The Irrationals

The irrationals

An irrational number is one that cannot be precisely expressed as the quotient of two integers. If written in decimal form, the decimal expansion proceeds infinitely without ever repeating. Using number theory it is usually possible to prove that a number is irrational without resorting to decimal expansion.

These decimal expansions exhibit unique properties. Some are more "random" than others, meaning that the nine numerals occur with more similar frequency. , for example, is more random than .

has been calculated to the 51 billionth decimal place, though only 50 are needed for measurments and calculations of any scale or complexity. It is said that using the 50-digit expansion of , it would be possible to draw a circle around the known universe and calculate its circumference correctly to within the radius of one proton.

Thus the decimal expansion of an irrational number is an absurdity. It should contain, in codified form, everything manifest and unmanifest, potential and actual, and would still be unknowable. It is more precise to think of as "", of as "", and so on.

To give another example of how ineffable the irrationals really are, John L. Casti, in Complexification, gives the Complexity Version of Goedel's Incompleteness Theorem (as discovered by Gregory Chaitin):

There exist numbers having complexity greater than any theory of mathematics can prove. (p. 146.)

What this means is, while we "know" that irrationals are random, meaning that there is nothing about one digit in the decimal expansion that will tell us anything meaningful about the next, we cannot formally "prove" this. Such complex numbers contain more information than could be codified if the entire Universe were dedicated to the task. The existence of such numbers chisels away at the idea that rationality could dominate Universe.

Irrationals were anathema to the Pythagorean philosophy, which held that whole numbers were literally the "stuff" of which everything was made. For a long time irrationals were not even fully accepted as numbers, similar to the way imaginary numbers are treated today (Terry, p. 68). One observation that might work in favor of this idea is the fact that an irrational cannot be expressed with numerals. It can be approximated, but not expressed; the distinction is of utmost importance philosophically. To correctly express an irrational number it must be described with an expression or formula, or represented by some glyph, usually a letter. Therefore, for most irrationals, practically all of them, there is no way to express or name them, ever. Thus, whether these unexpressable, unnamable irrationals "exist" or not is unknowable.

Irrationals greatly "outnumber" rational numbers. This latter group is of course infinite, but this is a" countable" infinity. Between any two rational numbers, however, lie an infinite number of irrationals -- implying that the number of irrationals is a "higher order" of infinity. This "next largest infinity" is refered to as "continuum," and is not countable. The idea of "orders" of infinity is a recent one (late 1800s) that clearly has not captured the mystic imagination, though an echo of the idea can be found in Aleister Crowley's "Star Sponge" vision (see The Law is For All, commentary on AL verse I:59).

pi

Defined as the ratio between the circumference of a circle and its diameter, this number equals approximately 3.14159. Archimedes approximated it as 22/7 using a 96-sided polygon -- a fairly good estimate.

Given the mystic importance attached to the line/circle dichotomy, has conquered the mystic imagination to quite an extent. If numbers describe the universe, this one is a key that will get one pretty far. A primary reason occurs so frequently is that it is amazingly convenient to measure angles in radians. The trigonometric values for key radian values are easy to remember.

360° = 2 radians, because if one presumes a diameter of 1, and one travels around a circle starting at 0, one covers 2 worth of circle. Travel halfway around, and one has covered worth of circle. Thus an angle, in radians, measures how much circumference one would have covered if the diameter was 1. For those not conversant in radians, this table will give you some perspective:

DEGREES	RADIANS	SINE	COSINE
0	0	0	1
30	/6	1/2	/ 2
45	/4	/ 2	/ 2
60	/3	/ 2	1/2
90	/2	1	0
135	(3/4)	/ 2	-( / 2)
180		0	-1
270	(3/2)	-1	0
360	2	0	1

So, wherever angles occur, so does . Angles, being one of the essences of shapes, are a sacred element in the Pythagorean and Platonic tradition. also frequently occurs "unexpectedly," as in differential equations that equate a function with its second derivative -- the sine and cosine functions match this description.

is a transcendental irrational, meaning that there is no finite algebraic expression that describes it. (The others are called algebraic: for example,

2 =

is an algebraic expression which describes .) However, it can be expressed in terms of continuing fractions, or infinite series.

e

Most people can conveniently ignore e, because its significance does not become apparent until the first semester of calculus. Suddenly e appears everywhere. e, which roughly equals 2.71828, is the basis for the exponential function, which is simple enough: is e to the power of x. What makes the exponential function so special is that it describes its own rate of increase.

The differential calculus is based loosely on the idea that one can compare small changes in any variable x to small changes in the values of some function using x as its variable, written f(x). If x changes just a little, so might the values of that function. (Recall, a function is just a set of commands telling what to do with x.) The difference between 1 and 1.001 is small, so can we assume the difference between and is as well? (This is true, in fact). Using a set of rules, once can define another function that describes the ratio between the rate of change in x and the rate of change in f(x). This second function is the derivative of x, or f'(x), and represents the slope of a line tangent to the graphed curve of f(x) at x.

Well, the neat thing is that ()' = . is its own derivative. It is thus the essence of change, the unchanging force behind the change that occurs everywhere. The exponential function can be expected to show up whenever one has to calculate continual change -- compound interest, population growth, radioactive isotope decay, numerous statistical problems -- all of these depend on e.

e is also transcendental, and can be expressed in many ways as an infinite sum.

phi

roughly equals 1.61803, and is the Golden Mean of ancient geometry and architecture. Suppose you want to take a line, and break it up so that the ratio of the whole to the big part is the same as the ratio of the big part to the little part, i.e.,

whole : big part : : big part : little part

that ratio will be the Golden Ratio, 1.61803 : 1. This Ratio is found in the works of sacred architecture all around the globe, as well as many important works of art.

Interestingly, there is a relationship between and the Fibonacci sequence, the various elements of which have been found to occur in nature, notably biology. The Fibonacci sequence is created from an algorithm that essentially copies the ratio between whole, big part, and little part that is used to define :

What this means is that, after 1 and 1 which are the first two elements of the sequence, every number will be the sum of the two before it. 1+1 = 2, 1+2 = 3, 2+3 = 5, 3+5 = 8, and so on. The ratio between each number and the one before it approaches as the sequence approaches infinity, but this approach is rapid: by the fifteenth element of the series, it is impossible to tell, using the average computer, that has not yet been reached.