![]() ![]() Thus, it was thought that this meant that the set of all two-dimensional points, known in set theory as $\mathbb R^2$, was larger than the set of all real numbers, $\mathbb R$, and specifically that $|\mathbb R^2| = |\mathbb R|^2$. It was originally thought that no $f(x)$ existed such that the set of its solutions for all real values of $x$ contained all points in $\mathbb R^2$ equivalently, that no bijection existed mapping all $\mathbb R$ to $\mathbb R^2$ existed. The function $f(x)$, in set theory, defines a "bijection" - a method of transformation between elements of two sets, for which every element in the "destination" set can be produced using one and only one element of the "source" set. ![]() As such, any one real number can be paired with any other in this manner, and only the points having coordinates $(x, f(x))$ for a deterministic, continuous $f(x)$ can be plotted along a line (this is what you learned in algebra). Now, two coordinates, making up a coordinate pair, are both real numbers. So, if the question really does refer to a "coordinate", that is, one half of a "coordinate pair" defining the location of a point, the answer is "always" every coordinate is a real number, and so it can be plotted on a one-dimensional number line. A "point" would have two "coordinates", each of them being real numbers. I will assume that by geometry you mean two-dimensional Euclidean geometry. ![]()
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