How can 0.999... be equal to 1?

Question

I don't get it, there different numbers/

Answer 1

They're not different numbers, they just look different.
Like 8/4 and 2.
C'mon, trust in the algebra here:


Here's an algebraic proof:

let N=.999...
multiply both sides by 10

10N=9.999...
now subtract:

10N=9.999...
-N = -.999

and you get 9N=9
divide both sides by 9, and you get N=1.

Remember, you started with N=.999... and did legal moves, and ended up with N=1, therefore they must be equal.

Note: We're not talking about rounding!!!!!

Answer 2

0.999 with an infinite number of nines can be treated as 1, for example the limit as X approaches 1 of (x-1)/(x-1) is 1, but the value of the function when X=1 is undefined, but you can see that the value gets infinitely close to 1 as x approaches 1 and therefore the limit is 1.

Answer 3

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.


Hope that helps, there's better diagrams on the website so check it out.

Answer 4

Hold a video camera on 1.
If 0.999 as a passing driver got to 1; then 0.999 becomes 1.
If did not reach to it;then it is 0.999....or 0.9999999.... .

Answer 5

A repeating decimal would be rounded off to the nearest digit which it approaches at infinity for practical purposes.
But there should be a dot over the final '9' to indicate that it IS repeating to infinity.

Answer 6

Simple:

1/3 = .333
2/3 = .666

1/3 + 2/3 = 1 while .333 + .666 = .999

Therefor, .999 = 1

Answer 7

Its not possible at all, but the 9 goes until infinity, so usually people round it to the nearest tenth or hundredth

Answer 8

Rounding.

Answer 9

No.

If you round then it will go to one but it is not eaqual to 1.

Now if you mean .9 repeating it is eaqual to one and here is a simple proof.

We all know 1/3 is equal to .3 repeating so

1/3 x 3 = 1

.3 repeating x 3 = .9 repeating

Get it?

Answer 10

you round it...its not equal, but in math rounding up is basically "close enough"