Zero had no place within the Pythagorean framework. The equivalence of numbers and shapes made the ancient Greeks the masters of geometry, yet it had a serious drawback. It precluded anyone from treating zero as a number. What shape, after all, could zero be? It is easy to visualize a square with width two and height two, but what is a square with width zero and height zero? It’s hard to imagine something with no width and no height—with no substance at all—being a square. This meant that multiplication by zero didn’t make any sense either. Multiplying two numbers was equivalent to taking an
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