Basics: Algebra
While I was writing the vectors post, when I commented about how math geeks always build algebras around things, I realized that I hadn’t yet written a basics post explaining what we mean by algebra. And since it isn’t really what most people think it is, it’s definitely worth taking the time to look at.
Algebra is the mathematical study of a particular kind of structure: a structure created by taking a set of (usually numeric) values, and combining it with some operations operate on values of the set.
One of the simplest examples of a kind of algebra is a simple group. A group a pair (G,*) where G is a set of values, and “*” an an operation on values in G such that:
- Closure: For all a,b ∈ G, a*b∈G.
- Associativity: For all a,b,c ∈ G, a*(b*c)=(a*b)*c
- Identity: There is a value i∈G such that for all a∈G a*i=i*a=i.
- Existence of Inverses: For every value a∈G, there is an inverse a^{-1}∈G, and for which a*a^{-1} = a^{-1}*a = i.
(In a moment of brain-death, I originally typed “Commutativity” instead of “Closure” in property 1 of groups. Stupid mistake on my part, pointed out by two commenters within minutes of posting! Thanks to both of you!
For an example of a group, think of (Z,+) – that is, the integers and the addition operation. Adding any two integers together always results in an integer. 0 is an identity element for integers with addition – for any a, a+0=0+a=a. Addition is also commutative and associative over the integers. And finally, for any integer i, there’s another integer -i such that i+-i=0.
It’s a very simple structure – a set of values with one operation and four simple properties. But that’s enough to capture the entire concept of symmetry. Exactly how it does that is well beyond the scope of a basics post, but some day I’ll resurrect my group theory posts from blogger, and post them here.
There are numerous kinds of algebra. A partial list includes groups, rings, and fields; linear algebra (an algebra which looks at the properties of vectors and matrices); algebraic topology (an algebra which looks at the properties of topological spaces), and too many others to mention.
The point of algebras is that they capture essential properties of structure in sets in terms of the ways that those sets can be manipulated using closed operations. Groups captures the concept of symmetry, and what it means for a set to possess a kind of symmetry. Fields capture the structure of real numbers (and other similar sets), and
what kinds of properties they have. Linear algebra explores the world of matrices, and what kinds of properties matrices have, and how you can manipulate them.
Algebras do this – and they do it in a symbolic way. Notice that in the definition of integers as a group, the only integer I needed to name explicitly was 0: for any other value, I could just use a symbol to represent any value, and talk about what it meant to do something to that value.
What we’re taught in high school as algebra is one very limited case of this. If you take the set of real numbers, along with addition and multiplication, and look at them as a field, then there’s a set of things that you can do symbolically with any statement about the real numbers. That set of things is what’s generally taught as algebra. But as I hope you’ve grasped from this brief post, algebra is much more than that one limited case.