same logic used in the article 0.999...8 is clearly nonsense, OTOH no one seems to be arguing that 0.888888888=1, so assuming that 0.9888...!=0.999... and that 0.9888...<0.999..., what number lies between them?

0.99988... maybe?

or wait, 0.99999888... is closer...

*am* I allowed to put an infinite number of 9s before my 8?

is there some mathematical way to find out the number less than, but closest to 1?

Posted by: worldsSmallestViolin | June 20, 2006 at 06:46 AM

No, such a number does not exist.

Thought experiment:

Draw a line, and put all real numbers on that line you like.

Now, you have two numbers, 1, and a number which is the closest number to one, but not actually one. Put these numbers on the line. Now take the average of these two numbers (1 + "closest to 1") / 2. This number is clearly smaller than 1, but larger than "closest to one".

This contradicts your premise that "closest to one" is the absolute smallest number.

Since for any number close to one you can compute the average of that number and one, which will be larger than the original number, you know for sure that there's no "closest" number.

Good question though.

Posted by: Pedantic | June 20, 2006 at 07:01 AM

Pi is exactly three!

Posted by: Eighty | June 20, 2006 at 07:10 AM