The exotic “number depths”
The numbers necessary for everyday purposes come from a rather narrow range that rarely goes beyond the order of a few millions. And most of us make little effort to really grasp the meaning of the billions or even trillions mentioned when the question comes for example to state economics, as such immense figures are thought of as belonging to a distant, exotic, incomprehensible range of numbers.
Since antiquity there have been several attempts to reach towards what Arno Schmidt, in his triptych “Leviathan” (1949), described as “number depths” and indeed the human mind has achieved going well beyond the ordinary and everyday numbers towards infinity. The ancient Greek scientist and inventor Archimedes of Syracuse (third century B.C.) compiled a number system based on the myriad (10000) that reaches as far as 10^(8*10^16), a number consisting of a 1 followed by 80000000000000000 zeroes, and actually used his numbers in order to proceed far from human scale and measure the universe, as it was understood in his time. Hindu cosmology describes a continually reborn universe and routinely speaks about extremely large numbers as Brahma, the god of creation, has a life span of 311 trillion and 40 billion years. Such numbers are now perceived either as curiosities or as completely restricted within the limits of some scientific field and, as such, detached from everyday life. However, it appears that, under certain circumstances, there may be instances when immense numbers actually do interfere with the everyday activities and do require a certain degree of comprehension by the average person.
The exponential growth
Things in nature and in society change and any measurable change over a specific period of time is often thought of as a multiplication by a factor and expressed as a percentage. For example, a 20% discount corresponds to a multiplication by a factor 1-20/100=0,8 while a raise by 20% corresponds to a multiplication by a factor 1+20/100=1,2. When such a change by a certain percentage is repeated over a constant period of time, the change is called exponential and finds quite a few applications in Physics, Economics, Biology and elsewhere. Note that a smaller than unit factor corresponds to exponential decrease and a greater than unit factor corresponds to exponential growth. For example, a 1500 euro salary raised each year by 3% corresponds to a growth factor 1+3/100=1,03 and would be raised to
after five years. The general rule for exponential growth is expressed by the equation B=A*p^n, where, A is the initial value (e.g. the 1500 euro), p=1+c/100 is the growth factor (e.g. 1,03), c is the percent increase and n is the number of periods of time considered (e.g. 5 years). The formula is the same for exponential decrease with p=1-c/100. Obviously, every exponential growth is bound to double the initial value after a specific amount of time d (doubling time) which is proved to be equal to d=log2/logp and approximately equal to 70/c. For example, the 1500 euro salary under the 3% yearly raise would thus double every d=log2/log1,03=23,45 years. And quite naturally, had the rate of change remained constant indefinitely, the salary would eventually go well beyond the ordinary, everyday numbers towards Schmidt’s “number depths” and far from a manageable range.
Exponential growth and inflation
Inflation is another example in Economics described by the law of exponential change (usually exponential growth) and is normally kept in levels that maintain prices within a manageable range. As has happened several times in the past in various countries of the world, under specific circumstances things about inflation may go quite astray and hyperinflation may appear. In such a condition when the growth factor p becomes very large for some reason, the doubling time d consequently becomes smaller and prices double too often, soon entering the “number depths” of extremely large numbers. In a short period of time everyone may become an impoverished billionaire, as prices soar and double every few days (or even hours), evaporating salaries only a couple of days after they were received. The state issues banknotes of enormous and ever increasing monetary denominations that soon are useless and the mere value they represent is far smaller than the expenses made for their printing. Over the past decade the unfortunate citizens of Zimbabwe had to plunge into Schmidt’s “number depths” and to learn that this exercise in extremely large numbers was only a manifestation of the claiming of their livelihood and welfare. Zimbabwe entered the condition of hyperinflation in 2001 when annual inflation reached 112% corresponding to a growth factor p=2,12 and doubling time of d=11 months, 7 days. By 2002 inflation was 199% with a growth factor p=2,99 and doubling time d=7 months, 21 days. During the next few years, and after a short spasm of ephemeral recovery in 2004, the Zimbabwean dollar plunged into a death loop with an ever increasing speed:
2003 annual inflation rate=600%, growth factor p=7, doubling time d=4 months, 10 days
2004 annual inflation rate=133%, growth factor p=2,33, doubling time d=10 months
2005 annual inflation rate=586%, growth factor p=6,86, doubling time d=4 months, 11 days
2006 annual inflation rate=1281%, growth factor p=13,81, doubling time d=3 months, 6 days
2007 annual inflation rate=66212%, growth factor p=663,12, doubling time d=1 month, 9 days
2008 annual inflation rate=231150889%, growth factor p=2311509,89, doubling time d=17 days
By some measurements, in November 2008 the monthly inflation rate may had reached 79600000000%, corresponding to a doubling time of approximately 20 hours; had this rate remained constant, prices would double 438 times within a single year. The Zimbabwean dollar collapsed as the value of the 100 trillion dollar banknotes issued in 2008 evaporated within hours and the citizens of Zimbabwe, having for more than 6 years been subjected to exercises on extreme numbers, were allowed to use more stable, foreign currencies for their transactions. Ironically, the eventually worthless Zimbabwean 100 trillion dollar banknote could fetch a few US dollars on Ebay when sold as a curiosity to collectors around the world. During the hyperinflation years, the Zimbabwean government repeatedly dropped several redundant zeroes from the banknote denominations in a futile effort to bring the level of prices as close to manageable proportions as possible. A total of 25 zeroes were dropped, making 10 septillion of the oldest version dollars equivalent to 1 dollar of the last version. Note that ten septillion equals 10 million million million million, i.e. a number ten times bigger than the diameter of the observable universe, expressed in kilometers. A one to ten septillion proportion is approximately the same as that of a single molecule of water compared to a big glass of water. Or, to change scale, like a droplet of water compared to all the water volume on planet Earth (oceans, glaciers, clouds, lakes, rivers and underground water combined).
Homeopathy and the infinitesimal
Such nearly ungraspable analogies may appear meaningless or absurd after a certain point, but how much more absurd would it be to seriously claim that such a droplet of any substance could mysteriously alter the consistency of the 1,36 billion cubic kilometers of water of the Earth, endowing it with miraculous curative properties? Yet such is the claim of a so called “alternative medicine” branch known as “homeopathy”, a pseudoscientific practice based on a quite remarkable conjecture. Homeopathic treatment of illnesses is based on dilutions of various substances prepared according to the so called “law of similia”, an axiom proposed by the German physician Samuel Hahnemann (1755 – 1843). According to the homeopathy doctrine the symptoms of an illness (e.g. headache) and the cure from them can be thought of as two opposite conditions, standing apart. If a higher concentration dilution of a specific active substance takes us towards the symptoms side, i.e. is causing headache for example, then, it is assumed, a lower concentration dilution should take us towards the other side and provide cure. This line of thinking may appear incredible already; however the most incredible part, and what is of interest here, is the claim that as the concentration of a dilution decreases, the curative effectiveness of the dilution actually increases. In order to achieve lower concentration dilutions (and consequently more powerful, according to the homeopathic doctrine), homeopaths apply a ritual of successive dilutions: first they dilute a small quantity of a substance in a quantity of a solvent (say water), then they dilute one drop of this to another quantity of solvent and so on. Between two successive dilutions, homeopaths apply “succussion” i.e. a thorough shaking of the dilution.
A common way of measuring the potency of homeopathic medicine is by measuring the number of successive dilutions, each dilution prepared in a ratio of 1 part in 100 parts. For example, a homeopathic medicine marked as 30C indicates thirty such successive dilutions, meaning that there is one part of the substance in 100^30=10^60 parts of the solvent. Obviously, after each step of the procedure, which is called “potentization”, there is less and less quantity of the substance present in the dilution. Hahnemann may have thought matter as infinitely divisible, assuming probably that, no matter how many successive dilutions are made, there will always be some infinitesimal quantity of the substance into the preparation. Though it is hard to understand how this assumed minuscule quantity could possibly endow the solvent with therapeutic properties, another obvious doubt is put forward by the simple fact that matter is actually not infinitely divisible and, after a certain point, the presence of even one molecule of the substance in the dilution becomes quite uncertain or even unlikely.
For example, in order to produce first a 1C potency (i.e. 1:100) dilution of graphite (an allotropic form of carbon) we need to dilute 1000/99=10,1 ml of graphite in 1 liter of water. Then, as the density of carbon is approximately 2g/cm^3, there are 20,2 grams of graphite in the dilution. The molar mass of graphite is 12g/mol and thus the quantity of graphite corresponds to 20,2/12=1,68 moles, i.e. 1,68*6*10^23=10^24 molecules, where 6*10^23 is Avogadro’s number (the number of molecules per mole of any substance). Therefore our 1C dilution contains 10^24 molecules of graphite.
To produce a 2C potency dilution, we need to drop 10,1 ml of the 1C dilution into 1 liter of water, shake thoroughly and thus get a preparation containing 10^24/100=10^22 molecules of graphite. Proceeding similarly, a nC potency dilution, where n=1,2,3,4,… etc., contains 10^24/100^(n-1)=10^26/100^n molecules of graphite. It is evident that the formula B=a*p^n for exponential growth described above applies here too, with a=10^26 and p=1/100, corresponding to a 99% decrease at each step. Just like the value of the Zimbabwean dollar, the number of molecules of graphite in the dilution decreases exponentially (as after each step it is multiplied by 1/100) until, for a 13C dilution, just 1 molecule of graphite is expected to be present. For even higher potency dilutions, the presence of even that one molecule becomes unlikely: a 30C dilution is expected to contain 10^24/100^29=1/10^34 molecules, meaning that there is a 0,00000000000000000000000000000001% probability to have any molecules of graphite present. In fact, finding graphite in such a preparation is far more improbable than winning a 6/49 lottery five times in a row. To put it differently, in a 30C dilution it is expected to find 1 molecule of graphite per 10 billion trillion trillion liters of water, a volume equivalent to a sphere with a radius of 1340000 km. Therefore, such a 30C potency preparation contains nothing but water and is chemically indistinguishable from any other sample of water that has not gone through the homeopathic ritual. Remarkably, not even homeopaths themselves can possibly distinguish between a 30C dilution and pure solvent.
However, a 30C potency is somewhat modest as homeopathic dilutions proceed even further away into the “number depths” with potencies as high as 200C (corresponding, in our graphite example, to 1 molecule of substance for every 10^374 liters of water, a mind bending volume, immensely greater than the observable universe). For the production of such homeopathic dilutions only the first 12 or so steps may still involve some molecules of the initial substance; all the rest are just about diluting pure water into pure water and shaking, with an obvious final result: pure water.
Homeopathic quackery and hyperinflation provide only two real life examples of the same underlying law, the law of exponential change, which inevitably puts forward the question of understanding immense numbers, at least to some extent. And, though indeed life is not only numbers, as is by many pointed out often, it is also true that understanding numbers may actually make easier the understanding of some of the not so unimportant aspects of life.