What is Algebra?

 

"Ordinary" Algebra

There are two main facets to ordinary algebra. In one sense, algebra is the generalized arithmetic of symbols. In a beginning algebra course you learn how to add, subtract, multiply and divide symbols or groups of symbols. You also learn how to factor certain groups of symbols; the arithmetic analogy would be something like factoring 15 into 3 times 5. And you learn how to handle exponents. The main purpose of learning this kind of arithmetic is so that you can simplify groups of symbols. Algebraic simplification has specific meaning applied to specific types of symbols, but the purpose is usually to shorten an expression of symbols so that it is easier to work with.

In the other sense, algebra uses the arithmetic methods developed, as mentioned above, to solve problems in which at least one quantity is unknown, This is algebra's power: to solve problems too complicated or impossible for ordinary arithmetic.

Put another way, algebra allows us to express certain problems in symbolic and numeric form, then to use algebraic methods to solve these problems. Algebra, then, is its own language with precise and consistent rules of grammar and syntax that allow us to restate algebraic sentences and phrases into equivalent sentences and phrases which result in solutions to problems.

Symbols represent numbers, so they must obey the same properties as numbers. Thus the rules of algebra must be consistent with the arithmetic rules for the number system the symbols represent.

At SSCC students study the ordinary algebra of real and complex numbers, and are also exposed to graphing, which is actually a blend of algebra and geometry called analytical geometry. All the algebra at this level was established by the seventeenth century, except for complex numbers, which followed a century later.

 

Abstract Algebra

But there are other forms of algebra, some of them quite abstract looking. At the graduate level mathematicians are interested in these abstract algebras. The three main types of abstract algebra, in increasing levels of complexity, are groups, rings and fields. Everyday numbers we use, known as the set of real numbers, comprise the field of real numbers. As such, they obey about ten rules--called axioms--governing all algebraic fields.

All these algebras may have infinitely many numbers in them, such as our field of real numbers, or finitely many numbers. Mathematicians are interested in these abstract algebras for their various properties which allow them to shed light on or to solve problems in other areas of mathematics.

Algebraic groups tend to describe systems in which there is some sort of inherent symmetry. The smallest non-trivial group could be illustrated with a coin. Doing nothing to a coin represents one number, and turning the coin over to the other side represents the other number in the group. Turning the coin over five times, for instance, would give the same result as turning it over once, so each action--turning the coin over once or five times--is the same number in this simple, self-contained algebraic system with just two numbers.

Groups have advanced applications in chemistry and, more so, in physics. An interesting, albeit not very important, group is represented by the Rubik's Cube. Each combination of twists to arrive at a unique combination of colors from any given starting position represents a separate number in this very large group of numbers. The Rubik's Cube is its own finitely-sized algebraic system.

 

Historical Background

People have done algebra for thousands of years, although the subject can be said to date to around 825 A.D. with the publication of a book by a Persian, al-Khwarizmi. The title of his book is quite long, but includes the word "al-jabr," which means "to restore." (When you solve an equation you are, in some sense, restoring it to its simplest equivalent form.) Several centuries later al-Khwarizmi was immortalized when a translation of his book into Latin spelled his name as "algorithm." The word algorithm means a generic method for solving a specific type of problem.

In fact, mathematicians in ancient times developed algorithmic methods for solving certain kinds of problems which cropped up in daily life, and some of these could rightfully be said to be algebraic. The Greeks even managed to solve certain quadratic equations by use of geometric construction techniques using a straight-edge and a compass. Progress in algebra was severely limited for two reasons. The ancients did not have a convenient computational system of numbers. (Try long division with Roman numerals, for instance!) And they lacked the simple symbols of computation that we take for granted, such as +, -, x, ÷ and =. They could only describe and work algebraic problems verbally with a lot of words.

The early, post-Mohammed world saw a flowering of mathematical and scientific thought in the Arabic world, then known as Persia. These early Islamics sought to advance the learning and ideals of the ancient Greeks. They developed a system of numbers which allows for decimals and easy computation and, of course, established a language for algebra. Some books will say that algebra dates to around 1000 A. D. instead of the aforementioned 825 A. D., as this is about the time that the language of algebra had sufficiently developed and caught on as a highly useful set of mathematical tools for solving problems unsolved through arithmetic means.

Algebra gradually migrated to the Western world--Europe--over the next several centuries. For instance, the numbering system we take for granted is a blend of the Hindu and Arabic decimal systems. Christian societies in that era were reluctant to consider anything developed by non-Christian societies, but pragmatism won out in the end.

Analytical geometry, which combines algebra and geometry, was developed by a westerner, the French mathematician and philosopher, Rene Descartes, in the early seventeenth century. Descartes realized that he could describe any position on a plane with a pair of numbers called coordinates, and quickly discovered he could also describe geometrical shapes using corresponding algebra equations. We tend to call this "graphing" in algebra courses. Calculus followed later in that century as a major extension of Descartes' analytical geometry. Arguably, Descartes' simple idea of what we call the "x - y plane" is the foundation for many subsequent technological advances. Without analytical geometry we might still be mired in the early Industrial Age. This was the missing link which allowed us to begin to bring the full power of mathematics to bear on understanding our physical world.

(Descartes also had a profound influence on science through his philosophy of dualism, which is that the mind operates outside the constraints of physical reality. The value of scientific experimentation as a way to verify scientific theories has some foundation in Descartes' philosophy, which implies that the mind, through reason alone, cannot reliably predict or explain the physical processes operating outside it.)

People learning algebra, and many of those who teach it, tend to take for granted many of the concepts which only came about through years of intellectual struggle. Negative numbers, for instance, were unacceptable to mathematicians for a very long time because you cannot visualize a negative quantity. After many decades, however, mathematicians began to see how these "artificial" numbers were quite useful for solving real-life problems.

The language of algebra did not fully evolve until around a hundred years ago. A textbook author (from Cincinnati) in 1870 lamented the fact that there was no universally agreed upon rule for calculating algebraic expressions. Sometime afterwards the familiar order of operations came to be the standard way in which we determine the order of calculation from a complicated expression.

 

Learning Algebra

In some sense the algebra one learns before learning calculus--"ordinary algebra"--amounts to learning a large collection of "rules" and techniques. These "rules" are not arbitrary, but based upon sound definitions, axioms and theorems. Algebra theorems are based in logic and thus are entirely consistent. Because of the sheer number of "rules" in algebra it is not enough to try to remember them; you really must practice them extensively in order to learn how to apply them and to reinforce that learning. It's not unlike practicing a musical instrument. You could memorize the fingering for each note and you could learn how to decipher written music, but without sufficient practice you will not be able to make good music.

Going to class and paying attention through taking notes and asking questions is vitally important to learning algebra. Seeing and hearing how an expert uses algebra is quite valuable. But you can learn even more by practicing on your own, so short-changing yourself on both practice and going to class are the two most damaging ways to ensure that you will do poorly.

Algebra may be difficult to appreciate when you are first learning it. It may appear abstract and irrelevant to the real world. And it is true that algebra alone is somewhat limited in its applications to realistic everyday problems, because most problems requiring more than arithmetic also require more than algebra. Mathematics gets much more interesting once you begin to learn trigonometry, calculus, statistics or finite math topics, but algebra is always part of the foundation for any mathematics you would learn afterwards. But the main reason students are required to learn algebra in college is because it makes people think analytically, and you're not going to become a better thinker without practice!

 

Algebra at SSCC

Southern State Community College offers four traditional lower-division and college preparatory algebra courses. For more information click on the following links.

College Preparatory Level:

Math 117 Elementary Algebra

Math 118 Intermediate Algebra

College Level:

Math 141 College Algebra

Math 250 Linear Algebra


To contact the author by e-mail click on this link: Jon Davidson