Math – Complicating the Simple, and Simplifying the Complex

As promised earlier in Upcoming Posts, I will know explain what I mean by “complicating the simply, and simplifying the complex”.  This article is intended for non-math types as a brief glimpse into the world of more advanced math.

I am currently pursuing an undergraduate degree in Pure Mathematics and I am currently taking an Introduction to Abstract Algebra course.  I would consider this among my first true math classes – and by true math I mean advanced math, rather than arithmetic.  Really, almost all math you learn in high school, and even college outside of upper division math, is really arithmetic or some variation.  You learn some rules, you manipulate some numbers, you get an answer.  Sometimes this can be quite difficult, no doubt about it – but it is still really just a lot of involved arithmetic.  Then you get to real math, and everything you thought you knew is turned on its head.

So 14+ years of math to get you back to – algebra?  And what is the first thing you learn in this advanced math textbook?  “Divisibility in the Integers“.  Isn’t that something you learn in elementary school?  Seriously, third year math course in college and you are learning how to divide integers?  Like 6 / 2 = 3?  Maybe advanced math isn’t that difficult after all.  But then you get to the definition of dividing –

A (nonzero) integer d is said to divide integer a (denoted d|a) if there exists and integer b such that a = db.  If d divides a, then d is referred to as a divisor of a or a factor of a, and a is referred to as a multiple of d.

That is what I call complicating the simple.  A whole paragraph to define division.  Believe it or not after defining division we learn about multiplication and addition.  But there is a method to this madness.

As it turns out what most of us think of as numbers is really only a small fraction of what can be considered as number systems.  Most of us probably have some idea of sets of numbers like the integers (“whole” numbers like 1, 3, -97, 102), rational numbers (number that can be expressed as fractions of integers such as 2/3, 101/270, -9/5) and even prime numbers (numbers that can only be divided by themselves and 1 such as 7, 13, 67).  When we think of adding together any of these numbers, we think of it the same way.  We know what the properties are for addition.  For example what order you add the numbers in does not matter -  i.e. 5 + 3  = 3 + 5.  If you had to prove that for each pair of numbers, it would be impossible.  What if you could prove it just once for any set of numbers?  That would really simplify things wouldn’t it?  That is what abstract algebra is all about – proving things once to use for a lot of specific cases.

Abstract algebra looks at a number of different things that seem completely unrelated – for example, even integers, polynomials with rational coefficients, and 2 x 2 matrices, and finds what is common between them.  Then using only these common properties, you derive certain properties which you can then apply to any number system that has the same common defining structure.  Prove a property once for an abstract set of rules and you can then use that for real specific cases.  That is what I call simplifying the complex.

That is where the term abstract comes from – if you are familiar with object oriented programming at all it works much in the same way.  You define a class of objects – say all sets where there is something called “addition” where there is some element “0″ that when added to any “number” gives you the same “number”.  This does not require a concrete (opposite of abstract) set such as integers – just some imaginary non-existent abstraction that follows those rules.  You can the instantiate an object of that type, as in programming, by defining something like our usual number system.

Things get really interesting from there – but the math is can be too hard to type out without a specialized math package and the background material necessary probably would not be interesting to many of you.

technorati tags:math, abstract, algebra

Blogged with Flock