What behind the idea of representing real numbers as points ?

AI Thread Summary
The discussion centers on the representation of real numbers as points on a line, questioning the foundational assumptions behind this concept. It highlights that the one-to-one correspondence between real numbers and points arises from the completeness of the real number system, particularly the convergence of Cauchy sequences. Participants note that representing real numbers on a line allows for a dual relationship between algebra and geometry, facilitating easier visualization and calculation. However, some argue that this correspondence is not inherently true but rather an axiom accepted without proof. The conversation emphasizes the importance of understanding these foundational concepts in mathematics.
mahmoud2011
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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
 
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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.
 
HallsofIvy said:
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
 
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.
 
And the same concept can be extended easily to complex numbers and points on a plane.
 
so my concept is ok , I thought I have something missing .

Thanks
 
HallsofIvy said:
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
 

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