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Numbers and Their Application - Lesson 14

Transcendental Meditations

Lesson Overview

Transcendental numbers have a long history, dating back to the ancient Greeks, even though they were not named or truly recognized until much later. As mentioned earlier, the ancient Pythagorean school discovered the existence of irrational numbers, with [the square root of two] being the prototypical example as the diagonal of a unit square. They then regarded it as a numberless magnitude—distinct from an arithmetic number—a concept which remained an essential element of Greek mathematics. The Pythagoreans had a pledge of secrecy so vowed to keep this discovery secret. According to legend, a man named Hippasus resolved to reveal to the world the irrational numbers. His associates, alarmed by his plans to "spill the beans" (a diet staple), conspired to throw him overboard the ship they were sailing. Soon other irrational numbers were found: the square root of every prime number, then the square root of most composite numbers. Irrational numbers, or incommensurables were well studied by the time Euclid wrote his Elements. However, it was not until 1872 when Richard Dedekind (1831-1916) published his Continuity and Irrational Numbers that a satisfactory theory developing such numbers was given, one devoid of geometric considerations. His Dedekind Cut was an essential part of that development and goes beyond what we can cover here.

The Story of [pi]

The concept of [pi] was invented to simplify calculations involving circles. The Rhind Papyrus, an Egyptian text from 1650 B.C., contains a statement relating as equals, the areas of a circle and a square whose side is 8/9 the circle's diameter. This value for [pi] of 256/81 [approximately equal to] 3.16049... is a much better value than the one recorded about 700 years later and given biblically in I Kings 7:23. "And he made a molten sea, ten cubits from one brim to the other...and a line of thirty cubits did compass it round about." These both recognize the need to relate the radius or diameter or a circle to its circumference or area. Euler was the one to attached the symbol [pi] to the concept.
[pi] is in fact defined as the ratio of a circle's circumference (C) to its diameter (d):     [pi] = C/d.
This gives the formulae: C = [pi]d = 2[pi]r, where r is the radius.     The area formula is similar: A = [pi]r2.
Archimedes first proposed a method of obtaining the value of [pi] to any desired accuracy by calculating the perimeter of inscribed and circumscribed polygons. By increasing (usually by doubling) the number of sides, the accuracy is increased—the true value of [pi] is squeezed between these two values. Using his crude numerical representation, Archimedes was able, by using polygons of 96 sides (bisecting the sides of a hexagon 4 times), to determine: 3 10/71 < [pi] < 3 10/70 or 3.140845... < [pi] < 3.142857... or [pi] [approximately equal to] 3.1418. Over the centuries this value was highly refined until hundreds of decimal places were know before the invention of computers and now billions of digits are known. An interesting challenge has been memorizing these random digits and the current record is about 83,000 digits, requiring many hours to recite. (The author had 750 digits well memorized and almost had one thousand at age 16 when he thought the record was only a couple thousand. He has since forgotten most all but the initial 50 which he memorized at age 11.)
[pi] = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 ...
Historically, the value [pi] [approximately equal to] 22/7 was used and is within 0.04% of the true value. Such a rational approximation was useful before calculators were invented and older geometry books have many problems which were done very easily using this value. The curious value [pi] [approximately equal to] 355/113 can easily be remembered because each of the first three odd number is repeated once and is even closer to the true value. [pi]2 [approximately equal to] 9.8696... is surprisingly close to 10, our preferred base. Extending the above definition of [pi] results in its most common usage: angle measurement. The radius of a circle seems like a useful unit to measure arc lengths or angles. Note how the circumference of a unit circle (one with r = 1) is 2[pi] [approximately equal to] 6.28318.... An arc the length of one radius is known as a radian and there are 2[pi] radians in one revolution or full circle (360°). Thus [pi] radians are 180° and 1 radian is 57.2957795...° or 57°17'44.806...". The conversion of radians to degrees is done by multiplying the radians by 180°/ [pi]. To convert degrees to radians, multiply the degrees by [pi]/180°. The circle below is partitioned into standard angle measure in both degrees and radians. It is important to know these. Mathematicians like to think of a radian as the proper serving size of pie, just ever so slightly less than 1/6.

The Story of e

Another important number to mathematics has a much shorter history than [pi]. It begins in Scotland, with the birth of John Napier (1550–1617). John Napier had wide interest including religion, fertilizers, water levels in coal pits, etc. Stories about him and drunk pigeons, black roosters, etc. abound. However, he is best know as the inventor of logarithms, which means ratio number. Although his usage was slightly different, the modern definition is:
logba = c   if and only if   bc = a, b>0, b1.
We thus see that exponentiation (exp) is an inverse operation of logarithm (log). Inverse operations have already figured prominently as in subtraction is the inverse operation of addition and division is the inverse operation of multiplication. Another important one is square root as the inverse operation of squaring. Inverse functions can have important restrictions which differ from the original function! Logs can be defined to any positive base (except 1), but two bases have become most prevalent: b = 10 (for common logs), and b = e (for natural logs). Both appear on most calculators. The base is often omitted and high school and chemistry students can usually assume log x = log10x. However, in college math and physics, log x = logex. ln x is fairly commonly used for natural logs (and now rarely looks like 1n). Napier's base was b=.9999999 = 1 - 10-7, which may be only slightly more understandable when you realize that decimal fractions were not yet widely used—Napier actually being the one to invent the decimal point! In making this choice, Napier came within a hair's breadth of discovering the limit of (1 - 1/n)n as n tends to infinity, which is merely the reciprocal of (1 + 1/n)n as n tends to infinity.
This latter value is e = 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 ...
Logarithms were quickly adopted by scientists all over the world because they simplified calculations by turning multiplication and division into table look-ups, addition and subtraction, and then another table look-up to find the antilog. One such important scientist was Johannes Kepler. Henry Briggs (1561–1631) was so impressed that he resolved to meet their inventor in person: "where almost one quarter of an hour was spent, each beholding other with admiration, before one word was spoke. At last Briggs said: 'My lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help in astronomy, viz. the logarithms; but, my lord, being by you found out, I wonder nobody found it out before, when now known it is so easy.'" (viz. is an abbreviation for videlicet, Latin for namely.) Briggs proposed two modifications which resulted in our base 10 or common logarithms. Briggs published tables accurate to 14 decimal places for all integers 1 to 20,000 and from 90,000 to 100,000 in 1624 in Arithmetica logarithmica with the gap filled in by someone else by 1628. This work remained the basis for all subsequent log tables up until 1924 when a 20 decimal place table was begun to celebrate 300 years of logarithms. About 1620, the slide rule was also invented which is laid out on a logarithmic scale and thus by adding and subtracting distances, multiplication and division are performed. For more on logs go to lesson 17. Like we saw in scientific notation, the decimal part of a logarithm is often called the mantissa. The integer portion is called the characteristic.

Geometric Constructions

The transcendental story really began with the restrictions the ancient Greeks (Plato) put on their Geometric Constructions. The only tools allowed were an unmarked straight-edge and a pair of compasses. (Most sources specify a compass, but some constructions require two.) In Geometry we still differentiate between constructing, drawing, and sketching. In a drawing, rulers and protractors are allowed, whereas a sketch may be a free-hand representation. The Greeks quickly mastered many constructions, such as for the perpendicular bisector, equilateral triangle, regular pentagon, etc., which must still be learned by high school geometry students. However, try as they might, they came up with four which defied solution. These four unsolved problems of antiquity remained so until the 1800's. They are:
  1. Squaring a circle (construct a square with area equal to a given circle);
  2. Duplicating a cube (construct a cube with twice the volume of a given cube);
  3. Trisecting an arbitrary angle;
  4. Constructing a regular heptagon (or actually all regular polygons).
During the 1800's, advances in mathematics enabled mathematicians to prove them all unsolvable. An important part of the solution was to couch the problem in terms of algebraic, rather than geometric terms. One soon discovers that constructions with straight-edge and compass represent rational operations and square roots, but not cube or higher roots. Thus if a cube root is unavoidable, the construction is impossible. The algebraic equations involved have what are known as algebraic roots. In 1844 the French mathematician Joseph Liouville (1809-1882) proved nonalgebraic or transcendental numbers existed. His proof was not simple, but allowed him to produce several examples, the most famous is known as Liouville's number and can be written either as 0.110001000000000000000001... or 10-(1!) + 10-(2!) + 10-(3!) + 10-(4!) + .... A favorite example is 0.1234567891011..., where the natural numbers occur in order. Integers of this form are known as Smarandanche Concatenated Numbers and work on their prime factorization can be viewed here. Although it had been already shown in 1737 by Euler that e and e2 and in 1768 by Lambert that [pi] were all irrational it took many more years before they were proved to be transcendental.
In 1873, Charles Hermite (1822-1901) proved e was transcendental.
He wrote "I shall risk nothing on an attempt to prove the transcendance of [pi]. If others undertake this enterprise, no one will be happier than I in their success. But believe me, it will not fail to cost them some effort."
But in 1882, Ferdinand Lindemann (1852-1939) proved [pi] was transcendental and coined the term.
Transcendental numbers are irrational numbers that are not the roots of algebraic equations.
The transcendance of [pi] finally solved, all-be-it in the negative, the problem of squaring the circle. Since [pi] is not algebraic, a segment of length the square root of [pi] is impossible to construct. In 1795 Gauss proved that it is possible to divide the circumference of a circle into n equal parts when n is odd, if n is either a prime Fermat number or a product of different prime Fermat numbers. He was 18. It was published in 1801 in his major work Disquisitiones aritmeticae. In 1837 Wantzel published a proof that no other regular polygons can be constructed, thus settling in the negative the question of the constructability of the regular heptagon. However, the regular heptadecagon (17-gon) is constructable! Wantzel also proved that the angle of 60° was not trisectable since the equation 4x3 - 3x = ½ has no roots which are rational or rational combinations of square roots. Wantzel is also responsible for the developments proving that the cube root of 2 is also not constructable with the same year usually given.

Many More Transcendentals

Although [pi] and e are the two most famous transcendental numbers, there are plenty more. Just as the reals can be divided into two disjoints sets, i.e. the rationals and irrationals, the irrationals can be similarily subdivided into algebraics and transcendentals. Another way to classify the real numbers is as any number that can be written as a decimal fraction. These decimals are of three types: 1) terminating; 2) nonterminating but repeating; and 3) nonterminating, nonrepeating. We explored the terminating and repeating decimals in lesson 8 and concluded they were all rational numbers. This last class, however, is another way to characterize the irrational numbers.
There are more irrational numbers than rational numbers.
This is fairly clear since the rational numbers were denumerable, but the real numbers, made up of the rational numbers and irrational numbers, were nondenumerable. Logarithms and the trigonmetric functions are examples of transcendental functions introduced and studied in the high school math curriculum.
Algebraic numbers are enumerable!         Almost all real numbers are transcendental.
It has been very difficult to prove numbers to be transcendental. David Hilbert (1862-1943) challenged the mathematical community in 1900 with a list of 23 unsolved problems in mathematics of utmost importance. The seventh problem was to prove that for any algebraic number (a 0 or 1), and any irrational, but algebraic number b, ab is always transcendental. The first in 1929 and the second a year later, the Russian mathematician Gelfond proved Hilbert's two examples, e[pi]=i-2i, and 2[square root of two] to be transcendental and in 1934 proved the general case. The status of many numbers remains unknown: [pi][pi], ee. Others: [pi]e, 2e, and 2[pi] have not even been proved to be irrational! The sin 1° is algebraic, whereas
The sin (360°/2[pi]) =    sin(1 rad)  =   1  –  1  +   1   –   1  +   1   –    1    ...   is transcendental.
1! 3! 5! 7! 9! 11!

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