# Introduction to Algebra

The branch of mathematics referred to as *algebra* can be split into a number of different specialised application areas, but in these pages we are only really interested in the basic principles. Algebra is essentially an *abstraction* of arithmetic (essentially, this means we are not just dealing purely with numbers any more) but follows, for the most part, the same rules of operation. The main difference between algebra and arithmetic is that the value of one or more of the terms in an algebraic expression is often unknown. We give these terms symbolic names that act as placeholders for the unknown quantities. The presence of these placeholders enables us to manipulate an algebraic expression (called an *equation* ) in order to isolate and find the value of any unknown terms. As you will see, algebraic equations can take various forms, depending on the kind of problem we are trying to solve. In fact, algebra has evolved over hundreds of years as a response to the need for repeatedly solving the same type of problem with different sets of known and unknown values. For that reason, a relatively small number of algebraic formulae can be used to solve a large number of mathematical problems.

Although a history lesson is beyond the scope of these pages it is interesting to note that algebra has its origins some four thousand years in the past. There is evidence, for example, that both the Egyptian and Babylonian civilisations developed algebraic methods of performing calculations. Babylonian algebra in particular was sufficiently advanced that it enabled the solution of problems that would today take the form of linear and quadratic equations (don't worry if these terms mean nothing to you - we will be looking at what they mean soon enough). The algebra used by the Babylonians was *rhetorical* in its format, which means that the equations were written using full sentences rather than the very concise (albeit far more abstract) notation found in modern algebra. By the beginning of the third century BCE a number of Greek, Chinese and Indian mathematicians were also using algebraic methods, although equations were still expressed rhetorically and this early algebra relied heavily on the use of geometric constructions.

Perhaps the earliest use of symbols to represent unknown values is found in the writings of the Greek mathematician *Diophantus*, who is believed to have lived in the third century CE. Diophantus wrote a series of texts entitled *Arithmetica* that deal with solving algebraic equations. The treatment of algebra as a branch of mathematics in its own right, however, did not really occur until the ninth century CE, when the Persian mathematician *Muhammad ibn Mūsā al-Khwārizmī* wrote a treatise entitled *Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala*. This translates roughly as *The Compendious Book on Calculation by Completion and Balancing*. In fact, Algebra gets its name from the term "al-ğabr" which is thought to mean something like "restoration", "completion" or "setter of broken bones". Despite the fact that al-Khwārizmī is believed to have been influenced by a number of ancient texts, including those of the Greeks, his algebra was fully rhetorical.

Nevertheless, al-Khwārizmī firmly established algebra as a mathematical discipline totally separate from either geometry or arithmetic, and developed a more generic approach to the solution of linear and quadratic equations. This meant that a single equation in algebraic form could be applied to a whole range of problems of a particular type. Over the next few centuries other Arab mathematicians would build on the work done by al-Khwārizmī, and by the latter half of the fifteenth century CE had begun to use letters instead of numbers, and abbreviated Arabic words to represent mathematical operations. By the twelfth century CE, however, many of the works of al-Khwārizmī and other Arab mathematicians had already been translated into Latin.

The thirteenth century CE saw the focus of developments in the field of mathematics shift away from the Middle East and into Europe. One of the most significant developments was the introduction of symbols to represent unknown values and algebraic operations. The French philosopher and mathematician *René Descartes* published the third volume of his major mathematical work *La Géometrie* in 1637, in which he describes an algebra that is remarkably similar to that used today. Towards the end of the seventeenth century CE, German philosopher and mathematician *Gottfried Leibniz*, coined the term *function* to describe the algebraic relationship between the *x* and *y* coordinates of any point on a geometrically derived curve such as a tangent or chord. The idea was later generalised to include any algebraic relationship in which the value of an *output*, *y*, is functionally dependent on the value of an associated *input*, *x*.