Why does 0.99999999999 (repeating) equal 1?

Question

We were always taught that repeating numbers never intersect the next number (i.e) 1.6 (repeating) will never equal 1.7, because the six repeats forever.

However 1/3 = 0.333333 repeating and 2/3 = 0.666666 repeating so therefore 3/3 = 0.999999 repeating but 3/3 is, of course, 1.

So how can these two rules exist in mathematics?

Answer 1

Tom has a fixation on this question and I believe you asked the question right on time to let him off some steam happily. I don't know what he has to show for his claims of high IQ and high this and that but there is no question in any mathematician's mind that 0.99999... = 1. This includes the guys who proved Poincare's Conjecture, Fermat's Last Theorem and alike as well as any half decent teacher of mathematics. Don't let anyone make you believe in his claims by bullying you with false credentials though. You should read and decide for yourself.

How do you see that 0.999... =1? Well one way is to observe that 0.9, 0.99, 0.999,... is a sequence of numbers which converge to 1, and in fact 0.9999... is a short hand notation for the limit of this sequence. This is not a proof exactly, but it can be made into a proof once we agree on the properties of the real line, which requires reading a chapter or two from a real analysis book together. But you can also think of it this way, if 1>0.9999...., then 1-0.99999.... must be a positive number. Can you write it? It must be a number less than 0.1, 0.01, 0.001,... is there such a positive number?

As for the two rules existing in mathematics: the problem is simple, our method of writing decimal expansions for real numbers is not perfect. Sometimes we just end up with two different ways of writing the same number. Their looking different is not necessarily an evidence of these two numbers being different. So yes 0.999999.....=1.

Note: You can't know Pi's exact value is also a very ignorant thing to say. Any digit that Tom wants to know in Pi's decimal expansion, I can go to my computer and run a program to find in a short while. The problem is, it would take infinte amount of time for me to write all the digits in Pi, but that does not necessarily mean I can't write them. And after all we know the exact value of Pi, it is the circumfrence over diameter length for a circle, our problem is in writing it down. As for the claim that there should be a unique representation property of decimals, I am trying to understand what the heck Tom is talking about when he can't even give one decimal expansion for his positive number
1-0.9999...


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Tom you are not even amusing anymore. You are now talking about real numbers that we can't even write in decimal form. How do you come to believe that they exist? All that you can do is to give a gross misapplication of induction mascarading as a proof. It is wrong, I explained it but either your mental capacity to understand math or your super sized ego is so big a barrier that you can't even read and make an honest reevaluation of your arguments. Not just that but now according to you there is no convergence and all the mathematicians on this world are stupid morons, although they are proving great theorems while you are shouting slogans and obscenities. Shout all you like, write all the erronous proof you wish, I came to the end of my time explaining anything to you. You are a beacon of ignorance that can't be helped and you have every right to be disappointed with people who awarded you a BS degree in math. If you were in my class you would not have gone past 2nd year. I sure do have better things to do with my time.

Answer 2

Before you are ready to accept a proof in mathematics, you need to have a clear understanding of what the terms involved mean.

What I want you to focus on is, what *is* .9999...? Or any decimal expansion, like 3.1415... or whatever.

A number doesn't change. It's not 0.9 one moment and then 0.99 the next, as if some outside person is changing it by contemplating it. It's stuck somewhere on the number line, no matter how you describe it.

And if I ask, what is the difference between 1 and 0.9999...., it's going to have to be a fixed number too, not some "infinitely small" thing. There has to be an answer; you can always subtract numbers. And it's easy to see that the difference, if positive, would have to be smaller than any other positive number. (its decimal expansion would start with more zeros 0.0000...) If not positive, it must be zero.

Can it be that there is a positive number smaller than any other positive number? No, then it would be smaller than itself (or half itself). Contradiction.

Here's another perspective: do you believe that 1/2 + 1/4 + 1/8 + 1/16 + ...=1? (Classic Achilles & the Hare problem.)
That's the same as saying 0.11111.... in binary is equal to 1. Your problem is the same idea base ten.

Answer 3

Excellent question. They are the same, absolutely.
Here is the demonstration.
0.9999... is a pure periodic decimal, so the rule says take the period and divide by a number composed of such 9s as the number of digits of the period. In this case the period has only 1 digit, therefore 0.999... = 9/9 = 1

Now, let's talk about infinite and limits. When we write 0.999... is meant to be infinite number of 9s. But the point that this number represents is exactly the same as 1, just the latest representation is simpler. The complication is because the real numbers are dense, and Mathematics allows such number of nines before they reach the 1. For example, you can have five hundred thousand of 9s and you still don't reach the 1. But it is said, with the infinite concept, that in the limit, they will reach the target (which is one in this case).

Answer 4

The Peter, Praveen S, and The Asker are all correct. All the other answers show some confusion or failure to understand the idea of an infinite recurring decimal. Many of the responders failed to understand that 0.9 (repeating) goes for ever and that we can only picture it going for a long way and then ending, which would be (as some suggested) only an approximation.

Also, "repeating numbers never intersect the next number" is not a rule I ever heard or taught in all my years of teaching Mathematics. I think it's your re-interpretation of something you've been told, but I can't guess what.

Answer 5

0.33333... is an approximation to 1/3, no matter how many 3's you put there. You'd need an infinite number of 3's to equal 1/3. Multiply that by 3 and you get 1 and nothing else.

It's the infinity that's confusing you.

Answer 6

Because the first "rule" is false. The concept of limits says that, in layman's terms, if a number is "infinitely close" to another number, then they are in fact equal. Not approximately equal, not rounded to equal, not very close to equal. Equal. This idea is necessary to resolve just the kind of paradox you are referring to. Read up on Zeno's Paradoxes.

Answer 7

3/3 is actually 1which is finite. The other numbers repeat to infinity so they are just an estimation. In other words .3333 continues to repeat to infinity when calculating 1/3 to let you know that there is no finite number for that problem.

Answer 8

It never will equal one because it goes to infinity. But the real answer is where we need to use the result and for most day-to-day applications we can use only a few decimal places. Also, mathematicians have agreed that we can round it off to 1.

Answer 9

0.9999.... infinity is close to one so that when you round 0.9 any number that is higher than 5 gets rounded up, any lower, it's rounded down, 0.9 is higher than 0.4 so it gets rounded up...

also, using fractions, one-third is obviously not a whole part of, lets say, a pie, but three-thirds is the whole thing, so therefore it is one "whole" pie

Answer 10

its just like you said...
.999999......... is the same as 3/3 in fraction form but 3/3 is also 1 so they are considered the same