What behind the idea of representing real numbers as points ?

[mahmoud2011]

I was wondering about the idea of representing real numbers as points on line , What is the basis of this assumptions , and as well the same question for Cartesian coordinates system ?
All books I have read , express the idea of Cartesian Coordinates in an elementary way like spivak's , Apostol , ... , many others that I have read this part in .

Thanks

 

[HallsofIvy]

What "assumption" are you talking about? That there exist a one to one correspondence between real numbers and points on a line? That comes from the "completeness" of the real number system. one expression of which is that every Cauchy sequence converges.

 

[mahmoud2011]

What "assumption" are you talking about? That there exist a one to one correspondence between real numbers and points on a line? That comes from the "completeness" of the real number system. one expression of which is that every Cauchy sequence converges.

yes , that is what I knew , but why we choose line exactly

 

[HallsofIvy]

Why? So that we can "algebra-ize" geometry! And, "geometrize" algebra. It is often easier to visualize geometry than algebra, easier to get precise values for algebra than geometry. To be able to convert from one to the other helps both ways.

 

[martix]

And the same concept can be extended easily to complex numbers and points on a plane.

 

[mahmoud2011]

so my concept is ok , I thought I have something missing .

Thanks

 

[SteveL27]

What "assumption" are you talking about? That there exist a one to one correspondence between real numbers and points on a line? That comes from the "completeness" of the real number system. one expression of which is that every Cauchy sequence converges.

I don't believe that's correct. The correspondence between the real numbers as defined in analysis, on the one hand, and the geometrical line on the other, is not necessarily true. It's an axiom; that is, it's assumed without proof.

http://en.wikipedia.org/wiki/Cantor–Dedekind_axiom

We use this visualization so often that we accept it as necessarily true; but it's not.

There's been some discussion of this on PF.

https://www.physicsforums.com/showthread.php?t=244274