I Like Abstract Stuff; Why Should I Care?
Abstract math makes some people flee in horror, but it's actually cool if you give it a chance. We're not saying you need to invite it to your birthday party or anything, but maybe you don't need to lie awake shaking at night because you're fearful it's in your closet. That's a baseless fear, since abstract math is notoriously averse to confined spaces. Hm. Now we're not sure what's more abstract: abstract math, or this paragraph?
When we put off division until the next unit, we talked about how integers are similar to polynomials. Integers form what's called a ring. You must take this ring into Mordor, and drop it into the top of Mount Doom. Wait, that's something else.
Here's what a ring means as far as integers are concerned:
- Addition is associative, meaning it doesn't matter how you group things. (2 + 3) + 4 = 2 (3 + 4)
- Addition is commutative, meaning you can order the numbers any which way. 2 + 3 = 3 + 2
- 0 is a magic number. For every integer m, adding 0 to m leaves m unchanged.
- Every integer m has an additive inverse -m, such that m + (-m) = 0.
- The distributive law holds.
- Multiplication, therefore, is associative.
How about that? We're back at associative, where we started. That process was rather ring-like, wouldn't you say?
Polynomials in one variable with integer coefficients also form a ring. We can do a lot of similar things with integers and polynomials: factor, look for primes, look at all the multiples of a certain number or polynomial, whine and complain about them to our algebra teacher, and so on.
There are many other kinds of rings. The things in a ring are called elements (1 is an element of the ring of integers; x + 3 is an element of the ring of polynomials, etc.).
Rings and their elements are studied in abstract algebra, which is like regular algebra, only weirder. Tough to imagine, we know. Some rings don't have a multiplicative identity (that is, no number "1"), and some rings have non-commutative multiplication (ab and ba might be different). The other rings only want to be loved. Is that so wrong?