Can anyone prove that 0.999 repeating to infinity is exactly equal to one?

Answer 1

To all those who say these are not the same: please don't answer questions where you don't know the answer! There are TWO ways of writing the number 1 as a decimal. One is 1.0000.....; the other is .9999....

The problem is to figure out what the symbol .99... means. It *means* the limit of a sequence. Which sequence? The sequence whose first term is .9; whose second term is .99; whose third term is .999; etc. As you go further out in this sequence, is there some number that the terms of the sequence are getting closer and closer to? Yes. That number is what the symbol .999... means. It is also clear that that number is also 1.

Answer 2

There are two good ways that I like to show these are the same.

One way is long division. Divide 1 into 1, but start with 0 on top, and see what happens.

The other way is to use decimal to fraction conversion. The way to change a decimal into a fraction is take the number of repeating decimals, divide by that many 9’s can simplify. For example
0.123123123… = 123/999 = 41/333
0.363636… = 36/99 = 4/11
0.999999… = 9/9 = 1

Answer 3

Yes, several proofs were given 1 day ago when the same question was asked:

http://answers.yahoo.com/question/index;_ylt=AqzjzgHALtONKR._O86.wzbsy6IX?qid=20060717085908AAiMIC7

See also my response to that question.

Furthermore, the people who say it is not the same as 1 are wrong. If 0.99999... has a value, then it is 1. Saying that it is not quite 1 is a conceptual error -- if we were to truncate at some point, i.e., at 0.999999, then that value is not quite 1. But the recurring decimal 0.9999999... is not the same as any of its truncations. Notice that none of its truncations are equal, so if it were the same as one of its truncations, how would you be able to decide which one?

The DEFINITION of 0.9999... is the sum of the series 9/10^1 + 9/10^2+9/10^3+..., an infinite series. Using the theory of such series, the value is 1. People who say 0.99999... is less than 1 would also think that pi=3.14, but it is not. It has infinitely many digits, and if you give me any number n, I could in principal give you the first n digits of pi. The same concept holds for 0.9999...

Answer 4

I think I do, let me give it a try...3 x 3 = 9 , so 3 × 0.3333… equals 0.9999…; but 3 × 1⁄3 equals 1, so it must be that 0.9999… = 1..

It's true though, there are more complicated algebraic proofs..one of those weird properties of infinity.

Answer 5

Nobody can. 0.9999999continuously dose not equal 1.
They are two different numbers, like 5 and 6.

Answer 6

if it's not 1 then it's not one, doesn't matter how many 9s follow behind it

Answer 7

No, because it isn't.