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How can we define area?

Anonymous Mon 14 Dec 00:17:09 2020 No.12462109 View Reply Original Report

Quoted By: >>12462155 >>12462405 >>12462426

Could we say that area is the number of [a,b]x[c,d] rectangles with b-a=d-c=1 that we can fit into a shape?
This could also account for fractional areas since we could say 1/n th of such rectangles fits inside a shape.

Anonymous

Anonymous Mon 14 Dec 2020 00:38:35 No.12462155 Report

Quoted By: >>12462193

>>12462109
>b-a=d-c=1
This makes no sense. Also "number of" doesn't work. You can't cover every space by a finite amount of rectangles.
Anyway google Lebesgue measure.

Anonymous

Anonymous Mon 14 Dec 2020 00:54:48 No.12462193 Report

Quoted By: >>12462202

>>12462155
>b-a=d-c=1 does not make sense
It's so we can talk in terms of "unit" squares

> You can't cover every space by a finite amount of rectangles

If you cant then the "area" diverges, no? So you can just make an argument for that

Anonymous

Anonymous Mon 14 Dec 2020 00:56:39 No.12462200 Report

Quoted By: >>12462203

Area is the number of small little 1x1 boxes you can fit inside. Final answer.

Anonymous

Anonymous Mon 14 Dec 2020 00:56:54 No.12462202 Report

Quoted By: >>12462210

>>12462193
can you cover the inside of a circle by a finite amount of rectangles?
does its area diverge?

Anonymous

Anonymous Mon 14 Dec 2020 00:57:38 No.12462203 Report

Quoted By: >>12462246

>>12462200
basically what I argued, is it the formal definition of area though?

Anonymous

Anonymous Mon 14 Dec 2020 01:00:18 No.12462210 Report

Quoted By: >>12462240

>>12462202
thats why i mentioned you could talk about something having area 1/n th of a rectangle. You can cover it with rectangles and then sum up the little pieces after your'e done with the rectangles that are in the interior of the set

Anonymous

Anonymous Mon 14 Dec 2020 01:10:10 No.12462240 Report

Quoted By:

>>12462210
You can take a cover of open rectangles and sum over their measures. then take the infinum of that and you get the outer lebesgue mesure. But you need to restrict it to lebesgue mesursble sets or else you end up getting bs.

Anonymous

Anonymous Mon 14 Dec 2020 01:11:13 No.12462246 Report

Quoted By:

>>12462203
amount of space enclosed by a boundary

Anonymous

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Anonymous Mon 14 Dec 2020 02:04:17 No.12462405 Report

Quoted By:

>>12462109
For simple shapes such as the circle, sectors of the parabola or the square, the following similar definition works.
You do the same thing but with smaller squares, say with b-a=d-c=1/n. You look at how many squares they fit. The bigger you make n, the smaller the squares will become, and hence more of them will fit (if any).
So what you do is define A(n) be the maximum number of squares of proportions b-a=d-c =1/n that fit into the figure. The idea is that as n gets larger, the area of the fitted squares, which is A(n) *(1/n)^2 should get closer and closer to the actual area. If it has a limit x, you call the area x.

Anonymous

Anonymous Mon 14 Dec 2020 02:10:01 No.12462426 Report

Quoted By:

>>12462109
see measure theory
see sigma algebra

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