Algebra

Algebra is one of the main branches of pure mathematics, along with geometry, analysis, topology, combinatorics, and number theory.  It is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Algebra is much broader than elementary (school) algebra and studies what happens when different rules of operations are used and when operations are devised for things other than numbers. Addition and multiplication can be generalized and their precise definitions lead to structures such as groups, rings and fields, studied in the area of mathematics called abstract algebra.  See this article from the University of Arkansas and the Wikipedia article for further reading.

Algebra has been developed over a period of many centuries but one of the seminal works describing this way of approaching and analysing problems was a book written in about 830 AD by an Arabic scholar named Al-Khwarizmi.  His books on mathematics were so influential that his name even became a mathematical term, algorithm, which describes a method or process for solving a problem! His second book, al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqabala, was a treatise describing algebra.  The term algebra comes from the expression al-jabr in the title of his book.  Keith Devlin, in his marvellous book, The Man of Numbers, translates the full title as The Abridged Book on Calculation by Restoration and Confrontation. Restoration (al-jabr) and confrontation (al-muqabala) refer to two processes used in balancing equations:

• Restoration (or ‘completion’) refers to the removing of all negative terms by transposing them to the other side of the equation to make them positive.  For example, ‘restoring’ the equation 2x = 11 – 4x would result in 6x = 11.
• Confrontation (or ‘balancing’) refers to the process of eliminating identical quantities from both sides of an equation.  For example, 8x + 1 = 3x + 9 would become 5x + 1 = 9 after removing 3x from both sides of the equation, and then 5x = 8 after removing 1 from both sides of the equation.

These two processes form the foundation for solving any algebraic equation and are often taught to students in the first year or two of High School.

Al-Khwarizmi did not use modern symbols, however (as they had not yet been invented).  He used words to describe the concepts … and unknown quantities (what we would call x and y today), were known by his contemporaries as shay (thing) and jidhr (root).  The term jidhr also means ‘the origin’ or ‘the root of a tree’ and may be the origin of our modern expression ‘the root of an equation.’  Note that the ‘root’ of a number comes from a different source … the Latin word radix.

Linear Algebra

Simultaneous Equations

Quadratic Equations

Polynomials

Matrices