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.
I have already bragged on 
I am sorry I have been gone so long. Between preparing for the Red Mountain Writing Project and going to Washington D.C. for the NWP Spring meeting, I have been trying my best to make time for writing.
