In the words of an unknown picketer’s sign from an episode of the Simpsons:
**You Are Making None Sense!**-
Humph. This is too easy. 0.999… is non-negative. We disagree as to whether it is equal to 1 or not, but I think we all agree that 0.999… is less than or equal to 1. Therefore, it is some number in the closed interval [0,1]. So it is finite.
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1 is finite by the same argument.
0.999… is not infinite. It is the limit of a sum as the number of terms goes to infinity. If you look at the epsilon definition of a limit at infinity, you will see that there is no actual infinity in the definition. While there is nothing wrong with saying infinity is a number, here it is not used as a number, but rather as a formal symbol. If you use the definition to determine the value of 0.999…, you will discover (without using infinity) that the value is exactly equal to one.
I’ve got two questions:
- Does 0 = 1-1? After all the left side has one symbol - 0, while the right side has three symbols.
- Does 1/3 = 0.333… If not, does 1/3 have a decimal representation?
Virtually yours,
DrMatrix — If I’ve told you once, I’ve told you 0.999… times.
On the contrary. The mathematical proof is the only way to look at this.
says who? Math? There are many things math can’t explain:
22/7, infinity, life, why some people hold automatic doors open (thinking it’s courteous), where my keys are, my incessant uneducated posts…
Can’t infinity be an abstract idea? Let’s not apply it to math, it just causes problems.
Math can’t explain 3 1/7?
nope.
You and you’re ‘pie in the sky’ notions. 
Peace,
mangeorge
And .999…=1, apparently, isn’t one of those things.
I can’t believe that this thread has reached three pages and nobody has simply suggested to kinoons that he expand in base 12 instead…
pan
I can’t believe thst thread has reached three pages, period! Only on the SDMB would you get this much response from such an apparently simple question. Wow!
Infinity is an abstract mathematical idea. It’s just not one that’s as simple as you like to think it is. Remember, there are as many even integers as there are integers.
Good grief, kinoons. This is a mathematical question, with a provable answer (“yes” is the answer), and the proof is the point. There’s no real debate here; the question’s been answered repeatedly already. Yet, you continue to maintain that “the only way to write 1 is 1” even though that statement has already been shown to be false!
Look, just to be clear, is there any type of evidence at all, showing that 0.9rep = 1.0, that you would accept?
Okay, I understand the proofs, I’ve even managed to follow through them and they make some sense to me…here is my disconnect…
Just as no two particles of matter cannot occupy the same space at the same time, no two numbers can be at the same point of the numberline at the same time. Even though the math can be manipulated to show that 0.999… can equal 1, I have a logical disconnect on how any number except one can be one. It is like when (excuse me if my terminology is a little off, it has been a while since a formal math class) if you have a quadratic equation, and one of the possible values of X does not make sense, it is thrown out. Even though the proof can be manipulated to be true, does it truly make sense? How can anything equal one but one?
No, it does not equal 1. End of story. Let it die. 3 pages too long.
The problem is, Bill, that it’s not that simple. that’s why the thread has gone so long.
Kinda like argueing the existence of god. The proof is right there, in the book. Obvious to anyone who believes.
But when taken outside their own proofs, both .9999… as 1 and the existence of god fall to ordinary logic.
As long as you keep having 9’s, which you must, there’ll never be a 1. Except in math.
Peace,
mangeorge
Do you have the same problem believing that .333… = 1/3?
OK, how about this:
- Do you agree that every number on the real number line must have an exact, although possibly infinite (and possibly non-repeating!), decimal representation?
- If your answer to number 1 is “no”, can you give an example of a number that does not have an exact decimal representation, and why you think so?
- If your answer to number 1 is “yes”, would you agree that the decimal representation of 1/3 is 0.3rep (i.e., 1/3 = 0.3rep)?
- If your answer to number 3 is “no”, can you tell me what is the decimal representation of 1/3?
- If your answer to number 3 is “yes”, would you agree that I can multiply both sides of the equation in question 3 to get 3/3 = 0.9rep?
- If your answer to number 5 is “no”, then can you tell me: does 31/3 not equal 3/3? Does 30.3rep not equal 0.9rep? What is the problem?
- If your answer to number 5 is “yes”, do you agree that 3/3 = 1? And thus 1 = 0.9rep?
I’m suspecting here that you’ll agree with number 1 and disagree with number 6, but I’d like to know just where the logical disconnect occurs.
And, then as an additional question:
8. You keep repeating that no number except one can be one. Are you comfortable with representing 1 as (3-2)? As 12[sup]0[/sup]? As 3/3?
“Ordinary logic” here seems to mean asserting that whatever you don’t like is false. That has no place in mathematics.
Now, if you want to claim that, in the real world, no matter how many 9’s you put after the decimal point, you’ll never get to one, then I’ll agree with you. If there are only a finite number of 9’s (which is all you can get in the real world), then it’s not equal to 9.
But as soon as you have an infinite number of 9’s, you do have 1. The take-home lesson is that strange things happen when infinities are introduced into the mix.
I am not comfortable saying that saying 3-2 is the same thing as saying one. I am comfortable saying that when you perform the operation 3-2 the result is one. It still comes down to the only thing that equals one is one.
And, by the way, the sci.math FAQ also covers this, for anyone who’d like a more formal reference.