1/9=0.1, repeating.
2/9=0.2, repeating.
.
.
.
8/9=0.8, repeating.
9/9=0.9, repeating!?
9/9=1, so 0.9 repeating equals one?
Of course this is intuitively wrong, but I imagine that explaining why it is wrong requires more knowledge of calculus than I can provide (viz, any). Can anyone “debunk” it in 500 words or less?
This has been done many times before on this board, but to some up:
0.99999… IS equal to 1.0000000 …
Many “intuitive” results turn out to be wrong when you start dealing with infinities and infite sets, as I discovered in real analysis classes 
Arjuna34
0.999… = 1.0
This has come up before. See .999 = 1?. This thread has links to other threads where this has come up. Some posters seem to disagree, but 0.999… does in fact equal 1.
You can’t “debunk” it. It’s true.
.9999… equals one . . .
In a nutshell, there is no number that comes between .9999… and 1.000…
This from two CS majors/math minors (as well as several intelligent dopers), so I would expect “proving” it is wrong is patently impossible.
.9999… isn’t a number, it’s an idea.
What’s the difference?
All numbers are ideas.
I prefer AWB’s provided proof from the previously linked-to thread:
Let:
_
N = 0.9
then
_
10N = 9.9
Since there's an infinite number of nines after the decimal:
_ _ _
10N - N = 9.9 - 0.9 = 9.0
10N - N = 9.0
9N = 9
N = 1
You may be confused by the finite difference between 0.9 and 1, 0.99 and 1, 0.999 and 1, etc. But the more places you go, the smaller the difference. If you go to an infinite number of places, the difference vs. 1.0 is infinitely small, which means zero.
Within the course of my life, I have
[ul]
[li]eaten three hot dogs at one sitting.[/li][li]crossed a distance of three miles.[/li][li]been hit on the head by one rock.[/li][/ul]
I imagine any of those experiences would have been quite different if the quantitative ideas (numbers) had been .999…
douglips, if you mathematically prove something, does that mean it’s true?
I know you just said ‘yes’, so let me show you mathematical proof that the number 10 doesn’t even exist:
amount of integers that are not 10 = infinity - 1 = infinity
% of integers that are not 10 = infinity/infinity * 100 = 100%
% of integers that ARE 10 = 100% - 100% = 0%
I just mathematically proved that the number 10 does not exist, does that make it so? I noticed that in AWB’s proof, he used the number 10 … he’ll have to think of something else, i just proved that 10 doesn’t exist. Throw me some more mathematical proof, and i’ll prove that the numbers 3 and 9 don’t exist either.
The point is that the concept of infinity can cause havok with mathematical equations. I don’t consider infinite numbers to be “real”. Yes, numbers may be just an idea, but you can have 2 of something or 1.445631 of something, but you can’t have .999… of something or 3.141592653… of something. 1 may equal .999… if you round off, but if you are dealing with infinity, there is an infintesmal difference between the two.
The number 0.999… is defined to be the number that 0.9, 0.99, 0.999,… approaches. The sequence doesn’t have to reach
the limit after a finite number of steps. If fact, since the sequence is increasing, it cannot reach its limit.
0.999… does equal 1.0 after an infinite number of steps.
Monocracy
What you have shown is that infinity cannot be thrown about carelessly. You just have to follow the rules. You are trying to use
cardinal arithmetic. All you have shown is that the size of an infinite set is not diminished by removing one element. One of the
definitions of an infinite set is that it is equal to part of itself.
There are several self-consistent systems that use infinity in arithmetic. You just have to know the rules.
Virtually yours,
DrMatrix - If I’ve told you once, I’ve told you 0.999… times.
Here’s a “proof” that I’ve read before:
Let X=.999…
10X = 9.999…
9.999… - X = 9
Therefore, 9X = 9
Therefore, X = 1
Oh yeah, and Monocracy, the error in your “proof” lies in the fact that not all “infinities” are equal. How can that be? Well, one infinity can be “smaller” than another infinity if it does not include elements that the other infinity does. In other words, a proper subset of the “larger” infinity.
Example:
The set of natural numbers : { 1, 2, 3, …}
vs.
The set of whole numbers : { 0, 1, 2, 3 …}
vs.
The set of integers : {… -2, -1, 0, 1, 2, …}
The set of whole numbers is larger than the set of natural numbers, even though both are infinite, because the set of whole numbers includes 0, which the set of natural numbers does not. The set of integers is larger than either of the other sets, because it includes negative numbers. Infinity is a concept, not a number. One cannot add or subtract “infinity” because there are many different kinds of “infinity.”
Oops, just noticed that douglips already posted the proof. How’d I miss that? I read the whole thread, or thought I did. Geez.
And yet in the post previous you were adding and subtracting with “infinity”. I think my point was that people are adding/subtracting/multiplying/dividing infinity to prove that .999… is equal to 1; and i was showing how silly it is to play with infinity in an equation. How can use arbitrary rules to discount one and not the other?
9.99… obviously isn’t infinity, but it is an infinite sequence, there is a difference. “infinity minus 1” and “9.99… - 0.99…” are not related. Also infinity/infinity is not defined, (1/0)/(1/0) = 0/0 = undef.
No. The set of natural numbers is a subset of the whole numbers. However, since they can be put in a one-to-one correspondence, they are the same size. In fact, the set of rational numbers is the same size as the set of integers. This is not true of all infinities. The set of reals is strictly larger than the integers. There is a whole system called cardinal numbers that uses infinities. There is nothing wrong with adding or subtracting infinities as long as you follow the rules. One result of transfinite cardinal arithmetic is:
Addition is very simple. If x or y is transfinite (infinite), x + y is the larger of the two.
If x is transfinite and n is finite, then
x + n = x.
You cannot subtract x from both sides and conclude n = 0. You can only subtract a finite number from both sides of an equation. Trying to subtract an infinity is like trying to divide by zero, it’s not defined. But that doesn’t mean we cannot use infinities; it just means we must make some adjustments in the rules to accommodate them.
Virtually yours,
DrMatrix - If I’ve told you once, I’ve told you 0.999… times.
[QUOTE]
*Originally posted by smugg *
**
In what way would the experience have been different? Would ‘0.999…’ of a hotdog have been less filling, requiring you to buy a fourth? It seems to me that your opinion about the nature of ‘0.999…’ is similar to that of certain elderly Canadians who, after the country switched from the Farenheit to Celsius scales, complained that the weather had gotten much worse since the metric system came in and lowered all the temperatures. :rolleyes:
I’ll admit that ‘0.999…’ looks funny, but that isn’t to say that it isn’t actually equal to 1.
My take on the whole thing is that ‘0.9…’ isn’t a number; It’s an approximation of a number. If you keep writing nines out to many thousands of decimal places, you’ll get better approximations, but you’ll never reach the actual value you want. Of course, since this is a rational number, we can express it exactly in fractional form- 9/9, which reduces to one.
How about a calculus proof that 0.999… = 1? The number ‘0.999…’ can be constructed as follows:
0.9 + 0.09 + 0.009 + ...
Which can be written:
9/10 + 9/10^2 + 9/10^3 + …
Or: Sum(9 * 1/10^n)
n = 1
For the value we want, n must go all the way to infinity.
Let’s find the difference between this number and one.
1 - 9/10 - 9/10^2 - 9/10^3 … = 1/10^N, where N is the
highest power of 10 used.
What is the value of (1 - 0.999…) as our approximation of ‘0.9…’ approaches the value we’re interested in? This is, fortunately enough, something we can find an answer for.
lim(n->infinity) 1/10^N = 0
So, as our approximation of the value of ‘0.9…’ becomes better, the distance between it and 1 shrinks to zero.
Please take me to task on any mistakes, obfuscations or handwaving that I may have committed above… I just got home from work and I’m not as sharp at this hour as I’d like to be.
Pyrrho12, I agree with everything you said except:
"My take on the whole thing is that ‘0.9…’ isn’t a number; It’s an approximation of a number. "
0.999… is a number. It isn’t approximately equal to one. It is exactly equal to one.
0.999… and one are two names for the exact same thing.
Virtually yours,
*DrMatrix - If I’ve told you once, I’ve told you 0.999… times.
Please do check the other threads on this topic, given earlier with a link.
Perhaps one confusion is the difference between a number itself and the “name” of the number. Let’s take the number 1. We can find many different expressions (“names”, if you will) for this number, including (6 - 5) or the solution to the equation 3*x = 3 or the multiplicative identity of the group of integers or the limit of the sequence {x/x} as x gets infinitely large. These are written in different ways, but they are all the same number, 1.
Some of these things can be written in a shorthand or simpler form. For instance, “multiplicative identity” is a shorter way of writing “integer value of x such that a*x = a for every a in the set of integers.”
Now, in this case, we have written an expression, namely “.9999…”. It’s meaningless because we don’t know what those dots mean until we define them. So, we are defining those dots to mean that the string of 9’s continues forever and ever, ad infinitam. That’s still kind of fuzzy, but in mathematics, we have a firm definition of what that means. In this case, as mentioned by several people, it means the limit of the sequence .9, .99, .999, .9999, etc.
And that limit can be shown mathematically to equal 1. Not to approximate 1 (although any finite subsequence of the initial sequence does approximate 1), but to equal 1, bang on the nose.
One more thought. Note that there is a difference between real life and mathematics, and hot dogs and rocks is as reasonable a way to express it as any. We can say that the circumference of a circle is pi times the diameter, and we can describe pi in mathematical terms. In real life, however, pi doesn’t exist – there are no measuring instruments sharp enough to draw a line of exactly pi in length. After a few hundred thousand decimal points, we are talking about distances much smaller than the width of electrons, and thus there is no physical reality to correspond to the mathematical concept.
OK, so it’s similar with .999… The number is a mathematical construct that is equal to 1. We draw a line draw a line of length .9, then add on to it a line of length .09, then add a line of length .009, etc. That would be doing the process that constructs .999… However, after drawing the first few million such lines, if we stopped, we’d still be at a number LESS than 1. To get to exactly 1, we could never ever stop the construction. Thus, the construction is not possible in the real world, it is only possible in mathematics.
Mathematics deals with infinity in two different ways. There’s the mathematics of infinity, as noted above, with the size of different infinite sets (and Dr Matrix is, as usual, right is correcting that the set of even integers and the set of all integers are the same size.) However, there’s the mathematics of the infinitely small, as well. We view the set of Real Numbers as forming a line, that is dense (mathematical sense) and things happen that are counter-intuitive when dealing with the real world. In the real world, I can make a dot; I can however never make a mathematical point (which has no length.)
Thus, the people who say .999… = 1 are the people who are working in mathematics. The people who say .999… can never actually reach 1 are the people who are visualizing in the real world. We’re in different definitional universes.
Hope that helps.
Aieee… you’re absolutely right. I guess what I meant to say was ‘when you write the number 0.999… repeating down on paper, you can only write it down as an approximation of the number you mean, which is more commonly known as 1, as long as you’re willing to completely ignore the word “repeating”, or that little bar that goes over repeating decimals, or any other thing that would let people know that you aren’t just copying out the number 9 over and over like some bizarre victim of Teret’s Syndrome…’
I guess that still doesn’t make much sense… 
Thanks for pointing out my mistake. I said that I wasn’t too sharp tonight.