Congruence as explained by Apostol in Calculus Volume 1.

In summary: And do we have to do this also for other geometric objects such as triangles?* axiom 5 - choice of scale - Every rectangle is measurable . If a rectangle has lengths of edges h and k then its area is hk.
  • #1
glb_lub
23
0
1) I am having trouble understanding some parts of congruence as is explained by T Apostol in the book Calculus volume 1.
In the chapter on integration , while discussing area functions he explains congruence as follows :-
Congruence is used here in the same sense as in elementary Euclidean geometry. Two sets are said to be congruent if their points can be put in one-to-one correspondence in such a way that distances are preserved. I.e ,if 2 points p and q in one set correspond to p' and q' in the other, the distances from p to q will be equal to the distance between p' and q';this must be true for all choices of p and q.

Now I can appreciate that concepts such as 'area' , 'congruence' are being defined in terms of sets. But , the definition of congruence makes use of the notion of 'distance' . And while calculating distance between two points one has to make use of the Pythagoras theorem. And as far as I know any proof of Pythagoras theorem makes use of congruence of triangles in one way or the other. I don't know whether I am being fussy here but aren't we having circularity of reasoning here. Or are there proofs of Pythagoras theorem not depending on congruence of triangles (or for that matter notions of distance not making use of Pythagoras theorem).

2) Furthermore , he defines rectangles as those sets of points which are congruent to a set of the form
S = {(x,y)| 0≤x≤h , 0≤y≤k}
where h and k are lengths of the edges of the rectangle.
Later by an axiom , area of a rectangle is taken as hk.
My question is , is the area 'hk' according to the axiom* only for the set S ? If so,do we have to show that more general rectangles are 'congruent' to the set S , by making use of 'congruence' as defined in terms of sets. (by general rectangles I mean rectangles which are rotated and/or translated versions of the rectangle represented by set S).

And do we have to do this also for other geometric objects such as triangles?* axiom 5 - choice of scale - Every rectangle is measurable . If a rectangle has lengths of edges h and k then its area is hk.Overall , I am having some trouble in seeing the motivation behind defining congruence and rectangles as is done in the book.
 
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  • #2
glb_lub said:
1) I am having trouble understanding some parts of congruence as is explained by T Apostol in the book Calculus volume 1.
In the chapter on integration , while discussing area functions he explains congruence as follows :-


Now I can appreciate that concepts such as 'area' , 'congruence' are being defined in terms of sets. But , the definition of congruence makes use of the notion of 'distance' . And while calculating distance between two points one has to make use of the Pythagoras theorem. And as far as I know any proof of Pythagoras theorem makes use of congruence of triangles in one way or the other.


*** Of the many proofs of Pythagoras Theorem I've seen, there is not even one which uses congruence of triangles or anything close to that, and I can't understand how this could possibly help in such a theorem's proof...Can you give some example(s)?

Please do note that the notion of euclidean distance in the plane (or in any Euclidean space) doesn't require congruence at all but only Pythagoras Theorem.

DonAntonio ****



I don't know whether I am being fussy here but aren't we having circularity of reasoning here. Or are there proofs of Pythagoras theorem not depending on congruence of triangles (or for that matter notions of distance not making use of Pythagoras theorem).

2) Furthermore , he defines rectangles as those sets of points which are congruent to a set of the form
S = {(x,y)| 0≤x≤h , 0≤y≤k}
where h and k are lengths of the edges of the rectangle.
Later by an axiom , area of a rectangle is taken as hk.
My question is , is the area 'hk' according to the axiom* only for the set S ? If so,do we have to show that more general rectangles are 'congruent' to the set S , by making use of 'congruence' as defined in terms of sets. (by general rectangles I mean rectangles which are rotated and/or translated versions of the rectangle represented by set S).

And do we have to do this also for other geometric objects such as triangles?


* axiom 5 - choice of scale - Every rectangle is measurable . If a rectangle has lengths of edges h and k then its area is hk.


Overall , I am having some trouble in seeing the motivation behind defining congruence and rectangles as is done in the book.

...
 
  • #3
DonAntonio said:
*** Of the many proofs of Pythagoras Theorem I've seen, there is not even one which uses congruence of triangles or anything close to that, and I can't understand how this could possibly help in such a theorem's proof...Can you give some example(s)?

Please do note that the notion of euclidean distance in the plane (or in any Euclidean space) doesn't require congruence at all but only Pythagoras Theorem.

DonAntonio ****
Euclid's proof makes use of "congruent triangles"
http://en.wikipedia.org/wiki/Pythagorean_theorem#Euclid.27s_proof
Since AB is equal to FB and BD is equal to BC, triangle ABD must be congruent to triangle FBC.
the above quote is from 7) in the link to Euclid's proof.

Other proofs that I have seen use similar triangles and so on. All these notions are not defined by Apostol in his book.
One proof I saw makes use of differential calculus but that concept is taken up in chapters after integration.

Perhaps , the notion of Euclidean distance(or the Pythagorean theorem) is taken as an undefined concept in Apostol's book. And the concept of 'congruence' derived using that.
After all , we are free to choose the 'undefined' objects and their axioms as long as they are consistent. My query is whether Apostol has done so with respect to the Pythagorean theorem.
 
  • #4
You're worried far too late in the game. Apostol assumes Pythagoras is true, so you don't have to worry about consistency at that point (it's not a geometry textbook). Your question should be: how does congruence of triangles make sense in the proof of Pythagoras, independent of what Apostol says congruence means? (He's surely allowed to re-define the term given that it's a different subject of mathematics)

In Euclid, triangles are congruent if all their side and angle lengths are the same (it should be clear that Apostol's definition gives this condition for triangles, but note that this does not depend on Apostol's definition at all). This doesn't require any notion of Pythagoras's theorem
 
  • #5
Office_Shredder said:
You're worried far too late in the game. Apostol assumes Pythagoras is true, so you don't have to worry about consistency at that point (it's not a geometry textbook). (He's surely allowed to re-define the term given that it's a different subject of mathematics)

Thanks. This helped. And yes , the re-definition was satisfactory as it was in terms of sets.
But I was having trouble seeing the motivation behind it.I only understood that he is expressing congruence in terms of sets of real numbers.

Could you also explain to me the query I raised in point 2) in post 1. (I hope I am making sense :shy:)
2) Furthermore , he defines rectangles as those sets of points which are congruent to a set of the form
S = {(x,y)| 0≤x≤h , 0≤y≤k}
where h and k are lengths of the edges of the rectangle.
Later by an axiom , area of a rectangle is taken as hk.
My question is , is the area 'hk' according to the axiom* only for the set S ? If so,do we have to 'show' that more general rectangles are 'congruent' to the set S , by making use of 'congruence' as defined in terms of sets. (by general rectangles I mean rectangles which are rotated and/or translated versions of the rectangle represented by set S).

And do we have to do this also for other geometric objects such as triangles?
 

Related to Congruence as explained by Apostol in Calculus Volume 1.

1. What is congruence in mathematics?

Congruence is a term used in mathematics to describe the relationship between two figures that have the same shape and size. In other words, two figures are congruent if they have the same dimensions and angles, and can be transformed into each other through rigid motions such as translations, rotations, and reflections.

2. How is congruence related to calculus?

Congruence is an important concept in calculus, especially when dealing with geometric shapes and their measurements. In calculus, congruence is used to prove the properties of geometric figures and to solve problems involving measurement and transformation of shapes. It is also used to understand the relationships between derivatives and integrals of functions.

3. What is the role of congruence in Apostol's Calculus Volume 1?

In Apostol's Calculus Volume 1, congruence is used as the basis for understanding geometric concepts such as area, volume, and arc length. It is also used to prove theorems and solve problems related to derivatives and integrals. Apostol's approach to calculus emphasizes the importance of geometric intuition, and congruence is a key aspect of this approach.

4. Can congruence be applied to non-geometric problems in calculus?

Yes, congruence can also be applied to non-geometric problems in calculus. In some cases, congruence may be used as a tool to simplify a problem by transforming it into a more familiar shape. For example, in optimization problems, congruence can be used to transform a non-linear function into a quadratic function, which is easier to work with.

5. What are some real-life applications of congruence?

Congruence has many real-life applications in fields such as architecture, engineering, and computer graphics. It is used to ensure that structures are built with accurate measurements and proportions, and to design three-dimensional objects on a computer. Congruence is also used in map-making and navigation, as well as in the study of crystal structures and molecular geometry.

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