.9~ = 1, and anyone who doesn't think so should go back to school and take advanced math.

Here are some proofs...pick one you can understand:
Let us assume x = .99999~
Now we know that when we multiply something by 10, we
move the decimal one place to the right.
so 10x = 9.9999~ There is no 9 lost by doing this
since there are an infinite number of them.
Now we do simple arithmetic
10x - x = 9.999~ - .9999~
9x = 9 This is allowed because every 9 after the
decimal will cancel with another 9.
x = 1 and x = .999~ so 1 = .9999~

.9999~ = .9 + .09 + .009 + ....
here we represent .9~ as an infinite sum
sum[i:0->inf.](.9*.1^i)
We know how to solve infinite sums.
sum = .9/(1 - .1) = .9/.9 = 1
Since we said the sum was initially .9999~, we can
conclude that .9999~ = 1

1/3 = .33333~
This is true, and can be proven with an infinite sum
as above.
3*1/3 = 3*.33333~
1 = .99999~
We are allowed to multiply by 3 because no part is
going to carry over to the next part. Thus, every part
of the decimal will increase by factor of 3, making it
a 9.

The real numbers are defined as limits of Cauchy
sequences of rational numbers.

*A rational number is a fraction of two integers
*A cauchy sequence is a sequence x(1), x(2), ... such
that for every integer n there exists an integer m
such that |x(j) - x(k)| =< 1/n for all j,k >=m.

Two Cauchy sequences x(1), x(2),... and y(1), y(2),...
are considered equivalent if for every integer n there
exists an integer m such that |x(k) - y(k)| =< 1/n for
all k>=m.

Let x(j) = 1 - (1/10^j)
Let y(j) = 1.

I'll leave it to you to check these are Cauchy
sequences.
These two sequences are equivalent:
Given some integer n, |x(k) - y(k)| = |1-(1/10^k) - 1|
= |1/10^k| =< 1/n if 10^k >= n. So we'll set m =
{smallest integer greater than log(n)}.

Thus the sequences .9, .99, .999, ... and 1, 1, 1...
are equivalent, so they have the same limit, namely,
.999~.

0.9~ = 0.9 + 0.09 + 0.009 + 0.0009 + ...
S = 0.9~
S = 0.9 + 0.09 + 0.009 + 0.0009 + ...
S = 0.9 + (1/10)[0.9 + 0.09 + 0.009 + ...]
S = 0.9 + (1/10)S
(9/10)S = 0.9
S = 1

Therefore, 0.9~ = 1.

If two numbers are not equal, there are an infinite
number of numbers between them. Give me a number
between .9999~ and 1.

All repeating and terminating decimals can be
represented as fractions. If .999~ is not represented
by 1, what fraction does represent it?

I like to point out that I lost all faith in the circle now with all this math and it's existence nonsense . So to help mess with people's mind's even more check out this wonderful info about the circle.
http://en.wikipedia.org/wiki/Circle

Of course through my researching this wonderful thought there are several other sites that would agree .9999~ = 1

here

http://digg.com/tech_news/.9999999=1

and here

http://polymathematics.typepad.com/polymat..._sorry_it_.html

But like I said you all ruin the beauty of the circle now I to rely on a triangle and a square to get me through life . but to stay on topic just a bit after reading these posts and those 3 websites I would have to agree especially with this part



The math teacher followed the same line of thinking, in which I conclude that .99999~ = 1 is both a true and false statement. Meaning that if you round up you will get 1; however, since the number is always repeating itself then it is not a true solid number (can't think of the word for it but you math geeks know what I am referring to). You call this a enigma in itself and odds are you would have to apply occam's razor to make the most sense out of this enigma.

This post has been edited by Saint_Michael: May 14 2007, 10:40 AM

Love this post. I go with the "smoking" answer, and it's winning so...
I can't discuss on this but I think a circle it's a circle and yes, it's also a polygon with so many angles that the eye can't see them and the final look is curved and sharpened. Both things, or one thing forms another, whatever. I need more inspiration to answer properly.

If you and your math teacher really think that, then neither of you understand what infinity is. .9~ is not a process; it's not growing. It's a number that's exactly equal to 1 without rounding.

Whoah! That was some...whooh! I don't think I can add anymore to what was already said. It's kinda hard to stomach, 0.999... being equal to 1 and all, really, but I suppose advocates of the Ptolemaic model felt the same with Copernicus' heliocentric model.

Personally, I don't believe in the existence of circles. They are, for all we know, nothing more than constructs in the mind of sentient species, like humans, for example. Why is that? Because circles have no thickness! They have height and width, being in 2D but they have no thickness. Their z-dimension is zero. In our world, it would have to be a loop of wire in the shape of a circle but having an infinitesimally small, even non-existent, thickness.

Oh yeah, for that matter, I don't believe in squares and triangles too. As a matter of fact, I could be audacious enough to claim that polygons do not exist. So, sorry, Saint_Michael, I suppose the beauty of triangles and squares have also been dispelled

I do concede, however, that what we have here in the real world are approximations of a circle, or polygons, for that matter

Well at least somebody finally understands what I'm trying to convey....

This post has been edited by alex7h3pr0gr4m3r: Nov 2 2007, 06:39 PM

That not my math teacher I am referring to, I was referring to the teacher in those links has follow the same procedure as stated by your previous post. Now just remember what I was referring to in my last post since .9999~ is not a whole number and not solid so a minor correction on that.

Well think about it though .9999~ is a fraction right? and if that 9 keeps repeating itself in just basic math then it will never equal 1. But I think it was mention somewhere you have to be high up there in math; like trig calculus and even physics. So I would say my state is true that depending on the situation that .9999~ is being used it will never be a true whole number, but it could be especially when you go into margin of error +/-. Taking this stats class last semester, .9999~ would become a whole number so to have none of those factions appear in statistics reports and thus the margin of error would go into effect. But hopefullly I can bump into this math teacher at college and ask him about it and see what he comes up with. I think he's been a math teacher since about 30 years so I would say he could give a good solid answer on this.



This isn't the matrix lets try and stay in the physical plane . It reminds me of a Star Trek episode where they mention a dodecahedron, which I believe is like 26 sides shape or something like that, but of course now we would be going into physics about containing energy and what not.