Why is area defined the way it is?

# Why is area defined the way it is?

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**terrytao.wordpress.com**

**math.stackexchange.com**

because it's run by someone who will send you to die if it means I am displeased with your presence for a moment. They believe it, it's only their minions don't.

Why is area defined the way it is?

You mean the Lebesgue measure?

That seems obvious.

it's purpose is to determine if something can 'fit' inside another thing in 2D

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.

Because the lebesgue measure is unique: (See exercise 23)**terrytao.wordpress.com**

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.

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

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.

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

Why is anything else needed?

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.