Investigations into the Irrationality and Transcendence of Various Specific Numbers

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Investigations into the Irrationality and Transcendence of Various Specific Numbers

Overview

Number theory has often been thought of as one of the bastions of "pure" mathematics, unsullied by application to real-world problems and in which mathematics is pursued for the sheer beauty of the concepts involved. Starting in the latter part of the nineteenth century, however, some actual applications of number theory began to emerge. In the first half of the twentieth century, this trend continued. In addition, mathematicians began to prove some important points about specific numbers, such as π, e (the base for natural logarithms), the square root of two, and others. This work led to some very interesting and important advances in the way these numbers are viewed. In addition, since many of these irrational and transcendental numbers are widely used in physics, engineering, and other practical disciplines, this work has also helped to shed some light on interesting phenomena outside the rarefied realm of number theory.

Background

The area of a circle is calculated by multiplying the circle's diameter by the number π. The area of a square is obtained by simply multiplying the length of one side by itself (squaring the length of a side). From the time of the ancient Greeks, professional and amateur mathematicians struggled to find some way, using only an unmarked straight edge and a compass (a device for making circles or for marking distances), to create a square having exactly the same area as a circle. For two millennia, this quest was unsuccessful. In 1882, Ferdinand Lindemann (1852-1939), a German mathematician, was finally able to prove conclusively that the quest was impossible because π is a transcendental number, a class of numbers that cannot be calculated by algebraic means. While it may not be immediately obvious, using a straight edge and compass are algebraic methods of solving a problem, although the algebra is disguised as numbers describing the length of the straight edge or the radius of the compass opening. Since π is transcendental (can't be calculated exactly using algebra) and irrational (the decimal neither repeats nor terminates), it is mathematically impossible to "square the circle," in spite of all those who felt they proved otherwise for over 2,000 years.

Irrational numbers are numbers that cannot be expressed as a ratio of two whole numbers. The square root of two, for example, is an irrational number. Another way to look at it is that the decimals neither end nor repeat (⅓, for example, is written as 0.33333.... with the number "3" repeating endlessly while ¼ terminates as the decimal 0.25). Transcendental numbers are numbers that cannot form the roots, or solutions, of algebraic equations in which the exponents are real integers. For example, if we look for the solution to the equation x² + x + 1 = 0 (that is, for what values of x is this equation true?), the answer will be two algebraic numbers, because the numbers are solutions to an equation with whole-number exponents. On the other hand, π is not the solution to such an equation, so it is a transcendental number. This was shown conclusively by Lindemann. By so doing, he also proved that the ancient problem of squaring the circle was mathematically impossible.

All transcendental numbers are irrational, but not all irrational numbers are transcendental. In Mathematics: The Science of Patterns, Keith Devlin notes that the square root of two, while irrational, is not transcendental because it is the solution to the algebraic equation x² - 2 = 0. Joseph Liouville (1809-1882), the famous French mathematician, was the first to demonstrate the existence of transcendental numbers when, in 1844, he showed the limits of how precisely algebraic numbers could be approximated by rational numbers (i.e., by numbers that are the product of two integers and, as such, either terminate or repeat). By going through successively more accurate approximations, Liouville was able to demonstrate that algebraic numbers could not provide exact solutions to some mathematical problems. This proof, in turn, required that transcendental numbers must exist because these problems (such as determining the value of π, which is the ratio of a circle's diameter to its circumference) were known to have solutions.

Many efforts have been made to develop very precise approximations for the value of π. One of the more accurate was also one of the earliest; the ancient Greeks realized that a circle could be approximated by a polygon with an ever-increasing number of sides. Since the length of each side could be calculated, this gave an ever-closer approximation of the value of π. Other approximations include expressing π as a fraction (22/7 is the most common). However, regardless of the degree of sophistication, these are only approximations. As of 1999, the value of π had been calculated to over one billion decimal places with no end in sight. In fact, mathematically, an end to the sequence is not possible.

In an address at the Second International Congress for Mathematics, held in Paris in 1900, German mathematician David Hilbert (1862-1943) posed a series of problems to the assembled mathematicians, problems whose solutions would help to advance the field of mathematics. A total of 23 such problems were eventually presented, although not all at this congress. Hilbert's seventh problem dealt with the mathematics of transcendental numbers and has yet to be solved in its entirety. However, in 1934, Russian mathematician Aleksander Gelfond (1906-1968) proved a specific case to be true, which may someday lead to a general solution.

Hilbert's seventh problem asks: given an algebraic number and an irrational number ß, will the number represented by ß always be a transcendental number? Gelfond was able to show was that, if ß is an irrational number (but not necessarily a transcendental number) and if α is not equal to either 0 or 1, this number will always be transcendental, but the more general case has yet to be proved.

Impact

One possible reaction to all of this is, of course, "So what?" To some extent, this response is not inappropriate. However, irrational and transcendental numbers are extremely useful in a number of scientific disciplines, are used extensively in engineering (including electrical and electronic engineering), and it may behoove us to better understand them.

One minor problem, that of squaring the circle, could only be resolved by understanding transcendental numbers and their significance. While the solution to this problem will not feed the hungry or lead to world peace, it does help to show some of the limits inherent in one branch of mathematics—and that we will never know π with perfect precision.

Many computer graphics programs, including those used for computer-assisted design, will calculate the area of a circle by adding up the number of pixels the circle covers on the screen and then assigning a unit area to each pixel. However, this is an algebraic method of calculating area and, since π is not an algebraic number, all such areas are inherently inaccurate. While we may be able to refine this algorithm (and have) to the point at which the error is negligible, there will always be error in this process. In fact, it is impossible to reach an exact numerical solution for any calculation using irrational or transcendental number because the numbers do not terminate or repeat.

Other commonly used numbers that fall into one or both of these categories are the square root of two (irrational), e (transcendental and irrational), and many other square roots, cube roots, and higher roots (irrational, but not transcendental). Roots in particular, whether square, cube, or higher, may be irrational, but are always algebraic (not transcendental). This is easy to see because an equation such as the one given above (x² - 2 = 0) is the same form as any equation representing one of these roots. Since, by their nature, such equations are algebraic, their solutions, the roots, cannot be transcendental.

This, in turn, implies that other problems cannot be solved using only straight edge and compass. For example, it is not possible to create a cube with exactly twice the volume of a unit cube using only these tools because the cube root of 2 requires solving a cubic equation (an equation in which one variable is raised to the third power). This is beyond the abilities of such simple tools. However, such a problem can be solved if a marked straight edge is permitted. Similarly, dividing an angle into three equal angles can be solved with a marked straight edge, but not with the simpler tools. Both of these problems have obvious implications for computer-aided design as well. In effect, this work helps to show the limits of what we can achieve with simple tools.

Gelfond's work was important to mathematics, as has been virtually all of the work done on most of Hilbert's problems. In fact, that was the whole point of posing these problems—to try to spark mathematicians to new and greater heights at the start of a new century. In the case of Gelfond's work, by developing his partial solution to Hilbert's seventh problem, he was able to advance the field of number theory enormously. His techniques and methods were accepted by his colleagues and found use in other aspects of number theory, while his construction of new classes of transcendental numbers helped to advance that particular sub-specialty of mathematics as well.

P. ANDREW KARAM