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When we talk about Euclidean space or n-Euclidean space, using notations such as $\Bbb R^n$ or $\Bbb E^n$ which are basically n-tuples of real numbers and if distance function is to be added, then we can certainly call it a euclidean space (metric space). But that's just a sort of Cartesian space if we are using n-tuples for our purpose. Is there any way to represent it in polar coordinates and if not than why do we exempt from calling n-Euclidean space as Cartesian space when it's already evident that points here are represented as tuples of numbers (ordered pairs or ordered triple in context of 2-D and 3-D resp.).

Manny46
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  • You can certainly choose to represent things in ways other than the usual. It is common to represent it how we normally do because addition properties are so easy to manage. It can be quite frustrating trying to add two numbers which are represented in polar coordinates. As for extending to higher dimensions, the three-dimensional version would be spherical coordinates and similar extensions can be made for even higher dimensions as hyper-spherical coordinate systems. – JMoravitz Mar 28 '18 at 04:24
  • As for the difference between Euclidean space and Cartesian space, see this related question. – JMoravitz Mar 28 '18 at 04:27
  • That's for the part of questions I've asked. But I want to clarify a doubt here. As we use n-tuples of real numbers while trying to represent euclidean space, so in a way there is a representation of points via ordered pair or triples or tuples as we go to higher dimension, then why do we differentiate Cartesian space and euclidean space like for example this question https://math.stackexchange.com/questions/112076/what-is-the-difference-between-euclidean-and-cartesian-spaces – Manny46 Mar 28 '18 at 04:28
  • My understanding of the difference is that when referring to $\Bbb R^n$ as a cartesian-space that we are acknowledging that there is some canonical "origin" and a canonical way to represent points. (We may choose to invent additional ways to represent points as we see fit, but at the very least this one canonical way to represent them is guaranteed). On the other hand, Euclidean space does not need to have a canonical predetermined representation for points at all, though we may choose to invent one after the fact (or indeed, multiple) and this may even mimic the one from cartesian. – JMoravitz Mar 28 '18 at 04:37
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    Imagine that I have a uniform solid-white ball that I can roll around. I stop the ball and point to a spot on it. You might ask me "where are you pointing to on the ball" and I can only really respond "I'm pointing to here." I start rolling the ball again and lose track of where I previously pointed. If you ask me to point to the exact same spot again, I may have a lot of trouble to do that. Compare that to if the ball had a grid-like pattern on it with a certain labeling scheme. I could then describe where I pointed the first time and could easily point to the same spot again if asked. – JMoravitz Mar 28 '18 at 04:41
  • Got the point. So when referring to euclidean space it's rather abstract but due to Descartes innovation of coordinate system we use terms like $\Bbb R^n$ to refer to euclidean space which is just a space as Euclid describes. And here we have things like real and complex numbers to supplement euclidean space which happens to form Cartesian space. Am I correct on that? – Manny46 Mar 28 '18 at 06:01

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