History-of-Algebra

From the book: Abstract Algebra

Author: Pinter

Algebra comes from ‘al jebr’ in arabic, which loosely translates to ‘reunion’
Omar Khayyam, a famous poet and artist is also a mathematician best known for defining algebra as the science of solving equations

Methods of solving linear and quadratic equations were knwon even before the Greeks
But no one had yet found a way to solve cubic or quartic(4th degree) equations
This is achieved in 16th century, in the backdrop of Renaissance
However, methods were often kept a secret and mathematicians used them against each other at problem-solving competitions, which were a spectacle
Girolamo Cardan - born 1501 - went on to be a mathematician, physician and an astrolger
- but also a compulsive gambler
- He wrote Gamblers’ Book On the Game of Chance - the first book on systematic computations of probabilities
- He published Ars Magna, in which he presented the current algebraic knowledge. A lot of it had to be coaxed out of practitioners who wanted to keep it a secret
Niccolo Fontana “Tartaglia” found a way to solve cubic equations in 1535. He kept this a secret until he succumbed to Cardan
Cardan’s personal servant Ludovico Ferrari soon followed with a method of solving quartic equations
In 1824, Niels Abel from Norway showed that there is no formula to solve a system of equations in 5th degree or higher - this also concludes what’s referred to as the classical age

Matrix Algebra
Boolean Algebra
- Named after George Boole
As new algebras began occupying the attention of Mathematicians, it became obvious that algebra can no longer be defined as a science of solving equations.
Algebraic Structures
- We needed new set of general principles that could be applied to all known and potentially all possible algebras
- What is it that all algebras have in common? All of them consist of a set, and certain operations defined on the set
- Hence algebra was now defined as the study of algebraic structures, where an algebraic structure was a set with operations defined on it

An operation * is Commutative for a set A if for its elements a,b, a*b = b*a
An operation * is Associative for a set A if for its elements a,b,c, a*(b*c) = (a*b)*c
There exists an e in A such that for every a in A, a*e = e*a = a
For every a in A, there exists an a^-1 such that a*(a^-1) = e
Let’s say there is a second operation %. For % and *, a*(b%c) = (a*b) % (a*c). In this case, we say * is Distributive over %

The process of selecting what is relevant and disregarding everything else is the essence of R:Abstraction