*(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.

Okay, non-math-geek, here. Isn't there some difference between a number that can be expressed with a single digit and one that requires an INFINITE number of symbols to name it? I've always imagined that infinity stretches out on either side of the number line, but also dips down between all the integers. Isn't .9999etc in one of those infinite dips?

Posted by: caitmcq | June 06, 2006 at 06:38 AM

I'm not 100% sure I understand your conception of the number line. But here goes:

The three main categories of numbers are:

integers: the number directly on the number line marks, like 5, 0, -7, 13, and .999999.... (otherwise known as 1).

rationals: the numbers that can be expressed as a fraction, like 6, -15, 1.5, .343434...., and any other number whose decimal representation either ends or repeats infinitely.

irrationals: the numbers that can't be expressed as a fraction, like the square root of 2, pi, .1010010001....

The nature of holes or "dips" between numbers on the line is a notion that requires some getting used to. The integers clearly have holes between them. The rationals do too (because the irrationals are missing), but the rationals nonetheless have the amazing property that between any two of them there's another one. The vocabulary is: the rationals are dense, but not complete. The real numbers (which comprise the rationals and the irrationals together) are dense and complete. There are no holes.

Also, thanks for caring (or pretending to) despite your non-math-geekiness!

Posted by: Polymath | June 06, 2006 at 08:02 AM

Huh. I don't see the dips as holes. I imagine an unbroken line that drops down between 0 and 1. Any point on that line can be described with a number. The line dips down infinitely. Except...wait a minute. Then...in this visualization, would there need to be a line that dips down between each pair of points on that dipping line, too? Okay. Maybe my "graph" doesn't work. But I have this belief that one can name infinitely decreasing fractions between 0 and 1. Don't have a place for the irrationals, really.

Posted by: caitmcq | June 07, 2006 at 06:44 AM

Haha not only are there holes in your logic, but there are holes in your mathematics.

First of all, by definition the number .99999999... cannot and never will be an integer. An integer is a whole number. .99999999... is not, obviously, hence the ...

The ... is also a sad attempt at recreating the concept of infinity. I only say concept because you can't actually represent infinity on a piece of paper. Except by the symbol ∞. I found a few definitions of infinity, most of them sound like this: "that which is free from any possible limitation." What is a number line? A limitation. For a concrete number which .99999999... is not. (Because it's continuing infinitely, no?)

Also, by your definition, an irrational number is a number that cannot be accurately portrayed as a fraction. Show me the one fraction (not addition of infinite fractions) that can represent .99999999...

You can't, can you?

Additionally, all of your calculations have infinitely repeating decimals which you very kindly shortened up for us (which you can't do, because again, you can't represent the concept of infinity on paper or even in html). If you had stopped the numbers where you did, the numbers would have rounded and the calculation would indeed, equal 1.

Bottom line is, you will never EVER get 1/1 to equal .99999999... You people think you can hide behind elementary algebra to fool everyone, but in reality, you're only fooling yourselves. Infinity: The state or quality of being infinite, unlimited by space or time, without end, without beginning or end. Not even your silly blog can refute that.

Posted by: removed by request | June 08, 2006 at 10:50 AM

Oh look, how cute...a "point 9 repeating" troll.

Let's see...where to start...your linguistic definition of infinity is not exactly relevant. As soon as you agree that a number like pi can have infinitely many digits (which I doubt you dispute), then we could change each of the digits after the decimal point to a 9 and get our "point 9 repeating" number. That concept exists completely independently of how you write it (with the dots or whatever) or of the English word "infinity" (which, incidentally, mathematicians use sparingly--mostly as the adjective "infinite" to describe sets). Lofty philosophical concepts of what infinity might mean have little to do with precise mathematical definitions.

If I had stopped writing the numbers, they would have rounded?? Numbers don't round by themselves. People round them because they decide they can ignore some of the digits for whatever purposes they currently require.

It looks like you're claiming that "point 9 repeating" not only doesn't equal 1, but that it's also not a rational number. That must mean that one-third of it ("point 3 repeating") is also irrational. But you didn't seem to think that the number 1/3 is irrational.

Holes in my math, huh? How about you bring my post and your response to 10 random mathematicians, and see whose they pick as having holes. Since your basic problem seems to be that infinity is too mystical to be quantified in algebra, I'll even grant you 10 highly spiritually inclined mathematicians.

You have also not answered the very basic challenge: If you claim that "point 9 repeating" doesn't equal 1, then the burden falls on you to find me the number halfway between the two. When you manage that, I'll listen more closely.

I'm not the one hiding behind algebra; you're the one hiding behind vague, non-mathematical definitions.

Posted by: polymath | June 08, 2006 at 12:25 PM

This is quite amusing. First off: I don't see how my 'vague' definition of infinity is irrelevant. It means the same thing whether you are talking about time, space, or just plain numbers. Never ending. Mathematics gives it a slightly more complex definition when used in complex equations, but you are using it in it's simplest form. Never ending. Am I right? .9999 repeating forever?

When you write out .99999999... you are giving it a limit. Once your fingers stopped typing 9s and started typing periods, you gave infinity a limit. At no time did any of your equations include ∞ as a term.

In any case, Dr. Math, a person who agrees with your .999999 repeating nonsense, also contradicts himself on the same website. "The very sentence "1/infinity = 0" has no meaning. Why? Because

"infinity" is a concept, NOT a number. It is a concept that means

"limitlessness." As such, it cannot be used with any mathematical

operators. The symbols of +, -, x, and / are arithmetic operators, and

we can only use them for numbers."

Wait, did I see a fraction that equals .9999 repeating? No I didn't. Because it doesn't exist.

And for your claim that I have to find a number halfway between .9999 repeating and 1 is absurd. That's like me having you graph the function y=1/x and having you tell me the point at which the line crosses either axis. You can't. There is no point at which the line crosses the axis because, infinitely, the line approaches zero but will never get there. Same holds true for .9999 repeating. No matter how many 9s you add, infinitely, it will NEVER equal one.

Also, can I see that number line with .999999999999... plotted on it? That would be fascinating, and another way to prove your point.

And is .99999999... an integer? I thought an integer was a whole number, which .99999999... obviously is not.

Have fun thinking you're right. Meanwhile, I'm going to find 10 mathematicians who agree with me.

Posted by: removed by request | June 08, 2006 at 01:28 PM

"When you write out .99999999... you are giving it a limit. Once your fingers stopped typing 9s and started typing periods, you gave infinity a limit."

So am I to gather that pi also doesn't exist, or that it also ends, since whenever anyone tries to write it, their fingers have to stop typing?

"Wait, did I see a fraction that equals .9999 repeating? No I didn't. Because it doesn't exist."

Ummm, how about 53/53. that's a fraction that equals .9-repeating.

"And is .99999999... an integer? I thought an integer was a whole number, which .99999999... obviously is not."

If I am claiming that .9-repeating equals 1, then I am indeed claiming that it is a whole number, and I gave a mathematical proof. It's "obvious"ness to you isn't relevant if it's a proof. Numbers can be written in many ways--1, 53/53, .999999....., 83-82, all the same.

"Also, can I see that number line with .999999999999... plotted on it?"

That's actually an argument for my claim, not against it. If that number exists at all (and I'm not 100% sure you think it does, but you haven't explicitly claimed it doesn't), then finding a place for it on the number line is indeed very difficult if you don't think it equals 1. I have no trouble putting it on a number line. It goes halfway between 0 and 2. If you think it exists and doesn't equal 1, then you find a place for it!

"Meanwhile, I'm going to find 10 mathematicians who agree with me."

I give my most solemn word that I will post an explanation on this blog that contradicts my claim if it

a) uses the standard real number system and

b) is verifiably written by a professor of mathematics at an accredited university.

I won't hold my breath waiting for that, though.

Posted by: Polymath | June 08, 2006 at 02:52 PM

Well, my brain still hurts a little from thinking about this, but heck, my 15 yo read it all and was explaining it to friends today. NOt that they were necessarily as amazed, but now he wants to take it and show it to his math teachers, too.

Thanks for the mindbend!

Posted by: Jen | June 11, 2006 at 12:36 AM

I was going to post a rebuttal with complete proof from 2(two) ASU mathematicians (who both agree with me), but upon review of all your posts, I came to the ultimate conclusion that you don't need proof. You will go to your grave believing with the core of your being that .9999999... does, in your mind, equal 1. However wrong I, or anyone else may think you are will not matter. Trying to convince you otherwise is like trying to convince an atheist that God exists.

I leave you now, respectful of your opinion, because whether I agree or not is a moot point. You may choose to use personal attacks, call me a fraud, a liar, and an idiot, but I leave taking the high road. I bid you farewell, good sir.

Posted by: removed by request | June 15, 2006 at 03:21 AM

I love teaching this one. But I always point out the weaknesses of my points of attack.

The easiest way to convince (three times one third equals one. 3 * 0.33333... = ? ) involves multiplying a repeating decimal by an integer... and I suspect this is not well-defined.

The subtraction business: as the number of 9's increases, the difference drops: 0.1, 0.01, 0.001, as small as you want to go, is effective in convincing my (young high school) students.

What's missing is a rigorous sense of deltas and epsilons, which kids and the public wouldn't necessarily buy anyhow.

And chiamera, if you have the names of those 'professors,' don't post them, especially if they tenure review coming up.

I cannot find this argument at mathworld, but the repeating decimal article does claim that 1.5 = 1.49999....

I don't love the article on Dr. Math, but it says the right thing.

Posted by: Jonathan | June 17, 2006 at 12:58 PM

Arrgh, typepad rejected my html. Links for mathworld and Dr. Math are:

http://mathworld.wolfram.com/RepeatingDecimal.html

http://mathforum.org/dr.math/faq/faq.0.9999.html

Posted by: Jonathan | June 17, 2006 at 01:00 PM

Let me ask you, if I start at one and take half, then keep halving infinitely, are you saying that I will eventually reach zero?

Posted by: Jason | June 17, 2006 at 02:06 PM

Ah, now I see it.

This isn't a problem about limits or infinity or the number line really.

It's a problem with symbolc representation.

1/3 = 0.333333....

or does it really?

Posted by: Jason | June 17, 2006 at 02:22 PM

Jason,

If you're talking about Limit[(1/2)^x, x -> ∞], I think you'll find that that *does* equal 0. However, if you look at an individual term in the sequence {1, 1/2, 1/4, ..}, no, no *individual* term will actually equal zero. There's a difference between the limit and any discrete term.

Does 1/3=0.3333...? Simple: try it with long division. It's easy to see that it does.

Posted by: Nick | June 17, 2006 at 03:16 PM

Polymath,

Excellent post! Of course, your next challenge is to explain Cantorian set theory and the difference between Aleph-0 and Aleph-1 infinities. The most common reaction I get when trying to explain it is "I see the math and how it works, but I just can't believe it..."

Posted by: Nick | June 17, 2006 at 03:19 PM

chimera,

There is no way you found a single professor of math the disagrees. You are definetly lying about professors at ASU claiming .99999.. !=1

In fact you even suggesting ASU has such professors actually is quite a grave insult to ASU.

Posted by: Brian P | June 17, 2006 at 05:52 PM

"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 ... "

Those do exist. But all the variants I know of will give the same answer in this case: 1. In any event, if someone is using non-standard defininions, it behoves them to give those definitions so others will have some clue as to what the heck they are talking about.

Posted by: Andrew Wade | June 17, 2006 at 06:00 PM

"I've always imagined that infinity stretches out on either side of the number line, but also dips down between all the integers. Isn't .9999etc in one of those infinite dips?"

Nope. There "are" numbers that correspond to concepts such as "infinitesimally smaller than 1" called hyperreal numbers, but the notation for those is different. Infinitesimals are sometimes used informally in physics, but it's easy to get the rules for them wrong. It's better to stick with limits.

Posted by: Andrew Wade | June 17, 2006 at 06:15 PM

I'd be interested to hear what your batting average is in convincing each year's skeptics in your class. It seems to me that the value of this exercise, when it's successful, is to achieve that "Aha!" moment when the student gets past the conviction that "the number .99999999... cannot and never will be an integer."

Enlightenment may come when the student meditates on the fact (first accepting it as a fact) that the statement "10 (0.9999...)=9.9999..." is valid precisely because, and only because, the series is infinite. If the series terminates, the left- and right-hand sides will differ by that last digit, and if it doesn't, they won't.

Your conceptual hooks may vary.

Posted by: jre | June 17, 2006 at 07:27 PM

You left out the multiplicative proof:

1/9 = .11111111...

(If you don't believe this, it is easily demonstrated by long division.)

Multiplying both sides by 9, we have

9/9 = .99999999...

But 9/9 is clearly equal to 1.

Some people on this thread are confusing a number with a representation of the number. 1 1/2, 3/2 and 1.5 are not three different numbers, they are three different ways of writing the *same* number. 0.999999..., like 9/9, 3/3 etc., is a different way of writing the number 1.

Posted by: Chris | June 17, 2006 at 07:49 PM

I've always found it helpful when dealing with this to start with something like:

(write "1 = 0.99999...." on the board)

"1" and "0.999..." are different strings. That is, they are different series of symbols. However, they represent the same real number in the same way that, here in class the string "Alice" and "person sitting to the left of Bob" represent the same person.

This isn't so unusual for numbers. The strings [five asterix three], [nine zero slash six] and [one five] all represent the number we commonly speak as "fifteen".

Anyway, I'm going to show you that these two different strings are both names for the number we say as "one" and along the way show you one useful technique for dealing with repeating decimals...

(and then continue with the 10x - x approach)

The useful thing here is to acknowledge up front that the two strings of symbols are different strings of symbols, which I find often removes much of the power of intial objections. Somehow, saying "these two look different, but are really the same" doesn't have the same psychological effect.

Posted by: Daniel Martin | June 17, 2006 at 08:27 PM

Jason asked:

"Let me ask you, if I start at one and take half, then keep halving infinitely, are you saying that I will eventually reach zero?"

Great question.

Can we switch it around a little. If we start at one half, and add half of a half, and then half of a half of a half, and we never stop, will we reach one?

Let's assume for the moment the answer is no, we won't reach one.

Let's call the final sum SUM, and let's call 1 - SUM Eppie (for end of the process).

Now, if we don't get to one, and we certainly don't go over one!, then Eppie must be a positive number. Really tiny, but a positive number.

Eppie is a positive number, we agreed. I don't think you can name her (state her value). Let's make it simpler. Since Eppie is positive, tell us a number that is smaller than her.

1 - SUM is a positive number, so you are being asked to name a smaller positive number (easy, there are lots of them).

But there's the problem. The process keeps going. If you said 1/1,000 and we only added the first 10 terms, Eppie would be smaller than one one thousandth

If you turned around and said, ok, 1/1,000,000 is smaller than Eppie, we would only need run the process 10 more times, and Eppie would again be smaller.

If you pick a number, and Eppie keeps getting smaller, she's going to get smaller than your number, no matter which number you pick.

You really can't come up with a positive number that works.

So 1 - SUM is not positive. What does that make it? And what does that make the SUM?

(If you are really encountering this for the first time it may seem frustrating. It all really comes down to this idea of not stopping. If we stopped, you could find the positive difference. But since we don't stop we can always get closer to 0 than any positive number you can come up with.)

Posted by: Jonathan | June 17, 2006 at 08:55 PM

VERY GOOD.

But there's a couple tricks

you missed.

First, simple pattern

completion

1/9 = .11111---

2/9 = .22222---

3/9 = .33333---

4/9 = .44444---

5/9 = .55555---

6/9 = .66666---

7/9 = .77777---

8/9 = .88888---

and therefore by logical

extension

9/9 = .99999---

but of course, 9/9 = 1.

And then there are the

SPIRITUAL implications

.9 a soul

+ .09

+ .009 adding experience

+ .0009

+ .00009

!

! infinitely increasing

!

or the infinitely

repeating process

of growing greater

i.e. life

--------

.99999---

which EQUALS

1

the finished, static, unity

characteristic of a single

never-created being

i.e. God

The equations show

IT MADE US TO BE ITS EQUAL

(the TRUTH behind all men

are created equal -

soul A = God,

soul B = God

transitive property -

A = God = B therefore A=B)

but .999--- being = 1

and 1 being a rational

number, 1/1 is the ratio

of two integers, as a guy

named Hugh Richmond asked

me, where are the two

integers whose ratio is

.999---

THERE ARE NONE thus

I coined an equation to

go with .999--- = 1 which

demonstrates our equality

x/y = .999---

where x and y are integers

THAT equation show the

DIFFERENCE between the

Creator and the Created -

only the Creator can make

it work.

Posted by: Fred Harry Wolnerman | June 17, 2006 at 11:53 PM

"Isn't there some difference between a number that can be expressed with a single digit and one that requires an INFINITE number of symbols to name it?"

Not really. The number of symbols required to express a number is merely a property of the numbering system you're using. For example, in base 9, 1/3 = 0.3 and 1/2 = 0.44444...

Posted by: The Science Pundit | June 18, 2006 at 03:52 AM

Thanks, Science Pundit, I was waiting for that to come up!

Posted by: Maria | June 18, 2006 at 09:07 AM