Is this really true?
The conjecture is that 0.9 recurring (i.e. 0.9999....9) is actually equal to 1
(For this exercise I will use the notation 0.999... as notation for 0.9 recurring,
the correct way would be to put a little dot above the 9)
Let X = 0.999...
Then 10X = 9.999...
Subtract X from each side to give us:
9X = 9.999... - X
but we know that X is 0.999..., so:
9X = 9.999... - 0.999...
or: 9X = 9
Divide both sides by 9:
X = 1
But hang on a moment I thought we said X was equal to 0.999...
Yes, it does, but from our calculations X is also equal to one. So:
X = 0.999... = 1
Therefore 0.999... = 1
Does anyone disagree with this? Let me know using the math is fun forum.
No, this is not true, properly following the initial equation set out we find that.
9X=8.991, and not in fact 9.999, so following through X=8.991/9 being .999 so X conforms to the expected solution.
But we're not saying that X = 0.999, we're saying that X = 0.999..., which means that the nines go on forever.
If they go on forever, then there isn't a limit to them, which means that it's impossible for X to have one more 9 after the decimal point than 10X.
This has already been discussed at length here.
Why did the vector cross the road?
It wanted to be normal.
Does there exist a real number between 0.999.... and 1? If there is, find it. If there isn't, then 0.999... = 1. This is because between any two different real numbers, there must exist another real number between them.
Another good proof is 1/3 = 0.333.... * 3 = 0.999.... but 1/3 * 3 = 1.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
I agree with Ricky. The fact is the number 0.99999999999...... doesn't exist. We may never come across this number in any mathematical problem. I had sometime back said here something similar to what Ricky stated.
Character is who you are when no one is looking.
I've spent a good 5 hours on this intriguing puzzle, reading the different links and past discussions and the such. I personally am not as adept at mathematics as some of you clearly are, but I would say that from all this my conclusion is that 0.999... isn't = 1 and in the original question the problem lies with when you multiply 0.999... by 10 to get 9.999... The reason being is that you just can't do it to start off with. When you multiply something, you first multiply the very most right value of the number, right?. For example 3 multiplied by 1.45678, you would first multiply 3 by 8 carry the 2 etc. In this case there is no very most right value as it is infinite and hence you cant just multiply it by 10 to get 9.999...
We always take the shortcut to move the decimal place, but is that a proven method or just a shortcut. I suppose the fact that we are using a decimal system for our calculations would indicate that you may be able to, but I thought I'd just pose that as a question.
Now, I don't know if I'm just looking at this too simply or not, but that's just a thought.
I see why you would say that, but I'm not sure I'd aggree(might be wrong though). Multiplying from right to left is also just a method of making it easier, just as 'moving the decimal place' is. You might just aswell say:
Basicly what you're doing is the same, but it's easier to do it the other way around. You're welcome to prove me wrong though, I like learning
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Hi Ausbomba. Yes, it is a great discussion point!
My view on it is that infinity has no end, so there is no "end" to 0.999..., nor is there an end to 10×0.999...=9.999...
It is the peculiar nature of infinity. You see inifnity is not a big number! It is a concept that says "not finite", or simply "no end".