I got stuck thinking about a question which seems to be very trivial. The question is that if we consider a string(basically a line segement) and now if we deform the string to change its geometry from straight line to something else suppose a circle. And if we are asked to calculate the radius of that circle then we just use the fact that the length of the string is equal to the perimeter of the circle. And from that fact we create an equation which gives us our required result.I have tackled this kind of problems in the chapter called mensuration. But I want to know why is this so? Though it seems that it should be but can it be proved ? I mean no matter how I deform my string maybe it takes the shape of some curve then also the length of that curve is equal to the lenght of that straight string.
After that I thought to see how mathematicians define the term length mathematically. I thought from there I can get my answer. After that I also searched about how mathematicians define area. There I saw a defination which helped me to create a very little sense about these. The defination is in the Wikipedia and I am just pasting that part down:
"An approach to defining what is meant by "area" is through axioms. "Area" can be defined as a function from a collection M of special kind of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties:
For all S in M, a(S) ≥ 0.
If S and T are in M then so are S ∪ T and S ∩ T, and also a(S∪T) = a(S) + a(T) − a(S∩T).
If S and T are in M with S ⊆ T then T − S is in M and a(T−S) = a(T) − a(S).
If a set S is in M and S is congruent to T then T is also in M and a(S) = a(T).
Every rectangle R is in M. If the rectangle has length h and breadth k then a(R) = hk.
Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. S ⊆ Q ⊆ T. If there is a unique number c such that a(S) ≤ c ≤ a(T) for all such step regions S and T, then a(Q) = c.
It can be proved that such an area function actually exists."
In the introduction it was written that area is a non negative real number which intuitively gives a measure of the region included within the boundary of a given geometrical figure like polygon for example( maybe some generalisation of these geometrical figures like polygon will be what they are calling as measursable sets as far as I have guessed).
But how can I get the above intuition of the area fucntion by seeing the defination? I mean I should get that intuition for the area function by analysing the properties of the function ( I know that I am unable to get that intuition it's totally my inability).But if I keep in mind that area function is a function which somehow quantifies the the included region within geometrical figures then some of those properties make sense like 'yes these properties should hold for such a kind of function.'
So here are my questions which I am listing down(and those large paragraphs I have written above are just background from where I came up with these questions):
1) Is it correct upto a little bit to think that mathematicians actually first think intuitively about something they want to define and then during defining that they jot down their obvious properties as axioms? I know that these are not at all so easy the way I am asking about them. So that's why I just want to know whether it will be totally wrong to think or correct upto a little bit.
2)Is there any defination for the length like the one I have pasted here for the area? I mean a defination using axioms. I tried to find such but I couldn't.
3) Lastly, if the length of the string is the real number that quantifies the distance between the two ends of the string when the string is kept in the form of a straight line , then when we change the geometric shape of the string to some another curve then why is the arc length of that curve equals the same real number which quantifies the straight line distance? To get some help I checked how is length of an arc defined. As it was given like first they approximate the curve by the joining some finite number of points taken on the curve. Then they add the length of those finite number of line segments formed. After that they increase the number of points and the approaching value of the sum of the lengths of the line segments formed as the number of points approaches infinity is taken as the length of the curve. But from this defination I am unable to conclude the answer of my question.
Thank you.
For the purpose of Euclidean geometry, area of a region $\Omega$ (for our purposes, an open subset) in a plane with rectangular coordinates is given by the integral $$ A(\Omega)=\int_{\Omega} dx dy. $$ It is a theorem (and not a very difficult one) that $A$ satisfies all the axioms you state with a small modification: The measure $A$ is not defined for arbitrary subsets (only for subsets, say, in the standard Borel sigma-algebra). By the way, measures are also required to be countably additive, not finitely additive: It is not enough to require that $$ A(U\cup V)= A(U) + A(V) - A(U\cap V). $$ There are "exotic" finitely additive measures on the plane (defined for the subsets in the Borel sigma-algebra). Surprisingly, once you go to dimension 3 and higher, you need only finite additivity to recover the standard notion of volume. But, again, this is all irrelevant for your purposes.
Yes, one can give a more axiomatic definition of length using the notion of 1-dimensional Hausdorff measure. One can make it even more axiomatic but I'd prefer not to do this.
The claim that "the length of the string is the real number that quantifies the straight line distance between the two ends of the string" is plain wrong, not sure where did you find it. For your purposes, it suffices to define length of a parameterized curve (with a piecewise-smooth parameterization $c: [a,b]\to {\mathbb R}^2$) as $$ \int_{a}^{b} ||c'(t)||dt. $$ Most calculus textbooks discuss motivation for this definition. For instance, if $c$ is piecewise-linear, equal to concatenation of straight-line segments ${\mathbf x}_i{\mathbf x}_{i+1}, i=0,...,n$, then the length of this curve equals $$ \sum_{i=0}^n ||{\mathbf x}_i - {\mathbf x}_{i+1}||, $$ i.e. the sum of lengths of the line segments forming the curve. In general, the length is defined via a certain limiting process using piecewise-linear curves which is then shown to be given by the above integral formula. (Usually, this is proven in a more advanced class.)