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This project illustrates a one-to-one correspondence between the entire real number line and the interval [0, 1]. By projecting the real number line above the blue dotted lines and drawing a half-ellipse connecting the endpoints, we create a visual representation of this concept. Each point on the real number line connects to the midpoint of the projection via red lines. Wherever these red lines intersect with the ellipse, we extend straight lines to meet the projection at [0, 1], demonstrating that the set of points in [0, 1] is equivalent in size to the infinite number of points across the real number line.
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-1 -.03 0 1 1.3 2 Project the real number line from [0,1] up above (the blue dotted lines), and draw a ½ ellipse connecting the end points of the projection. Pick any point on the real number line and draw a straight line from it to the midpoint on the projection (these are the red lines). Wherever the red lines hit the ellipse, draw a straight line up to his the [0,1] projection (these are the black dotted lines). Thus, every point on the real number line can be shown to have a one-to-one correspondence to a point on [0,1]. This proves that the infinity of points on [0,1] equals the infinity of points on the whole real number line, stretching infinitely in both directions
