No matter what you say about mathematics, it really all comes down to numbers. Here we discuss two special kinds of numbers: algebraic and transcendental.
When the famous mathematician Gauss said, “Mathematics is the queen of science, and numbers the queen of mathematics,” he was not kidding. No matter how abstract up the ladder of mathematics we go, numbers ultimately are at the root of all. We simply cannot escape them. The world of mathematics is filled with all kinds of interesting and glamorous numbers. Here we look at two special kinds: algebraic and transcendental.
If you consider the quadratic equation x^2 – 2 = 0, the solutions are x = +/-(sqrt2). The square root of 2, or radical 2, (rad(2)), as we shall call it, are the two zeros of the quadratic equation just given. That is, if we substitute x in the given equation by either -sqrt(2) or +sqrt(2), we get a true statement (the equation becomes 0; that is why these values are called zeros). Because the coefficient of x in the given quadratic equation is an integer, namely 1, we say that rad(2) is an algebraic number. A formal definition of algebraic numbers are those numbers which are solutions to polynomial equations with integral coefficients.
Examples of common algebraic numbers are the rationals (the common fractions), and the roots of quadratic polynomials, that is equations of the form ax^2 + bx + c = 0. The latter algebraic numbers are called quadratic surds, because they derive from the quadratic formula. The constructible numbers, those that can be constructed with a ruler and compass, such as sqrt(2), sqrt(3), etc., are also algebraic. Less commonly known algebraic numbers are the golden ratio, or the positive root of the quadratic equation x^2 – x – 1; and the so called Gaussian integers, or those complex numbers a + bi, where both a and b are integers.
Unlike algebraic numbers, transcendental numbers are those which are not roots of polynomials with integral coefficients. The most famous of these numbers are pi, the ratio of the circumference of a circle to its diameter, and e, which is approximately equal to 2.718, and is the base of the natural logarithm.
Showing that a number is transcendental is quite the task. Proving that e for example is not algebraic requires some advanced mathematics and analysis. Once this result is established, however, the transcendence of pi follows quite readily. A very curious fact about transcendental numbers is that finding them is very difficult, yet they are not denumerable. What this means is that although both the set of algebraic numbers and transcendental numbers are infinite, the infinity displayed by the set of transcendental numbers is greater than that displayed by the countable algebraic numbers. If the previous statement has left you agasp, then welcome to the fascinating world of mathematics!
If there is any doubt to the various uses of this breed of numbers, just consider pi: anything involving circles, wheels, cylinders, cones, spheres, or the trigonometric functions relies almost exclusively on this famous transcendental gem. Or take the base of the natural logarithm e: this curiosity occurs in problems of growth or decay, such as compound interest growth or radioactive disintegration; this gem is also the key ingredient in the most famous curve in statistics: the Gaussian or Normal Distribution, also known as the “bell curve.” This entity is used to describe everything from IQ’s to income to personality traits.
So the next time you hear the word number, remember that mathematics is full of many special ones, two of which we have examined here. Maybe if we got to contemplating some of the mysteries of something as majestic as algebraic and transcendental numbers, we just might get lost enough to forget about some of our more mundane problems.
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