You’ve almost got it. The problem you’re hung up on is the infinite series of 9s. No one is placing the 9s there. They are just there. Whether it’s physically possible to actually write them all our or not is immaterial, sice that’s not what we’re discussing. there is an infinite number of 9s after the decimal point. Accept that.
Such a small mind that cannot encompass the notion of infinity.
No, he doesn’t have any formal training, but soon he will be introducing a revolutionary machine that produces more power than is put into it and that proves Einstein was wrong!
Phage
Phage, like many mathematicians and the mathematically inclined here, we’ll have to agree to disagree. A limit, properly applied, can provide an exact answer. You can disagree, if you like, but in doing so, you either define your own mathematics, or are arguing with the basis of established mathematics. The former is perfectly reasonable. The latter requires extraordinary proof, and counterexamples that have already been given to you have to be shown to be flawed. You have yet to do so. You can define what 0.999… means to you, but it doesn’t mean that to mathematicians, and they have a better reason for doing so than you do – its consistent with the axioms of mathematics and everything deriving from them.
As an example, if limits did not produce exact values, then it would not be appropriate to say that:
d( x^3 )
----------- = 3x^2
dx
(That the derivative (slope at a point) of x cubed is three times x squared)
for all values of x, not just for some values of x, or with some error component. The reverse, that
S 3x^2 dx = x^3 + c
(I use S as a cheap integral symbol, can’t figure out how to do it)
is true for all values of x, not just some x, or with an error component.
None of these would be provable without using some concept of limits, and allowing those limits to achieve exact values. They are not approximations.
Because of the way reals are defined, and limits are defined, the result that 0.999… = 1 is a consequence, and an unavoidable one based on the axioms involved.
If you accept the axioms, and still don’t accept the result, your logic is flawed, intuition aside. If you don’t accept the axioms, you aren’t using the same mathematics as we are.
Since it is an iron-clad property that between two reals a and b there is at least one other real c such that c=(a+b)/2, there can never be a ‘next real’ or a ‘previous real’ in the real number line. Ever. This has already been discussed. There is no value difference between 0.999… and 1. There can’t be on the real number line.
It may be non-intuitive, but sometimes that’s the way it goes.
But as you say, we’re just going over old ground, and I’m ill-equipped to prove right from first principles, and the true mathematicians here won’t bother as the result is already well established in countless existing mathematics tomes, courses, and web-pages.
First, the cite listed is an answer appropriate for an 8th grade math class. The page cited BY that page for more information matches exactly at least two of the arguments presented against you.
The problem with your thinking is at step 3. The elipse is defined as the representation of an infinite # of 9s. Since the elipse is designed to represent an infinite # of 9s, .999… is equal to 1, by your own fourth point.
-lv
Actually, the concept of infinity is a very complex and subtle idea, so I’m willing to give someone the benefit of the doubt who doesn’t grasp it. You guys have fought a good fight. You’ve explained it as many ways as there are to explain it. Six pages worth.
It may be time to let the naysayers simply ruminate on the principles involved.
I knew there’d be a problem when I read somebody say that 0.333… does not equal 1/3. 1/3 is precisely equal to 0.333…
Therefore 0.999… is precisely equal to 3/3, which is precisely equal to 1.
Achenar, of course 1/3 = 0.3333… is defined as having infinite decimal places otherwise if it has finite decimal places it is an inequality not an equality.
I am the 18th person pointing this out, but what you or I or the entirety of humanity can do is irrelevant. We cannot create the long hand number of .9 repeating infinately. But we can create the shorthand representation of.999~. Now we have infinite 9s to work with.
For example we have 2 circles one with a radius of 2 units and one with a radius of 4 units. now lets compare areas in base 10.
circle one: pi r[sup]2[/sup] = 4pi square units
circle two: pi r[sup]2[/sup] = 16pi square units.
Circle 2 has exactly 4 times the area circle 1.
Do you disagree with this calculation? It is only possible because we accept the shorthand ‘pi’ for the value it represents. In base ten the number cannot be written out because it would be infinite. Does this make the shorhandt calculation invalid?
I love proofs like this 
Ok, here’s what I wrote to a friend:
Let X = 1 + 2 + 4 + 8 + 16 . . . . + 2^n
then X - 1 = 2 + 4 + 8 + 16 . . . . + 2^n
so (X-1) / 2 = 1 + 2 + 4 + 8 + 16 . . . +2^(n-1)
so ((X-1) / 2) + (2^n) = X
and (X – 1) + 2^(n+1)= 2X
and X = (2^(n+1)) - 1
so 1 + 2 + 4 + 8 + 16 . . . + 2^n = (2^(n+1)) - 1
Just because it’s an infinite series, doesn’t mean to ignore certain terms. An infinite series is just a very very very very very very long finite series that doesn’t like to end.
X(0)= 1 =2-1=1
x(1)= 1+2 =4-1=3
x(2)= 1+2+4 =8-1=7
x(3)= 1+2+4+8=16-1=15
Also, I pointed out earlier that an arrow peforms this operation (9/10+9/100+9/1000…) to infinity in terms of l, it’s length, every time it is fired into a target (or indeed any moving object moving across an arbitary length, l)
Phage said:
(Emphases added.) Here’s the crux of the disagreement. When mathematicians talk about 0.999…, they are talking mathematics. When Phage talks about 0.999…, he’s talking about “reality”. It doesn’t matter what reality is, it has no effect on mathematics. Mathematics is a formal system, deriving theorems from axioms, using a logic rule-set.
In his point above, Phage basically admits that the mathematical proofs of 0.999…==1 are correct, but denies they are applicable to the real world. He can of course claim this, but should be aware that when most people write 0.999…==1 (or 0.333…==1/3) they are implicitly talking mathematics, not “reality”.
Is there still a GQ here? I humbly suggest a moderator either close this thread or move it to GD.
Do you even read this stuff? Here are some quotes from that exact page:
And of course…
I mean, really. Why do you quote this stuff when it doesn’t support your position?
The evidence certainly seems to agree with this statement.
After perusing 4+ long pages of (interesting yet time-consuming) mathematics, I think the problem is that the supporters of Phage-math and the supporters of actual math are talking through each other. Phage, et al, keep bringing up “properties” of 0.9~ that no Real numbers have - ie, it’s the “closest number to 1”, and such. Basically, Phage looks at 0.9~ and imagines something that isn’t a Real number (note here that I’m referring to “Real numbers”, the rigidly defined class of numbers, as opposed to “real numbers”, which are just some random “number” that can possess any property you throw at it).
And that’s fine, I suppose. Phage, go ahead and create some new class of numbers. It can include your version of 0.9~ (the closest number to 1, as you approach from the left). It can include 1.00…1 (the closest number to 1 as you approach from the right). It can include Bob, defined as 2<Bob<1. It can include G, defined as “infinity + 1”, and G*, defined as “infinity + 1, no backs”.
The thing you have to keep in mind, though, is that these Phage-numbers are not Real numbers, are not Integers, are not Complex numbers, and, frankly, are of absolutely no use to mathematicians, engineers, scientists, or anyone else who is in a profession that uses rigidly defined mathematical concepts. They are in a class by themselves, and that class does not intersect with any commonly used set of numbers. Hey, maybe someday we’ll find a use for Phage-numbers, but we don’t have a use now (and I wouldn’t hold my breath).
So to summarize: according to the specific rules that have been laid in place by mathematicians, the Real number 0.9~ = 1. If you want to talk of some other mythical number represented by 0.9~ that doesn’t equal 1, that’s fine, but make sure you know that this “number” isn’t of much use to anyone, and certainly isn’t a Real number.
Jeff
This is exactly what the “…” means, we can write the “…” in this case because all of the values are known.
Reached no, expressed symbolically yes.
Not “as more 9s in 0.999… are considered,” all of them are being considered here.
That is what the “…” allows us to do.
You yourself are saying that “as more are considered the closer the number comes to one,” the trick here is to realize that “more” is meaningless here because there are an infinte number of 9s being considered.
Which is exactly what the “…” means and exactly what everyone here has been trying to show you.
Ah but in reality, .999… = 1. In order for them to be two different numbers you have to show that there is a difference between them. What is that difference?
You are confusing yourself when you say “infinity cannot be reached.” When you say that you mean that an infinite amount can never be reached. However, when we are talking about an infinite number of nines after the decimal point we are talking about an infinite degree of precision. That means that the number we throw out is defined to be exactly as it is represented.
Just like the number …0002.000… or …00027.34333… or even …000.999…
If that doesn’t help then think back to the earlier example involving a triangle with area equal to 1 being divided into three parts. Each part will have an area equal to 0.333… and the sum of those three parts can be written both as 0.999… and as 1. If 0.999… != 1 then it seems you have a real world problem where a given real world area can be changed simply by performing math upon it.
Pleonast said:
While I understand your frustration, I think many of us would be worried by an official statement that mathematical quandaries were matters of debate rather than factual questions.
Well, how about an "Asked and answered
…and asked, and asked, and asked, and asked, and asked…"
This is going to take a while.
Hey, that brings up a good point! Phage’s pattern of posts are meant to mimic the infinite sequence of 9’s! He’s a GENIUS!
<sings>
This is the thread that never ends,
It just goes on and on my friends,
Some people started writing it not knowing what it was
And they’ll continue writing it forever just because
This is the thread that never ends…
JRootabega said:
On that one you would get no argument from me.
I humbly request a cite, preferably from a text on Analysis or some other math of a similar level of advancement.
I am not saying that the notation 0.333… represents any finite number of 3’s. I am saying that the notation 0.333… represents the limit of a sequence whose Nth term is 0.333…333, where the number of 3’s is N. In your mind, this may be the same thing as an infinite number of 3’s, but in my mind, it’s not. Specifically, exactly what “an infinite number of 3’s” means mathematically is not well defined.
This is incorrect. An infinite series is an infinite series. It has more terms than any finite series. The “proof” that you quoted is incorrect because laws of arithmetic only work for convergent series, but there are contexts in which it is a valid argument: add all the powers of 2, and you get -1.