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Good Math by Mark C. Chu-Carroll

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An Annoyingly Difficult Number

Even now, when we recognize zero as a number, it’s an annoyingly difficult one. It’s neither positive nor negative; it’s neither prime nor compound. If you include it in the set of real numbers, then the fundamental mathematical structures like groups that we use to define how numbers apply to things in the world won’t work. It’s not a unit. Units don’t work with it—for any other number, 2 inches and 2 yards mean different things—but that’s not true with zero. In algebra, zero breaks a fundamental property called closure: without 0, any arithmetic operation on numbers produces a result that is a number. With zero, that’s no longer true, because you can’t divide by zero. Division is closure for every possible number ...

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