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Number line geometry7/27/2023 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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