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True or False: 1=.999...

TRUE	FALSE
What else could .999... possibly be?	It's infinitely close to 1, but not quite there.
If it's not 1, there must be a number between it and 1. What number would that be?	It's obviously not 1. Just look! It looks different!
It looks like a decimal expansion of 9/9 to me!	It looks to me like it's less than 1 by exactly .000...1.
It's 1! It's 1! It's 1!	How can it be 1? It never quite gets there, now does it?

Question: Why can't we all just get along?

This is not a page about what 1 is. I think we can all agree that we know what 1 is. This is a page about what on earth point-nine-repeating is. I hope we can all agree that it's a number. Otherwise the discussion ends before it begins! .999... doesn't look like 1, but then again .333... doesn't look like 1/3, does it? For that matter, 2/14 doesn't look like 1/7. But of course they're the same. Here are a few tricks designed to convince you that .999...=1. See if you can spot any errors.

Trick #1	Trick #2	Trick #3
Let x=.999... Then x/3=.333... That is, x/3=1/3. Multiplying by 3, x=1.	Let x=.999... Then 10x=9.999... Subtract the 2 equations. 10x-x=9.999...-.999... 9x=9 So x=1.	Let x=.999... Using the subtraction routine we learned in elementary school 1-x=0.000... So x=1.

Most people, upon seeing one or more of these demonstrations, feel half-convinced. There's nothing wrong with the arguments, but they all seem a bit too slick. They all look like swindles.

For the first trick, who's to say in fact that I can go willy-nilly dividing all the digits by 3 in order to divide x by 3. Did this introduce some sort of infinitesimal inaccuracy? There is no way to know. Even worse, who's to say after all that 1/3=.333... We've all heard it before, but maybe indeed .333... is only infinitely close to 1/3!

As for the second, again who's to say that the multiplication by moving the decimal point is valid in this most strange case? Maybe that computation only works for normal numbers, but not for .999... The subtraction also should provoke suspicion.

The third trick is the worst of all. The elementary school subtraction routine was most certainly not designed to go on for infinitely long, nor was it designed to begin on the left hand sides of the numbers. From whom shall we "borrow the one?"

The reason all of these tricks fail is that they don't address what .999... really is, but instead they apply some well-used operations of arithmetic to it to try to find out. But if it were some incredibly special, new kind of number (say, infinitely close to 1, whatever that means), then these operations might not apply, so the proof wouldn't work. It's like trying to user a hammer and nails to build a lake.

In order to truly get to the heart of the matter, we have to "walk directly to the shore." In other words, we have to say directly what .999... means, without any fancy tricks. .999... is after all a decimal expansion of some number. Each nine fills a certain decimal place, and indicates a fraction which is part of the whole number .999... itself. The first '9' appearing is in the tenths place, and stands for 9/10. The second is of course in the hundredths place, and stands for 9/100. The number .999... itself is the sum of all these fractions, so

.999... = 9/10+9/100+9/1000+.....

Are we out of the woods? Hardly. The addition problem on the right is an infinite sum. If we tried to add them directly we could never finish, so we're doomed! What if we tried really hard? After an infinitely long time, where would we be? Well, after an infinitely long time, the universe would not be around anymore, so we're doomed again!

Well, not quite doomed. The problem is that we haven't yet said what we mean by an infinite sum. Fortunately, mathematicians can tell us what we should mean:

The sum of an infinite list of numbers is obtained in this way: Add up more and more of the terms, producing finite sums which can be thought of as approximations to the infinite sum. If these finite sums do indeed get closer and closer to some number, then that number is by definition the sum of the infinite list.

Does this seem like a reasonable definition of an infinite sum?

Here's what it tells us to do in our case:

Subtotal Number:	Terms added	Answer
1	9/10	.9000...
2	9/10+9/100	.990000...
3	9/10+9/100+9/1000	.9990000...
4	9/10+9/100+9/1000+9/10000	.99990000...
5	9/10+9/100+9/1000+9/10000+9/100000	.999990000...
		Do these numbers get closer and closer to anything?

The numbers in the "Answer" column do indeed get closer and closer to some number, namely 1. The fact that they never get there is irrelevant. 1 is by definition the exact answer to the infinite addition problem.

So in conclusion, Point-Nine-Repeating is indeed equal to one. In order to see it, we needed to know how to add up infinitely many small numbers. By the way, this knowledge itself is probably more important that the little fact that .999...=1.

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The page last modified Thursday, 13-May-2004 15:02:59 EDT