Key Mathematical Symbols Begin to Find General Use

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Key Mathematical Symbols Begin to Find General Use

Overview

Mathematical symbols seem impenetrable to most nonmathematicians. A seemingly arbitrary collection of shapes and letters in several alphabets, each with arcane meanings (some with multiple meanings), these symbols often seem designed to obscure meaning rather than lead to a deeper understanding. However, to a mathematician, these symbols are as easy to read as this paragraph is to the average student. In addition, by representing complex or sophisticated concepts or mathematical operations in a short, easily-recognizable form, mathematical symbols make it easier to concentrate on the actual mathematical arguments taking place rather than the words describing what the symbols represent. Many of today's commonly used mathematical symbols came into being in the eighteenth century, and a disproportionate number of them were introduced by the great mathematician, Leonhard Euler. By formalizing the language of mathematics, Euler helped set the stage for the great flowering of mathematical thought that began in the latter part of the eighteenth century and carried forward to this day.

Background

A symbol represents something. Typically, a symbol is a way to quickly represent an object, an idea, or a concept. The first mathematical symbols were the numbers, which were a simple way to represent how many of any object there might be. The idea of using written symbols to represent numbers of objects dates back at least a few tens of thousands of years, and may be even more ancient. In fact, it is possible that numbers were the first symbols used by humans.

It wasn't until relatively recently that other mathematical symbols came into use. A few thousand years ago, the ancient Egyptians wrote some of the first arithmetic problems, using their equivalent of +, -, and fractions. Gradually, with time, other simple mathematical symbols came into being, all designed to make the communication of calculations more consistent and understandable.

As the field of mathematics grew in complexity, so did the need for ever more sophisticated symbols, describing these new mathematical concepts and operations. The greatest flowering of mathematical symbolism followed the invention of the calculus by Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716). Each man developed calculus largely independently of the other, and each had his own preferred notation for writing problems and their solutions. It was, in a way, left to the great Swiss mathematician, Leonhard Euler (1707-1783) to standardize the notation and mathematical symbolism, giving rise to most of the symbols taught in college calculus classes to this day.

In a series of papers published during his long career in mathematics, Euler became the first to introduce use of the letter e to represent the base of the natural logarithms, the Greek letter π to represent the ratio of a circle's circumference to its diameter, and the first to use the symbol i to represent the square root of -1. In addition to these symbols, Euler was apparently the first to represent a mathematical function as f(x), which is instantly recognizable to anyone who has taken high school algebra. Other symbols and conventions pioneered by Euler were the abbreviations sin, cos, tan, sec, csc, cot, and so forth for the trigonometric functions and the Greek letter Σ to represent a summation of terms.

Each of these symbols carries as precise a meaning to those using it as the numeral 2 does to anyone writing it down. Each of these symbols, too, allows mathematicians to convey their thoughts to other mathematicians, physicists, engineers, and others in a completely unambiguous manner, so that their thoughts can be read, understood, and appreciated. By developing these symbols and giving them their current accepted meanings, Euler helped to advance the field of mathematics because, without clear communication, progress is much more difficult to achieve and convey to others.

Impact

It is difficult to understate the importance of mathematical symbolism on both the field of mathematics and on the other fields that use mathematics. The impact on mathematics will be discussed first, followed by the other effects of this development.

As mentioned above, symbols give us a way to quickly and accurately convey a complex idea to someone when we are communicating. If someone sees a Star of David, for example, they will think of the Jewish religion and everything that goes along with Judaism. Similarly, a cross symbolizes Christianity, an elephant symbolizes the American Republican Party, and a blue flag with a yellow cross symbolizes the nation of Sweden. All of these symbols, mathematical included, depend on recognition to succeed; a person unschooled in mathematics will see π simply as a Greek letter or, depending on their educational background, as simple a grouping of three lines. If a symbol is not recognized, then it is without meaning.

Similarly, a symbol must be unambiguous to succeed. If a symbol can have more than one meaning, then the reader has to be able to figure out the meaning from the context in which they see the symbol. For example, a symbol similar to the swastika was used as a symbol by Native Americans and in parts of Asia long before the rise of Nazi Germany. To most of the world, the swastika represents all of the evil that came from Adolf Hitler, in spite of the fact that, for several thousand years before Hitler was born, it was associated with peace. It is only through the consensus of those who experienced, read about, or in some other way know of Nazi Germany that the swastika has come to have the meaning it does for most of the world.

In the case of mathematical symbols, Euler's contribution was to take complex mathematical concepts and express them as simple, recognizable symbols. Because of his stature as one of the greatest mathematicians ever and because his symbols were easy to use, Euler's symbols became widely used.

To give a concrete example of the significance of this, consider the following equation: For this equation to have meaning, we must be able to understand what each symbol means. We know that, for example, f(x) means that this is a function where x is the variable. Thanks to Euler, we also know that the capital Greek letter sigma (Σ) means a summation and the zero and ten tell when to start and stop the summation. So, taken together, this short equation means this: "A variable, x is mathematically manipulated such that we add up (sum) all of the values of x from 0 to 10." This is a somewhat more cumbersome statement, however, than the elegant equation above.

Or, consider the statement "To find the surface area of a sphere, you multiply the number four by the ratio of a circle's circumference to its diameter, then by the square of a circle's radius, and divide this by three." Obviously, it's easier to write, instead, the equation: which is much simpler to express and to decipher.

The symbols introduced by Euler are among the most important and most commonly used symbols in mathematics, but they are also among the most important in many other fields that depend on mathematics. Physics, engineering, chemistry, and some aspects of geology, biology, social sciences, economics, and other disciplines that use mathematics all use Euler's notations. By making mathematics simpler to understand and communicate, these symbols make research in all of these fields easier to share with colleagues, even colleagues in other fields. This, in turn, has helped make possible many of the great advances in these fields that have taken place in recent centuries. Consider, for example, if only mathematicians used Euler's symbols and if every field of science had a unique way of expressing itself mathematically. This would make it nearly impossible, for example, for a geologist to learn from a physicist, and vice versa. The science of geologic dating uses concepts from both disciplines and is the foundation upon which our understanding of terrestrial and lunar evolution rests as well as providing the basis for our understanding of plate tectonics, evolution, and other important events. Would these concepts have gone undiscovered in the absence of consistent mathematical symbols? Of course not. But their discovery may well have been delayed by decades, waiting for the one person who could understand the mathematical symbolism of physics and the concepts of geology.

Many of the most fruitful human endeavors have relied on the ability of scientists in different specialties to understand the work of those in other fields. A team of geologists and physicists was the first to discover that an asteroid struck the earth, ending the dinosaur's reign. Geologists working with physicists, chemists, and biologists made a convincing set of arguments that the crust of the Earth was composed of plates that moved hither and yon through the ages. Scientists and engineers from several nations collaborated to build the international particle accelerator beneath the Alps that continues to generate important information about the structure of matter, itself. A team of chemists and physicists teased out the properties of radioactivity and radioactive elements, and mathematicians, chemists, and physicists developed the uranium, plutonium, and hydrogen bombs. While the benefits of some of these discoveries may be questioned, their impact on society is not. And the commonality of all is that scientists from disparate fields and, in many cases, from different nations were able to communicate complex, sophisticated ideas to one another using a single set of mathematical symbols and notation.

P. ANDREW KARAM