4 ms·

Can't you divide any number by itself (multiply by its inverse) to get 1? Or does that not count?

codethief 8y ago
With the exception of the number 1, the inverse of any natural number with respect to multiplication is not a natural number.
- davnicwil 8y ago
  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 :-)

With the exception of the number 1, the inverse of any natural number with respect to multiplication is not a natural number.

davnicwil 8y ago
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 :-)