It's easy to imagine that eating 1 out of 3 slices of pie is "the same" in a very specific sense as eating 333333... out of 1000000... slices of the same pie (a

It's easy to imagine that eating 1 out of 3 slices of pie is "the same" in a very specific sense as eating 333333... out of 1000000... slices of the same pie (although that would be infinity out of infinity slices, which is meaningless). What slips people up is that ignoring everything but the total pie consumed (taking the limit) is embedded in the definition of real numbers. There's an analogous story with rationals: Suppose x1 = 1, y1 = 3, x2 = 2, and y2 = 6. If we plot them, (x1, y1) and (x2, y2) are clearly different points, but x1/y1 "equals" x2/y2 because they lie on the same line through the origin. We decide that we don't need to know about those individual points.