Quote:
Originally Posted by DWarrior
Nope, not made up, it's a result of R and R^n having the same cardinalities.
For one example of a method that produces a unique single-coordinate identifier given two coordinates is this: imagine your X coordinate is .abcde... and your Y coordinate is .ABCDE... You can form a single real number that'll be unique to these coordinates: .aAbBcCdDeE... you thus have one coordinate that completely defines your position.
The pre-requisite for the above method is that each coordinate is some real number x so that 1>x>=0, but you can see that any real number can be uniquely identified by such coordinate using the same trick. For example, say X is a number ...EDCBA.abcde... and that it's either positive or negative, then your unique coordinate may be defined as .1aAbBcCdD... if X is positive, or .0aAbBcCdD... if X is negative or zero.
You can even extend it to all of space: X given by .abcde..., Y given by .ABCDE..., Z given by .klmno... your single coordinate is .aAkbBlcCm...
and so on. You can find other ways to uniquely represent stuff by one coordinate, and they'll all be fine as long as you use one method. As I said, it's not necessarily any more useful, but it's possible. It demonstrates that no dimension is "bigger" than another, they're all the same (as far as the number of points is concerned).
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We use this all the time now in air travel. (Thought it is used for a two dimensional point on a map, not a three dimensional position of an aircraft with altitude.)