# What to do about mathematical concepts with different definitions?

You must be wondering, why is this a so important subject? Well, take the example where a challenge asks you to take a "natural number" as input. What's a natural number? A non-negative integer, or a positive integer? The same would happen with Fibonacci numbers, do we include the 0 or not?

Another example would be for a challenge asking you to handle a case where you have to do 0⁰. Does that equal 1 or 0?

In both cases, and generally in a case where a mathematical concept has many accepted definitions and the question doesn't specify which one to use, one would either

• wonder what definition to follow, or
• perceive it as "too obvious" and use the definition they see themselves fit.

In the first case, one can simply leave a comment asking the original poster what definition to use. In the second case, they will use a particular definition to make an answer, and then there are many possibilities, so that the definition used can be:

• free for the answerer to choose,
• the one the original poster intended, or
• different from the intended definition.

I emphasize on the last case, since there can be the one answer with the wrong definition, or a swarm of answers with wrong definitions, the latter being more difficult to face, so more of a problem.

To avoid such cases of ambiguity, what should be done about it?

P.S. This discussion actually arose from a CMC!

• Another (less ambiguous) example could be for people who include 1 in the primes – Stephen Jun 27 '17 at 14:14
• @StephenS I thought of that but...they do? I don't think so, the only definition I could find is "a natural number greater than 1 that only has divisiors 1 and itself". – Erik the Outgolfer Jun 27 '17 at 14:15
• I don't think they are anyone who's studied it, but it's natural to assume, before studying prime theory or whatever, that 1 is prime since it doesn't have any factors (it's only not prime by definition and because that's how mathematicians use primes) – Stephen Jun 27 '17 at 14:17
• @StephenS I think that's another issue though. – Erik the Outgolfer Jun 27 '17 at 14:18
• AFAIK, it's just as accepted by mathematicians that 0^0=1 and 1 is not prime – Stephen Jun 27 '17 at 14:21
• @StephenS Mathematically, 0^0 is as undefined as 0/0. – Martin Ender Jun 27 '17 at 14:21
• @MartinEnder Well, the latter could've been just ∞, hadn't been defined as undefined... – Erik the Outgolfer Jun 27 '17 at 14:24
• @EriktheOutgolfer 0/0 = ∞, 0/-0 = -∞, so it's undefined – Stephen Jun 27 '17 at 14:24
• @EriktheOutgolfer Or it could have been defined as 0 (because it's 0/x) or as 1 (because it's x/x), which is why it's undefined (same for 0^0 which could be 0 or 1, depending on whether it's 0^x or x^0). – Martin Ender Jun 27 '17 at 14:25
• @MartinEnder I think x/x where x≠0 is defined as 1 since x=x*1...for 0, it can even be 0*∞...wait, division is borked, since it could be 0*x=0 for any x...pretty much that's the definition of undefined. – Erik the Outgolfer Jun 27 '17 at 14:27
• @MartinEnder, both of those are contextually defined. In practice you never see 0/0: you see f(x)/g(x) where lim x->a: f(x) = g(x) = 0, and then you evaluate f(a)/g(a) using l'Hôpital's rule; similarly you never see 0^0 but f(x)^g(x). Ask a combinatorialist what 0^0 evaluates to and they'll say 1 because in any combinatorial context in which it occurs that is the appropriate limit. – Peter Taylor Jun 27 '17 at 15:38