3 ms·

By inverse I was meaning 1/x, i.e. x * 1/x = 1

• ColinWright 8y ago
  But we're working in the integers, so 1/2, for example, doesn't exist.
  • davnicwil 8y ago
    I'm just saying to get to 1 through multiplication you can always multiply any number by its own inverse (which is another way of saying divide any number by itself) I.e. X * 1/X = X/X = 1, always. I guess, technically, it's a division needed to get the inverse, so it's perhaps it doesn't count :-) Edit: Ah, yeah, now with help from the reply below I see my mistake - you're saying the inverse itself cannot exist since we're only dealing with natural numbers, that makes sense. I guess that's pretty much the reason that only multiplication in allowed in the OP's original question, allowing division lets us get all kinds of fractional numbers which wouldn't be allowed.
    • jfarmer 8y ago
      It's important to separate the structure from the objects. There's 2-the-natural-number, 2-the-integer, 2-the-rational, and 2-the-real. In each context, it's the "same" 2, but the structure we put around it is different. The structure is what makes it interesting. 2-the-rational is not prime. There are no primes when every non-zero element has a multiplicative inverse. An element with a multiplicative inverse is called a unit and for the same reason we exclude 1 from being prime we exclude units from being primes in other structures. Every non-zero real number is a unit. Every non-zero rational is a unit. Only -1 and 1 are units in the integers. Only 1 is a unit in the naturals.
      • davnicwil 8y ago
        Thanks for this detailed response! You made me realise the flaw in my reasoning :-)