Q.E.D. has already expounded on this a bit, but let me elaborate as well.
Given any two distinct real numbers, there is always a third real number lying strictly between the first two. Between 0 and 1, there is 1/2. Between 1 and 1/2, there is 3/4. Between x and y, there is (x+y)/2.
By the same token, if there is no such third number then the first two numbers can’t be distinct. There’s no number x such that 1 < x < 1, which is why 1 and 1 are the same number.
Now, what about 0.999… and 1? I claim there is no number between them. Indeed, suppose 0.999… < x < 1; I claim this leads to a contradiction.
For starters, it must be true that 0 < x-0.999… < 1-0.999… . But 1-0.999… is clearly less than 1-0.9=0.1. It’s also less than 1-0.99=0.01, 1-0.999=0.001, 1-0.9999=0.0001, and so on. In fact, given any positive number y I can show that 1-0.999… is less than y by writing out enough 9’s. But somehow 1-0.999… isn’t less than x-0.999…, even though x-0.999… is a positive number. This is the contradiction.
So there does not exist any such number x.
The same thing happens with the sequence 0.1, 0.01, 0.001, … . No term in the sequence is 0. But the limit must be zero. After all, the limit certainly isn’t negative. And the limit can’t be postive either, for if it were then it would be smaller than 0.000…00001 for some number of zeroes, and how can the limit of a decreasing sequence be smaller than any of the terms in the sequence? Zero is the only possibility remaining.
X-Ray Vision, the whole point behind infinite arguments is that is you go on forever you do get 0.
These threads always come down to this one basic factor. Either you get the idea of infinity and understand this equality or you don’t. What’s ironic is that your argument - that we keep getting smaller and smaller but never reach zero - is arguing for an infinitesimal. But infinitesimals and limits are incompatible ideas in our arithmetic.
No, you can’t. That’s just it. You’ll die, eventually.
0.999… has an infinite number of 9s.
Using .9, or .99, or .999 is an approximation, which is why you don’t get exactly zero. But the approximation gets closer by a factor of 10 for each digit you add. So, what happens when you use an infinite number of digits?
Your conclusion is based on an intuitive understanding of what 0.999… means, an intuitive understanding which is simply wrong. 0.999… is generally understood to mean the limit of the sequence 0.9, 0.99, 0.999, 0.9999, … and so on. That limit is 1, as inarguably as any mathematical fact, no matter how strange it seems to you.
.9 differs from 1 by (1-.9)/1, which is .1 or 1/10.
.99 differs from 1 by (1-.99)/1, which is .01 or 1/100.
.999 differs from 1 by (1-.999)/1, which is .001 or 1/1000.
Let n be the number of 9s you are using in the approximation. Thus, the difference between 1 and .9999… is 1/(10[sup]n[/sup]).
For n=infinity, the difference is 1/(10[sup]infinity[/sup]) or 1/infinity.
Since 1/infinity equals zero*, the difference between .999… and 1 is zero. Saying that the difference between two numbers is zero is equivalent to saying that they are the same number. QED.
*Yes, I know that it isn’t technically correct to say that 1/inf equals 0, but rather lim[sub]x–>inf[/sub] 1/x = 0. Small technicality.
Let’s leave infinite arithmetic out of this, and not cloud the issue with technicalities.
Others have already said most of what I would, and I’ve been there and done that on this topic, so let me just add that the fact that .9… = 1 is a direct consequence of the most important property of the real numbers, and if they’re not equal, everything falls apart.
Q.E.D. your wrong again!!
If you went .999 of the distance to 1.0, then guess what?
You never got there!!
Terribly obviously wrong again!!
If you went .99999999999~~ way around the world you never reached your starting point so .99999~ will never equal 1!!
I don’t care who tells you it does!!
WRONG!!
To all the mathematicians out there give it up; those who don’t understand limits and calculus and choose not to study will never understand. To all of the math challenged out there let’s discuss something a bit more on your level: How many angels can dance on the head of a pin?
I’m tryin’ real hard to be open minded about this because I respect the intelligence of you guys, but some of the rationale you guys are giving I just can’t jive with.
Allow me to repeat the quote from the second web-page I quoted from earlier.
It seems the writer is contradicting himself. First, saying that they never get there, then saying 1 is the exact answer to the infinite addition problem.
I know Q.E.D. tried to explain this by writing that the writer is talking about a finite number of 9s, but how is that so? We’re talking about an infinite number of 9s.
I’m not looking for any more explanations, but I am going to delve into this further on my own. I’ll get back to you all when it sinks in, or I have proof I was right all along.
Ok dejahma, I’m no math whiz, but if you tell me how big each angel’s feet are, how close each angel can stand next to another, and how big the head of the pin is, I bet I can figure it out.
It’s making that transition from understanding what a finite number of 9s after the decimal point means (which clearly you do, x-ray vision, to understanding what an infinite series means. RealityChuck’s algebraic proof is pretty good towards making this step. The hard part to accept about it is that even though the decimal point has been shifted, the number* of 9s to the right of the decimal point hasn’t changed. It’s still infinite, since the expression infinity - 1 is meaningless.
A.) You’re wrong.
B.) We’ve explained why this is over and over again, as the previous threads cited in the second and third posts demonstrate.
C.) The writer doesn’t say anything that contradicts anything. By definition, the fact that “they never get there” is irrelevant. If you’d like to come up with some new definitions to use in mathematics, feel free to.
Let me put it to you this way: if 0.999…<1, then what number comes between them?
No, he’s not. He’s saying that successive iterations of the sum .9 + .09 +.009 +.0009… will never equal 1 with a finite number of iterations. And it won’t. But with an infinite number of iterations, it will exactly equal 1. This is the difficult part of this concept, mainly because there is no smooth, simple transition from finite to infinite. There is no end to the series I just outlined, but it’s limit is clearly 1. It cannot be less than 1, because any finite number x of iterations will always be exceeded by x + 1 iterations. So that .99999999999999999999 can be exeeded by .999999999999999999999 and so on. It cannot be greater than 1 because that would be silly. So, if it’s not less than and not greater than, it must be equal to. There is no other choice.
I think the biggest thing to realize here is this: Infinite does NOT mean “a lot”. It does not mean “more than you can imagine.” It does not mean “Three lifetimes worth.” It means that it simply DOESN’T end.
You know how you start subtraction on right hand side of a number, and then carry to the left? With .999… there IS no right side. None. Because no matter where you decide the right side is, there’s infinitely more 9s to the right. Not a lot more 9s. Not more 9s than you can imagine. An infinite number. The same number as are to the right of the decimal point.
