Area of a line

[Anon No.6129004]

Discuss

[Anonymous No.6129017]

>>6129004
Errors in steps 2, 3 and 4.

[Anonymous No.6129027]

>>6129004
'Area' applies to 2D shapes.
A line is 1D, and therefore does not have area
You saying that the area is '1' is completely meaningless.
1 what?
centimetre squared? meter squared?

You also can not use infinity as a number in equations, and you can not divide by it.
It is a concept, not a number.

[Anonymous No.6129028]

What is lenght? You mean length? You didn't spell check your theorem?

[Anonymous No.6129029]

>>6129017
Saying that there are errors is not helpful unless you specifically point out what the errors are.

[Anonymous No.6129032]

>>6129004
<span class="math">\tfrac 1 \infty[/spoiler] is undefined.

[Anonymous No.6129037]

>>6129004
infinity * 1/infinity is not 1.
for a more correct but not quite answer:
area of point = 1/x^2 with x-> 0
area of line = x* area of point = x/x^2 = 1/x = 0

[Anonymous No.6129041]

>>6129037
*x-> infinity i mean

[Anonymous No.6129042]

>>6129028

I'm truly sorry, dear Anon. The only excuse I can find for this pitiful situation is that I must have overlooked it and Microsoft Paint does not contain spell checking function.

Once again, I'm sorry.

[Anonymous No.6129058]

>using infinity as a real number
Assuming you're not trolling, you might benefit from looking up the mathematical terms group, ring and field, as well as reading up on the way real numbers are constructed from natural numbers.

[Anonymous No.6129060]

>>6129042
Apply yourself

[Anonymous No.6129067]

>>6129027

Infinity is a number if you work with the one-point compactification of the real line. The problem is this is no longer a field, and certain operations are no longer defined, and division no longer has its original algebraic meaning. For example, 1/infinity = 0 but 0*infinity is undefined, which is what OP used in his "proof" and that is the error.

[Anonymous No.6129071]

>>6129004

>Hurr durr guys look at me I don't understand the real projective line or measure theory!

1/infinity*infinity is undefined, dumbass.

http://en.wikipedia.org/wiki/Real_projective_line

[Anonymous No.6129072]

>>6129071
There is no need to be rude.
You can correct an incorrect statement without having to be mean about it.
:(

[Anonymous No.6129076]

>>6129072

You want me to call you a whaaaambulance?

[Anonymous No.6129078]

>>6129076
?
That is not a real word.
Ambulances are for people who are ill.
I have no idea what you are talking about. :/

[Anonymous No.6129090]

This is OP, just to clarify, I wrote these posts:
>>6129042
>>6129004

and I have no connection with these:
>>6129078
>>6129072
>>6129029

I will consider your replies while fixing my theorem, thank you for your time. If there is still something you want to say, I'm continuing to follow the thread and will answer questions if necessary.

[Anonymous No.6129091]

Shut the fuck up.

http://en.wikipedia.org/wiki/Measure_(mathematics)

[Anonymous No.6129098]

>>6129090
>I will consider your replies while fixing my theorem...
It is not a theorem.
How many theorems do you know that are made in MS-paint, in less than 2 minutes, with spelling mistakes and very obvious incorrect use of mathematical laws??

>and I have no connection with these:
No, those are mine.

[Anonymous No.6129106]

>>6129090

It's not a theorem. You assumed that the real projective line is a field. It is not. You cannot assume that the rules of real number arithmetic apply to the real projective line.

Basically, I'm saying you're a fucking retard and this thread is cancer.

[Anonymous No.6129109]

a point has no area, so a line could not have an area

[Anonymous No.6129115]

>>6129106
That is not very nice.
If you do not like the thread, then just do not post in it.

[Anonymous No.6129119]

>>6129115

> Hey stop telling me my math is wrong!

[Anonymous No.6129128]

A line can't have a thickness, dumbass. How does it have an area? 0*infinity = indeterminate

You have no proof.

[Anonymous No.6129131]

>>6129119
It is not my work; I am not the OP.

And you can them that they are incorrect without being rude about it. (As I did)

[Anonymous No.6129130]

you forgot to ad QuED. I fyou dont add that, you dont get your norbel prize

[Anonymous No.6129139]

>>6129128

OP here: line's thickness strives to 0, meaning that it is almost 0 but not exactly 0.

[Anonymous No.6129140]

>>6129128
>dumbass
Again, unnecessary rudeness.
/sci/ is not usually this bad. :(

>A line can't have a thickness
It depends what kind of line.
In a mathematical sense, usually not.
A line drawn with a pen obviously does.

[Anonymous No.6129145]

>>6129139
>line's thickness strives to 0
>strives

You keep using that word. I do not think it is the optimal one for the sentence.
I think one would say: "A line tends towards zero"

[Anonymous No.6129156]

>>6129145
OP: english is not my native language, I know some of it but I've never had any math in english so I had to check it up in vocabulary.

I must agree, "tends" sounds a lot better.

[Anonymous No.6129159]

>>6129140
Yes, obviously in order to draw a line for graphical/purposes it will have a literal thickness, but that's like saying 2 has a thickness or = has a thickness.

Mathematically it's understood that lines/points have no thickness.

Even if OP wants to argue its thickness tends to zero, whatever. It's not exactly true, but you can present an argument.

The problem with his proof is dividing and multiplying by infinity. Infinity is not a number and shouldn't be treated as such.

[Anonymous No.6129162]

Operations on inifnity are not defined (at least that's the default assumption, if you're using certain definitions please tell us beforehand)

The area of a point is not "striving" for 0, it is 0.
Therefor a sum over this area is also 0.

Even if we used the definition of the area of a point like this:
<span class="math">
A_{point} = \lim_{\epsilon \rightarrow \infty} \frac{1}{\infty} = 0
[/spoiler]

The sum over the these points still is 0, so the length of a line would be zero, as would be it's area.

[Anonymous No.6129168]

>>6129004
>Treating infinity as a number

[Anonymous No.6129170]

>>6129162
This guy sums it up nicely.

[Anonymous No.6129212]

>>6129168

You can treat infinity as a number. There's nothing wrong with using infinity as a number (extended reals, projective line) if you understand what the rules are. But OP is a dipshit and can't be bothered to actually check that the projective line is not a field, so certain operations such as 0*infinity are undefined.

[Anonymous No.6129224]

>>6129139

Infinitesimals are nonstandard analysis. I seriously doubt you know anything about using infinitesimals.

http://en.wikipedia.org/wiki/Non-standard_analysis

[Anonymous No.6129249]

<span class="math">
\lim_{\epsilon \rightarrow 0}\int_0^\epsilon dy \int_a^b dx \lim_{\delta \rightarrow \infty} \frac{1}{\delta}=0
[/spoiler]
Class dismissed.

[Anonymous No.6129250]

Btw for anyone who actually cares about learning about infinitesimals, Kiesler has a free online version of his calculus text that uses rigourous infinitesimals.

http://www.math.wisc.edu/~keisler/calc.html

[Anonymous No.6129298]

>>6129004

>area of the point is (1 / {infinity})
>area of line amounts {infinity} * (1 / {infinity})
Why is the area of the point not 0.001 / {infinity}?
Why is the area of the line not ({infinity}^1000) * (1 / {infinity})

You have to understand that there are many different "inifnities", broadly speaking. Google "countably infinite" and "uncountably infinite".

The second problem is, length and area are different DIMENSIONS.
And anything that "exists" in (n-1) dimensions is always a null set in (n) dimension - ask good old Lebesgue, he knows.

HTH.

[Anonymous No.6130303]

Let's try to find the area of the line using actual mathematics.
Let A be the area of a line (i.e. A is some non-negative real number).

Now let ε be any positive real number.
For each integer n, let R(n) be a rectangle of length 1 and width ε/(3·2^(-|n|)) (so that the rectangle R(n) has area ε/(3·2^(-|n|))).
The combined area of the rectangles is then
Σ_[n=-∞..∞] ε/(3·2^(-|n|))
= (ε/3)(Σ_[n=-∞..∞] 1/2^(-|n|))
= (ε/3)(Σ_[n=-∞..-1] 1/2^(-|n|) + Σ_[n=0..∞] 1/2^(-|n|))
= (ε/3)(1/(1 - 1/2) - 1 + 1/(1 - 1/2))
= ε.
Observe that we can cover the line with the rectangles if we place them together lengthwise and side to side.
The line is then completely covered by the rectangles, so whatever its area is it must be less than their combined area (i.e. A < ε).

Conclusion: the above argument did not depend in any way on a particular choice of ε, so A is less than every positive number.
Now, what is the only non-negative number that is less than every positive number?

[Anonymous No.6130443]

>>6130303
finally a good answer.

>>6129027
numbers is also a concept. also http://en.wikipedia.org/wiki/Riemann_sphere

[Anonymous No.6130498]

a line has no area
it's a one-dimensional shape and therefore by definition has no area

also stop treating infinity as a real number

[Anonymous No.6131125]

>>6130303
>measure theory ftw

>>6130498
Saying that a line doesn't "have an area" makes it sound like the notion of area
is simply undefined on lines. A line DOES have an area, it's equal to zero.

[Anonymous No.6131156]

It doesn't make sense. What's the area of a cube? It's meaningless unless you're referring to its surface.

What's the midpoint between California and Los Angeles? Same type of question.

[Anonymous No.6131157]

>>6130303
>ε/(3·2^(-|n|))
don't you mean ε/(3·2^(|n|)) ?

[Anonymous No.6131600]

>>6131157
Yes, you are entirely correct.

[Anonymous No.6131661]

>its area is striving for amount of 0

I chortled.

[Anonymous No.6131696]

>>6129072
>There is no need to be rude
This is 4chan, not fireside girls.

[Anonymous No.6131730]

>>6129004
in your first point you have 1/inf. the units there are 1D side over 1D side. when you multiply the inf again in the 3rd equation its an infinite 1D unit. So you end up finding the length, not area