Why is area defined the way it is?

Gigastrength
Gigastrength

Why is area defined the way it is?

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All urls found in this thread:

terrytao.wordpress.com/2010/09/09/245a-notes-1-lebesgue-measure/
math.stackexchange.com/questions/800699/why-square-units

Deadlyinx
Deadlyinx

What do you mean? "Area" is just the word representing the definition

BlogWobbles
BlogWobbles

Idk why is it measured like it is

Illusionz
Illusionz

Why wouldn't it be?

Lunatick
Lunatick

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.

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takes2long
takes2long

Because it's physically meaningful and helps engineers get what we want.

Skullbone
Skullbone

le engineer "because its useful" justification

TechHater
TechHater

Do you have a better answer?

TurtleCat
TurtleCat

Why is area defined the way it is?
You mean the Lebesgue measure?
That seems obvious.

AwesomeTucker
AwesomeTucker

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

Ignoramus
Ignoramus

Why is "is" defined the way it is?

Supergrass
Supergrass

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.

Soft_member
Soft_member

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.

Burnblaze
Burnblaze

can't tell the difference between reason and purpose

holy shit good luck dude

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JunkTop
JunkTop

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

MPmaster
MPmaster

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

kizzmybutt
kizzmybutt

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.

DeathDog
DeathDog

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

Poker_Star
Poker_Star

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

TreeEater
TreeEater

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.

hairygrape
hairygrape

Because the world is the way it is?

Sir_Gallonhead
Sir_Gallonhead

learn calculus

askme
askme

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

Snarelure
Snarelure

[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

Skullbone
Skullbone

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

Firespawn
Firespawn

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

WebTool
WebTool

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

Evilember
Evilember

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.

TalkBomber
TalkBomber

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

RumChicken
RumChicken

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

Why is anything else needed?

Emberburn
Emberburn

can you explain more?

CouchChiller
CouchChiller

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

MPmaster
MPmaster

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.

happy_sad
happy_sad

Measure of product is product of measures

idontknow
idontknow

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

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