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.
A set you can do “arithmetic” with: there’s adding, subtracting, and multiplying (and they “play nice” together, i.e. distributive law).
It doesn’t require division because a ring is meant to capture the essence of what makes something a “number system” and, like in the integers, division doesn’t always work. The natural numbers are missing the subtraction requirement.
Mathematician specializing in abstract algebra here!
For context you should probably realize that Mathematicians are mostly doing things with arbitrary logical constructions rather than just numbers and geometric shapes.
We figured out that some results are much easier to think about when we look at the set of all things sharing a property rather than just the property alone. For example, people spent a long time knowing self evident results like 2+3= 3+2, and understanding that it doesn’t really matter which numbers you pick because it’s true for all of them, but leaving it as just that, a property of numbers.
But at some point people stopped and thought that maybe it’s not really something true about numbers, but rather something that is true about the way we’ve decided to represent numbers. And so people basically kept repeating the same thought experiment over and over: what is the smallest amount of assumptions that anyone would need to deduce how numbers work? In doing so they started to make discoveries, because if you have too few assumptions you can actually come up with really weird (and interesting!) patterns that operate under those same assumptions, but are clearly very different from how numbers work.
These patterns became so interesting and popular that we felt the need to package the assumptions together so it was easier to talk about the patterns. One of these packages is called a “ring”. There is no reasonable explanation for why exactly it’s called a ring, it’s not ring shaped or anything, it’s just a name. Another comment talked about the other packages we call groups and fields, and with a more detailed explanation. If you relax the assumptions in a field you can find what’s called a ring, and if you relax the assumptions in a ring you can find what’s called a group.
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