I’ve heard things like the integers form a ‘ring’ while natural numbers do not, but I don’t quite understand what they *are*.
In: Mathematics
Ignoring the details of exactly which properties we require each to have, you can think of groups, rings and fields like this:
– A group has one operation and its inverse, which we’ll think of as addition and subtraction.
– A ring adds a second operation which interacts “nicely” with the first, but the second operation need not have an inverse. We’ll think of the second operation as multiplication.
– A field also has an inverse for the second operation (except that you can’t divide by zero).
So the natural numbers aren’t a group under addition (you can’t guarantee that subtracting two natural numbers gives another natural number), and therefore they’re not a ring either. Moving up to the integers gives you a ring, but not a field (dividing two integers may not give you another integer). The rational numbers form a field, since you can always divide two rational numbers and get another.
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