Why is it defined that way?

There are numerous conventions in mathematics that student continually question.

  • Why isn’t 1 a prime number?
  • Why is 0! defined to be 1?
  • Why is an empty sum 0 and an empty product 1?
  • Why can’t you just say 1/0 = ∞?
  • Etc.

There are good reasons for the existing conventions, and they usually boil down to this: On the whole, theorems are more simply stated with these conventions than with the alternatives. For example, if you defined 0! to be some other value, say 0, then there would be countless theorems that would have to be amended with language of the form “… except when n is zero, in which case …”

In short, the existing conventions simplify things more than they complicate them. But that doesn’t mean that everything is simpler under the standard conventions. The next post gives an example along these lines.

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6 thoughts on “Why is it defined that way?

  1. “Why isn’t 1 a prime number?” might be the most interesting question historically. Apparently, most mathematicians did consider 1 to be prime until recent centuries. G. H. Hardy was apparently the last major mathematician who still considered 1 to be prime. He apparently didn’t come around to the majority view until 1938.

  2. The first three bullet points are the same thing. The number 0! is 1 because it’s the empty product. And 1 isn’t prime because there’s a list of natural numbers that multiply to give 1, none of which are 1 (the empty list).

  3. There are good reasons for the existing conventions, and they usually boil down to this: On the whole, theorems are more simply stated with these conventions than with the alternatives. For example, if you defined 0! to be some other value, say 0, then there would be countless theorems that would have to be amended with language of the form “… except when n is zero, in which case …”

    This is an answer you can give, but it’s not a good answer. There are reasons that these values make the statement of theorems simpler while other values would turn theorems into multi-part special-case affairs, and those reasons are what the people asking the questions are asking you to tell them.

    A very simple example that other commenters have already pointed out is that 0! is equal to 1 specifically because 1 is the empty product – when you multiply zero numbers together, 1 is what you get. No other value of 0! would make any conceptual sense.

  4. “Why can’t you just say 1/0 = ∞?” makes another very obvious example. You can say that, and people frequently do. That one isn’t about the statement of theorems at all; it’s about the space you’re working with and your goals for what division should do.

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