A New Realm of Numbers

Overview

Great strides were made in the 1800s toward moving back to a rigorous, logical base for mathematics. Essential to this effort was progress in number theory. Joseph Liouville (1809-1882) expanded the understanding of real numbers when he proved the existence of transcendental numbers. Later, Charles Hermite (1822-1901) demonstrated that e, the natural logarithmic constant, was a transcendental number. In 1882, Ferdinand Lindemann (1852-1939) answered "no" to the classic challenge, "Can the circle be squared?" when he proved that pi (π), the ratio of the circumference of any circle to its diameter, was also a transcendental number. Julius Wilhelm Richard Dedekind (1831-1916) completed the view of real numbers by explaining them in terms of irrational and irrational numbers. The establishment and characterization of real numbers extended the rigor of mathematics, improving the quality of proofs. It affirmed the concept of limits and allowed the rigorous development of analysis, which is essential to solving many difficult problems of engineering and science.

Background

During the 1800s, mathematicians took on the challenge of making their discipline more rigorous. While the Greeks carefully defined terms and worked out proofs based on logic and consistency, their successors took a more practical approach. For over 1,000 years, most mathematicians relied primarily on intuition and geometry to expand their understanding of the quantitative world and to solve practical problems in science and engineering. Galileo (1564-1642) used geometry to formulate and express his understanding of the motion of bodies. René Descartes (1596-1650) developed graphs to synthesize algebra and geometry, creating analytical geometry. Even calculus failed the test of rigor. In 1734, the philosopher George Berkeley (1685-1753) criticized the use of limits by Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716) because they did not have an adequate logical basis. Mathematics had had a long run of successes based on visual and intuitive approaches to mathematics, but, by the eighteenth century, the rigorous base provided by the Greeks was insufficient to the task supporting detailed analysis of data in maturing sciences. A return to rigor was needed if progress was to continue.

A New Realm of Numbers

Overview

Background

In 1821, Augustin Cauchy (1789-1857) started the move toward making mathematics more rigorous with his work Cours d'analyse de l'École Royale Polytechnique. But his work, and the work of his colleagues, presupposed a number system which did not yet exist. Calculus had demonstrated that there were precise results that were not solvable by rational numbers (integers and fractions). It relied on the notion of limits, but no one had created a number system in which convergent sequences had limits. What mathematicians needed to develop was a deep understanding of real numbers, one which divided these numbers into algebraic and transcendental, and, alternatively, into rational and irrational. This would provide what was needed to support rigorous proofs in analysis.

Paris was a hotbed of scientific and mathematical thought in the nineteenth century. In addition to Cauchy, mathematicians like André Ampère (1775-1836) and Pierre Laplace (1749-1827) and physicists like Jean Baptiste Biot (1774-1862) and Dominique Arago (1786-1853) collaborated, and often competed, for recognition, funding, and public attention. Joseph Liouville (1809-1882) rose to become one of the leading mathematicians of this period. Liouville was much influenced by Cauchy and was a student of Ampère. He moved to the center of activity in mathematics as a scholar, teacher, and publisher.

When Liouville read the correspondence of Christian Goldbach (1690-1764) and Daniel Bernoulli (1700-1782) from the previous century, he became intrigued by their speculation on existence of transcendental numbers. A transcendental number is a real number, that is, it is the limit of a convergent series of rationals. But unlike algebraic numbers, which are real numbers that can serve as solutions for polynomials with rational (integer or fractional) coefficients, transcendental numbers can never serve as solutions for these equations. Liouville decided he would prove that the natural logarithmic constant e was a transcendental number.

This was an interesting target. The natural logarithmic constant e is critical to calculations of compound interest and to the description of natural phenomena like radioactive decay and population growth. It is also a number that intrigues mathematicians because the derivative of e^x is e^x, and is part of e^iπ + 1 = 0, an equation that links the five most important numbers in mathematics—e, i, π, 1, and 0. Johann Lambert, the man who proved π was irrational, had speculated that both π and e were transcendental numbers.

Though Liouville never succeeded in proving e was transcendent, he did succeed in 1844 in becoming the first person to prove that transcendental numbers existed at all. He did this by showing that in some instances where the number of algebraic solutions must necessarily be finite, you can still find cases where the number of solutions is infinite. These numbers, which clearly are both non-algebraic and real, are transcendental. Liouville identified an infinite class of such numbers. He also took the first steps toward proving that e was transcendental, but he was not able to complete the job. This work was to be accomplished by his student, Charles Hermite (1822-1901).

Hermite was one of the leaders in adding rigor to mathematics in the nineteenth century. He provided the first solution to the general equation in the fifth degree, and he was a key contributor to the development of the theory of algebraic forms. Taking up where Liouville left off, he set out to prove that e was transcendental. Essentially, while Liouville had shown that e could not solve some polynomial equations, Hermite showed in 1873 that e could not solve any polynomial equations. It was a transcendental number.

Just a year earlier, another important view of real numbers had been established: rational and irrational. Pythagoras (c. 580-500 b.c.) had discovered the first irrational number (the square root of two) over two millennia before. Legend has it that his followers executed the man who let this secret out, and irrationals had troubled mathematicians thereafter. But in 1872, the German mathematician Julius Dedekind (1831-1916) looked at irrationals and rationals as cuts in a line. This concept allowed manipulation and use of both irrationals and rationals in a consistent way. The irrationals and rationals were thereby brought together to comprise all the real numbers.

In addition to being transcendental, e was known to be irrational. Was the same true for π? The answer would prove to be of both mathematical and cultural importance. Arguably, π is the number that has most fascinated both amateur and professional mathematicians throughout history. Famous for the expression π r², π is the ratio of the circumference of any circle to its diameter. It is also central to one of the classic challenges of classical Greece (included in Euclid's Elements), popularly known as "squaring the circle." Using only an unmarked ruler and a compass, is it possible to construct a square with the same area as a circle? In 414 b.c., Aristophanes mentioned squaring the circle in his play, The Birds. Since that time, squaring the circle has been used as an expression for doing the impossible. But great minds were not discouraged. Hippocrates (c. 460-377 b.c.), Archimedes (c. 287-212 b.c.), and even Leonardo da Vinci (1452-1519) had taken on the challenge. For centuries, squaring the circle had provided an intellectual stimulus and led mathematicians to a deeper understanding of their discipline. In the seventeenth century, François Viète (1540-1603) used a polygon with 393,216 sides to approximate squaring the circle and determined π to 10 decimal places, the most accurate measure of π to that date. Proving that π was transcendental would prove that squaring the circle could not be done, since then, no line could be π (or its multiple) units in length.

By 1873, π was known correctly to the 527th digit, and no pattern was evident in the decimals. This is what would be expected from a transcendental number, but could it be proven? Ferdinand Lindemann (1852-1939) decided to find out. Using the methods of Hermite, he succeeded in doing so in 1882. Lindemann had shown Aristophanes was right. For all the effort that had been put into squaring the circle, it could not be done. The answer to Euclid's challenge was "no."

Impact

From a popular standpoint, the biggest impact of nineteenth century was the end to the pursuit of squaring the circle for most reasonable people. Like the building of perpetual motion machines, squaring the circle is one of those challenges that delights amateurs and annoys professionals. Even before Lindemann's proof of the impossibility of the challenge, both the French Academy (a century earlier, in 1775) and the Royal Society in London had declared that they would no longer review "solutions." Although there will probably always be a cadre of people who will continue to try, Lindemann forever shut the door on serious attempts to square the circle. (Approximations of the solution, however, are still fair game. The brilliant mathematician Srinivasa Ramanujan (1887-1920) provided a solution in 1914 that would correlate to less than an inch error in each side of a square constructed from a circle of 8,000 miles [12,872 km] in diameter.)

The work of Liouville, Hermite, Dedekind, and Lindemann helped make mathematics a more rigorous discipline. As Gödel was to prove, every mathematical system contains unresolvable paradoxes. One expression of this is in number theory. Natural numbers become incomplete in the face of subtraction—negative numbers must be postulated. Integers (both positive and negative) are insufficient in the face of division. Rational numbers must be postulated, which include fractions and take everything into account except division by zero.

Real numbers, including transcendental numbers, moved the boundary forward, allowing for rigor and consistency in analysis (and, of course, moving mathematics forward to new paradoxes and inconsistencies). Working with transcendental numbers introduced new approaches to approximation and made limits an accepted concept in mathematics. This opened the door for the solution of many important problems in science and engineering and brought the mathematical discipline of analysis into its own. However, the practical utility of real numbers has been limited. Using real numbers to calculate answers to problems in applied mathematics is very difficult. For instance, while real numbers help provide a theoretical basis for computers, they cannot be used by a computer. Exact real numbers are infinite, so they can't be stored or manipulated. Computer scientists are forced to use floating point numbers to approximate answers, something which can lead to errors (many of which are not easily anticipated). This lack of precision is part of our heritage in the mismatch between the number system and the tools we inherited from the nineteenth century. It is an area of active investigation.

Perhaps the most far-reaching effect of nineteenth-century work in number theory was the forceful shift of science and mathematics from the visual to the textual. The rigor of exact equations came to be a prerequisite for any theory or proof to gain acceptance by professionals in mathematics, science, and engineering. It is only as the visual capability of computers has risen to a high level that confidence in visual demonstrations has begun to reemerge.

PETER J. ANDREWS