Why is area defined the way it is?

Why is "is" defined the way it is?

Area is defined the way it is so that people can use simple definitions and simple methods in many real world practical cases, and complex definitions and arcane methods in in a small minority of problems.

Because it works. If u want a carpet for the whole room then doing it the way u are supposed to do it will work.

>can't tell the difference between reason and purpose

holy shit good luck dude

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Because the lebesgue measure is unique: (See exercise 23)
terrytao.wordpress.com/2010/09/09/245a-notes-1-lebesgue-measure/

>That seems obvious.
that would make it easy to explain, no?

The basic Idea is really simple.
It is pretty obvious to see what the area (or volume) of an interval *should* be, that is just applying a rule you should have learned very early in your life, about how to compute the area of a recangle.
From there on, you cut other "nice" sets, into intervals, so that you can calculate their area, this is a rather complicated construction and will probably seem confusing on the first glance.

Once you understand the basic principle behind it, it really is easy to understand why the construction works that way.

>about how to compute the area of a recangle.
why are rectangles' areas calculated like that, user

>the difference between reason and purpose
So what is the difference, and why does it matter in this context?

describing the conditions for which something is not only viable but desirable over something else, is what gives it purpose. reason is the ability to convey how something was constructed. in this case it's the actual mathematical content.

Because the world is the way it is?

learn calculus

This is the right answer imo. At some level it really just is an arbitrary definition.

[eqn]\int_\Omega f_{\left[\alpha_1, \alpha_2\right]}(x^1,\dots,x^n)\ \text{d}x^{[\alpha_1}\wedge\text{d}x^{\alpha_2]}[/eqn] How else would you define it

Because the square tiles nicely and the whole rectangles are just multiplication grids thing

math.stackexchange.com/questions/800699/why-square-units

>why does x mean y
>"what do u mean y just means x"

It has lots of nice properties.
>simple formula
>Congruent rectangles always have the same area
>a rectangle with no length or no width (i.e. a line) has zero area
>It works arithmetically. For example, if you combine two rectangles of the same length into one, it will have twice the original area.

>if you combine two rectangles of the same length
*of the same area

Fun fact: Arabic, Golden-age mathematicians used to denote the area of two combined lines by simply three points.

Why is anything else needed?

can you explain more?

>denote the area of two combined lines
What does that mean?

>why are rectangles' areas calculated like that, user
Because ultimately we want math to be able to model reality, thus we have to accept *some* things which seem to be intuitively clear.

Of course you can ask reductionist question to a point where the answer is "because I say so", but that doesn't really matter.

Measure of product is product of measures

area would be better defined as just being the determinate. prove me wrong