algebra life 4

I am reading a lot of maths lately, trying to get as much as possible before my birthday.

Algebra and analysis.

Algebra and geometry.

Algebra and number theory.

What is algebra? It comes from arabic, meaning the reunion of broken parts. “In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics.” Wikipedia.

So far in reading algebra, I have been reading about groups, different types of groups, subgroups. Basically a lot of definitions and theorems and proofs regarding groups which follow from the group axioms.

Fairly abstract.

Groups, along with rings and fields (which I have not read yet) are descriptions of the simplest algebraic structures which allow equations like polynomial equations to be solved.

My life is broken, can algebra help to reunion the broken parts of my life?

algebra life 3

Whenever I read some sort of text, like mathematics, the beginning is usually enticing and interesting enough to keep my attention.

And then it gets tough, and some things like abstract algebra gets very abstract quickly, the learning curve is fairly steep for someone with no guidance.

I then try to skim through it and see if I can find any beacon of light where I can latch onto or motivate me to persevere through the text.

For modern/abstract algebra, this will have to be Lagrange’s theorem which states that for any finite group G, the order of every subgroup H of G should divide the order of G.

I understand what it means, but I don’t think I truly appreciate it yet. Which means I will have to backtrack and see if I can get a better intuition through means of maybe drawing some pictures or construct a more complete picture (story) of group theory.

algebra life 2

Associativity

A lot of algebraic structures, like the addition and multiplication of real numbers or the rational numbers have the associative property, which is often taken for granted.

It basically says that given a binary operation ∗ , is associative if for all three elements x, y, and z in a given set S, (x ∗ y)  ∗ z = x ∗ (y ∗ z). Then the products can be unambigiously written as x  ∗ y  ∗ z.

From this, the general associative law follows in that for any number of elements, their products can be unambiguiously written without any parentheses.

This can be proven by mathematical induction, but it is not the easiest one to formulate.

As a non-math major, it took me some time to absorb this fact.

Non-associativity

Without the associative property, as the number of elements increase, the number of possible products increase quickly (look at the Catalan numbers for a formula).

Operations that don’t have associativity include subtraction, division and exponention.

For example, we can see that (2/3)/5 does not equal 2/(3/5). The first equates to 2/15 and the second equates to 10/3.