eli5 What relationship does the Cartesian product R² and R³ have with the Cartesian plane and three-dimensional space?
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R is the set of real numbers. It can also be represented by a number line.
If you pick a real number — call it x — you can plot it on the number line.
R² is 2 sets of real numbers. It’s the same set of real numbers, but twice.
You can pick a real number from each — call them x and y — and plot each one on its own corresponding number line.
OR you could rotate one of the number lines 90°, make it the Y axis to a Cartesian plane, and plot both x and y at once as a single point (x, y).
If you have N full sets of real numbers, you have a coordinate system in N-dimensional space.
You can think of them as the same thing.
R^2 is the set of all pairs of real numbers. The plane
Is a way to visualize the set geometrically as a space, where a pair of real numbers are coordinates.
Same with R^3. It’s the set of all triples of real numbers, which can be visualized as coordinates in 3d space
Abstractly, a Cartesian plane/space describes a class of objects with particular characteristics (Euclidean plane/space with a Cartesian coordinate system). R^2 /R^3 is a particular realization, a particular instance of that class.
The distinction don’t really matter though, since any Cartesian plane/space can be identified with R^2 /R^3 in an obvious manner.
There are some small benefits in distinguishing Euclidean plane/space (in the abstract sense) from R^2 /R^3 , but there are no benefits to distinguish Cartesian plane/space from R^2 /R^3 , because every Cartesian plane/space already come automatically with an identification.
Well you can have collections of 2 numbers as well as 3. You can represent the collection of 1 numbers on a line. Now for R² it can be represented on a plane. Pick every R² has a position on a plain. Now same with R³ but add another axis. Every element of R³ can be represented as a point in 3D space.
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