This essay is not about deviations from the use of reason, but about irrational numbers, meaning those that are not ratios of integers, i.e. are not fractions.

The integers 1, 2, 3,…form an infinite series; but it is only the first order of Cantor’s infinities. The fractions, such as 1/2, 2/3, or 501/499, form an infinite set between any two neighbouring integers, so that the total number of fractions is a second order of Cantor’s infinities. Nor is this the end of the story, because between any two neighbouring fractions (whatever “neighbouring” might mean) there is an infinite number of irrational numbers, i.e. decimal numbers with an infinite sequence of non-repea-ting decimals; irrational numbers would thus be a third order of Cantor’s infinities.

Of this unimaginably huge number of irrationals, a few stand out as being of deep, almost mystical significance. Some of these are discussed below.

We can start with the hypotenuse of a square, which is 1.44…, the square root of 2. Then there is the altitude of an equilateral triangle, which is 0.86…, half of the square root of 3. And then there is the ratio of the circumference of a circle to its diameter, which 3.412…or pi. The ancient Greeks knew all about these irrationals, though they knew them to only a few decimal places. Modern computers can grind out millions of decimals places for these irrational numbers, without ever coming to an end. All of these are related to simple geometrical (Euclidian) figu-res in the plane, though they turn up also in the geometry of solids, e.g. pi occurs in the surface area and the volume of a sphere, a cylinder, and a cone.

Another famous, useful and mysterious irrational number is e (= 2.732…), which is “true compound interest”: what you would get at the end of one year if you invested 1 dol-lar at the interest rate of 100%, compounded not once a year, not 2 or 4 times a year, not monthly, or weekly, or daily, or every hour or minute or second, but all infinite-simal moments of time an infinitesimal fraction of the 100%. It does not seem possible that you would get so little, only 2 dollars and 73 cents. If compounded annually, you would already get 2 dollars; if 4 times a year, you would get 2.44, quite an increase; why does it not get over 2.73 even if compounded infinitely often? Well, the series we get by binomial expansion of (1+1/n) to the nth power, as n increa-ses without limit, converges very quickly to the limit e = 2.732… The same number e when raised to the power x (e ) is also the only function of x which is its own derivative in differentiation, i.e. the only function whose rate of increase is equal to itself (“the bigger it gets, the faster it grows”), and the rate of the rate of increase likewise, and so on. It is this property which makes it suitable as the base of natural logarithms, since taking logarithms is the opposite operation to raising to a power, and thus the derivative of ln x is simply 1/x, without having to multiply by any conversion constant.

If we carry out the binomial expansion of (1+1/n) as n approaches infinity, we obtain the number e, as already noted. If we do it for (x+x/n) , as n approaches infinity, we obtain e . If we write out the terms of this series, we have e =

where 2!, 3!…etc are the factorials, e.g. 3!=1.2.3, 4!=1.2.3.4, etc. When we compare this with the series for sin x and cos x, derived from the very general Taylor series (the series for e can also be so derived), we note the Euler relationship

e = cos x + i sin x

where i is the imaginary unit, the square root of minus 1. This looks very simple, yet mysterious – why should this be so? Until we remember that exponential and sine/cosine functions are often alternative solutions to simple second-order differential equations. Physically, this means that a system like a steel spring or a pendulum, when stretched or swung out and then released, tries to return to its original position either by asymptotic exponential decay or by a si-nusoidal oscillation, if it overshoots.

Thus our number e occurs in many diverse (but probably related) situations in physical nature and in mathematics. I tend to regard it with a certain awe as a magic number. The true magic numbers, I feel, are not integers like 3 or 7 or 11 (the magic primes), or “perfect numbers” like 6 or 28 (whose addends are the same as its factors), though these are interesting. The truly magic numbers are the irrationals like pi, e, and our next two candidates, phi and the Feigenbaum number. Indeed, why shouldn’t a few irrationals be very fundamental, when there are so many more irrationals than rationals?

Phi ( ) = 1.628… is a number which bears the same ratio to 1 as 1 does to 0.628…, i.e. /1 = 1/1+ , or

+ -1 = 0, which means that = 1/2.(1 + 5). The ancient Greeks called this the Golden Section, a way to divide a line segment into two parts in the most esthetically har-monious way. However, phi emerges again as the limiting ratio between two consecutive members of the Fibonacci series, 1, 1, 2, 3, 5, 8, 13…, formed when each member of the series is the sum of the two preceding members. The ra-tio is not phi for the lower members of this series, but ap-proaches it in the limit as the series is lengthened. Why this should be so is not clear to me. Phi is also the sum of the continuing fraction

Again, why? The Fibonacci series occurs widely in natural structures, such as the shell of the Nautilus, and whorls of leaves and flowers. Norman Alcock (“The Seventh Seal”) has used it to model all of human, biological, and cosmic evo-lution. He also uses the series which starts with phi and continues with each member the sum of the two previous members; this alternative Fibonacci series finally merges with our first-mentioned one, after a series of spirallings around each other, in a pattern strongly reminiscent of the double helix.

Another interesting irrational number has emerged quite recently: the Feigenbaum universality constant, 4.66920… This number was discovered first in the successive doubling of the attractors as the logistic equation y=rx(x+1) is repeatedly iterated, and as the parameter r is gradually increased. There is at first a single attractor to which the iterations converge, then (between r=3 and r=3.43) there are 2, shortly after 4, and in rapid succession 8, 16, 32 etc., until at about r=3.57 the attractor goes chaotic. Feigenbaum then discovered that all equations whose graph has a single hump do this – it does not have to be the logistic (whose graph is a parabola). And the scaling factor at which the finer and finer detail is observed is always the same, 4.66920…, regardless of the precise form of the equation.

In fractal geometry, the “dimension” of the Koch “snow-lake” is another irrational, 1.2618…, which is log4/log3; and other similar irrationals occur as the dimensions of other fractal curves, usually a ratio of logaithms. (It does’t matter whether these are decimal or natural loga-ithms, because the proportionality constant cancels out in the ratio.) The Koch snowflake is generated by starting with an equilateral triangle, then sprouting smaller (1/3) equilateral triangles from each of its sides, and repeating (iterating) this ad infinitum. Thus we no longer have just dimensions 1, 2 and 3, as in Euclid, or even just dimensions 4, 5 etc., which are mathematical, not spatial dimensions; we also fractional dimensions in between; and I should not have said “fractional”, because these are not fractions (rational numbers), but irrationals. Dimensions get blurred, just as “truth” and “falsehood” get blurred in fuzzy logic. (See my essay Fuzzy All Over.) I cannot help thinking that this irrationality and fuzziness and half-truth is a cultu-al symptom of our times, a “Zeitgeist”; even though these concepts are also valid in themselves. The cultural aspect comes from their having been discovered in this particular historical time. It shows that we are inclined to think in a certain way, and therefore discover the things that are in consonance with this way of thinking.

Now about irrational numbers in general: each sits on an infinitely sharp knife-edge in the set of real numbers. Similarly, at an infinitely sharp knife-edge of time, de-cisions are made in a world of chaos; if the initial con-ditions are just slightly off this edge, even in the mil-lionth decimal place, the results will diverge. This diver-gence may make a difference in evolution, even between the extinction or continuation of whole species or genera or families or orders. It is not chance, but deterministic chaos. Yet the initial point, though of strictly zero di-mensions, must lie SOMEWHERE, so one or the other outcome MUST HAPPEN. At the same knife-edge of time, the particle is at a definite place in the wave packet of an electron or a quark (I visualize it vibrating extremely rapidly within it), and that’s where we catch it in our experiment, when “the wave packet collapses”. The Schrodinger cat lives or dies on the knife-edge of time, as if between two irrational numbers. And we are all Schrodinger cats.