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Topic: Analytic vs. Synthetic Geometry
Replies: 7   Last Post: Sep 5, 2013 10:55 PM

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 lite.on.beta@gmail.com Posts: 134 Registered: 2/21/06
Re: Analytic vs. Synthetic Geometry
Posted: Sep 5, 2013 9:30 AM

On Monday, September 2, 2013 4:35:04 AM UTC-4, William Elliot wrote:
> On Mon, 2 Sep 2013, lite.on.beta@gmail.com wrote:
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> > [Yet another geometry question that involves no geometry exercises]
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> >
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> > So after reading various sources on the internet, it seems like Analytic Geometry is an extension of Synthetic Geometry (or a more tool rich replacement). That is, all the theorems of Euclid are automatically true in the R^2 plane or space because the coordinate systems are simply just something "overlaid" (like grid marks) on the regular Euclidean plane or space.
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> >
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> > The advantage of overlaying these grid marks is that we can use analysis and algebra to solve harder geometry problems, correct? And correct me again, anything we prove (where the fact doesn't involve the coordinates explicitly) using this extra machinery is automatically true without coordinate systems.
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> > -----
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> > Are the Cartesian coordinates more "fundamental" then other coordinate
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> > systems?
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> They're simpler.
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> > When someone says R^n do we mean the space or the space+coordinate
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> > system?
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> The space.
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> > Sometimes I read "Cartesian space" for R^n, but what about calling
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> > R^n "polar space" (would that be slightly more silly or way more silly or
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> > not silly at all) ?
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> R^n is not a polar space. Polar coordinates can only be used with
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> the real plain R^2 and were you to use them, then R^2 is still
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> the real plain.
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> That's weird. Where did you see Cartesian space?
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> Often R^n is referred to as Euclidean space.

I read it somewhere, but today I see :

From the modern viewpoint, there is essentially only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the real coordinate space (Rn) of the same dimension. In dimension one this is the real line; in dimension two it is the Cartesian plane; and in higher dimensions it is a coordinate space with three or more real number coordinates.
{From http://en.wikipedia.org/wiki/Euclidean_space}

Also, R^n is the Cartesian product. So isn't it by default that the n numbers in R^n are cartesian coordinates?