While I agree that the example dosn't show that circles don't exist, the above has me scratching my head for two reasons.

First, a line contains an infinite number of points.

Second, the only value math has as a formal system is it's ability to help us model "objective" reality, whatever "That' is.
Goodel showed the limits of formal systems for capturing "Truth".
I suspose what I'm trying to say is that I don't see how an equation equaits to proff.


Anyway, Rucker used this as a springboard into Transfinite Numbers.
Aelf-One and such.


I didn't mean to imply no Circles.

Sorry.






Maou

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Circles exists for two reasons:

1.) They are a conceptional idea for something that is round and connected in 2 dimensions. These obviously exisit because we see them everywhere everyday. However, how are they made and defined? This leads to reason 2.

2.) Circles exist because an equation can be determine in in which for every x value of a circle defined by a radius r, there is a value for y. Since an equation exists, a circle does exist. Your point about there being an infinite number of points is so wrong because if you use that argument, a line doesnt even exist because a line contains an infinite number of points.

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This is hitting closer to the point. Pardon the pun
Infinagons don't cut it as infininty is just a way of saying I don't know. If you wanna throw infinities into the thing then you can renormilize it afterwards.
I believe the first time I ran across the little example for .9_ = 1 was in a book called "Infinity and the Mind" by Rudy Rucker.
The one I liked better from the first chapter is this;

Take a center-point.

Extend a line of unit 1.

Make the Circle r=1.

Now back to the point bit. What is a point? Well we simplly say they are dimensionless. Something like ghostmarkers in space. There can be an unlimited amount. Inifinate amount.

Okay then, what does this mean?

Take center-point from above.

Extend a line of unit 2.

Make the Circle r=2 (congurent?)

Now the questions are;
1 How many points make up the Circumfrence of Circle r=1?
2 How many points make up the Circumfrence of Circle r=2?
3 What is the ratio of the Circumfrence of Circle r=1 to the Circumfrence of Circle r=2?

In other words how can these figures Circumfrence consist of the same number of infinate points (radi from center-point for proff) when r=2's Circumfrence is twice that of r=1?





I found it a nifity read,




Maou

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Perfect circles are impossible to create, so technically they don't exist because they can't be created. For Pi, the best we can ever come up with is an approximation.

As a calculus student, I have run into the .999... = 1 argument multiple times with multiple people. I always try to explain to them that it is not debatable, but accepted definition by mathematicians for the last 400 years. I will be more than happy to help you explain why. There have already been some proofs thrown around here varying from simple to somewhat more algebraic. Allow me to sum these up:

1/9=.111...
2/9=.222...
3/9=.333...
4/9=.444...
5/9=.555...
6/9=.666...
7/9=.777...
8/9=.888...
9/9=.999...

Since this is inductive in theory it's not a proof. So, let's use algebra to deductively prove this. This was done beautifully by alex7h3pr0gr4m3r:

1. x=.9... (Set x to .9...)
2. 10x=9.9... (Multiply both sides by 10)
3. 10x-x=9.9... - x (Subtract X)
4. 9x=9.9... - .9... (X is equal to .9 repeating, we stated that in step one, so this is legal)
5. x=1 (Divide by nine)
6. .9...=1 (Substitute)

Most people refute step 2 and 4, saying it does not correctly respect the concept of .9... however if one took the time to look in any algebra book this is taught and used by 8th graders everywhere to convert repeating decimals to fractions.

Still don't buy it? Ok then, let's assume .9... and 1 are different numbers. This means they have to have a difference.
1 - .9 = .1
.1 - .09 = .01
.01 - .009 = .001
On and on and on...
So in order for them to be different there must be 1 at the end of a set of infinite zeros. Yet this is logically impossible.

For anyone who has an elementary understanding of calculus, what is the value of (1/2)^x as x approaches infinity? Yes, it is indeed zero. The difference here is that .9... isn't approaching anything. It is simply itself, 1.

Finally, head over to Wikipedia and type in .999:

But Wikipedia is inaccurate right? People are changing it all the time, so there's no way .9... equals 1! Right

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i don't really quite understand that yet...i must be having a blonde moment because i didn't understand that cartoon thingy either

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