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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 .

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- Thread starter mahmoud2011
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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

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HallsofIvy

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

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HallsofIvy

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

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

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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.exista 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.

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

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