Results 1 to 6 of 6

Math Help - Area is unique.

  1. #1
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9

    Area is unique.

    In Euclidean Geometry we definie area as (for polygons) a positive real number satisfing:
    1)Two congruent polygons have the same area.
    2)The area of a rectangle is the product of base and height.
    3)A polygon which is a union of two polygons not having an interior point is the sum of the areas of the two polygons.
    Prove:The area of a polygon is well-defined.

    Meaning no matter how you disect a polygon as smaller polygons and sum them you always end up the the same real number.

    This problem seems extremely difficult. It seems to be connected to graph theory (get it? connected to graph theory ). How do you prove such a monster problem.
    ---------------
    However, if you define area not in terms of polygons but rather in terms of function. Then the problem is easier. You define it area to be the limit of the Riemann sum. And the fact is that no matter how a Riemann sum is partitioned as long as the norm is convergent to zero the limit (if exists) is unique. Thus, area is well-defined.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    At my house.
    Posts
    527
    Thanks
    7
    The definition of area for a surface, should go as follows:

    Given a polygon, consider a (disjoint) partition of it into rectangles. We can find the area here but adding up the elements of the partition. The axiom guarantees this.

    For a general surface, we can consider a disjoint covering of it, consisting of polygons. We can find the area of this covering, call it S. Then define the area of the surface as liminf{S}, where the limes inferior is considered over the set of all coverings of S.

    This is well defined, and easily seen to comply with everything a measure of area should be Plus, it does not need anything more for its definition, but the area of rectangles!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    My 10th Grade math teacher defined (and I still remember this), after he asked the class what area is, that area is the measure of a mathematical region. I just find it funny, he manages to bring Measure Theory into 10th grade.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,664
    Thanks
    298
    Awards
    1
    Quote Originally Posted by ThePerfectHacker View Post
    My 10th Grade math teacher defined (and I still remember this), after he asked the class what area is, that area is the measure of a mathematical region. I just find it funny, he manages to bring Measure Theory into 10th grade.
    I have a problem with this definition: what does the term "measure" mean? I have the same problem with the formula d = rt (distance equals rate times time.) What the heck "rate" are we talking about? Not specific enough for me.

    -Dan
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by topsquark View Post
    I have a problem with this definition: what does the term "measure" mean? I have the same problem with the formula d = rt (distance equals rate times time.) What the heck "rate" are we talking about? Not specific enough for me.

    -Dan
    I think he meant the Lebesque Measure.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,664
    Thanks
    298
    Awards
    1
    Quote Originally Posted by ThePerfectHacker View Post
    I think he meant the Lebesque Measure.
    Ah! That will do it.

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. unique solution
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: January 2nd 2011, 09:34 AM
  2. Unique identity
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 7th 2010, 01:10 PM
  3. Two unique Derivatives
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 30th 2007, 07:12 PM
  4. Unique Geodesic
    Posted in the Advanced Math Topics Forum
    Replies: 5
    Last Post: June 8th 2007, 03:24 PM
  5. Unique factorization
    Posted in the Number Theory Forum
    Replies: 8
    Last Post: November 2nd 2006, 09:22 AM

Search Tags


/mathhelpforum @mathhelpforum