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What is the modern way of defining these concepts? (Real coordinate space, Cartesian coordinate system, Euclidean space).

As far as I can tell:

  • Real coordinate space of dimension n, $\mathbb R^n$, is simply the set of n-tuples of real numbers (constructively built using Cauchy sequences, or otherwise)
  • To get to Euclidean space, we need to endow $\mathbb R^n$ with maps $a, d: \{\mathbb R^n, \mathbb R^n\} \to \mathbb R$ to specify angles and distances between elements. If we make $\mathbb R^n$ into a vector space (trivially), we can use the standard inner product for this. However, if we just need to satisfy the original Euclidian postulates, are we able to do this in any other way?
  • No idea how Cartesian coordinates fit in. It seems in the definition (from Wikipedia), we already have a magical notion of perpendicular axis and distance
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  • $\begingroup$ @JnxF Thanks for the tip! Updated! $\endgroup$ – Zach Sep 1 '16 at 23:49
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  • Euclidean space can have many meaning on what kind of space you consider, because you may have to define more or less things for the space to work. Usually we're speaking of the Euclidean vector space, over which you must define a sound operation $+$, $-$ and $\cdot$ that must obey the axioms of a vector space. Usually the easiest way is to use $\mathbb R^n$ and do the addition and scaling component wise for the tuple, but you could use any set with well-defined operations $+$, $-$ and $\cdot$ that is isomorphic to that. For instance, $\mathbb C$ with its standard definition of $+$, $-$ and $\cdot$ is a 2-dimentional Euclidean space.

  • You could also speak of the Euclidean affine space, which is made of a vector space and a set of point with no definite center. For example the real world (actual physical considerations apart) with the vector space $\mathbb R^3$ and an operation that convert the direction given by two points to a vector and another that translate a point along a vector. When informal and using the same underlying set for the vector space and the set of points, mathematicians may not distinguish points from vectors clearly.

  • A real vector space is synonymous to an Euclidean vector space, with the added convention that the underlying set is $\mathbb R^n$ for some $n$. A real affine space is an affine space build unpon a real vector space.

  • Cartesian coordinates, in a affine space, is a coordinate system over points using a orthogonal system. A point coordinate in the system is determined by the distance it has to each axis. Other systems include polar coordinates, if you have defined a dot product over your vector space, that uses angle and the distance to an "origin" point.

I've never heard of real coordinates. I guess it's synonym to cartesian coordinates or coordinates of a real (affine) space.

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