Googol. It is a large number, unimaginably large. It is easy to write in exponential format: 10100, an extremely compact method, to easily represent the largest numbers (and also the smallest numbers). With the smallest of effort, you can also present it in the full format: a “one” followed by one hundred “zeros”. However, in its exponential format, it can be easily read; in the full format, you may loose count of the amount of times you need to use the term, “billion” in “ten billion of billions of billions of billions of …. etc.”. In any case, we cannot even begin to appreciate the extent. Even with just one googol, we are faced with a number which is larger than anything used to describe the Universe which we understand. Our Galaxy, for example, is formed of about a hundred billion stars. In exponential form, some 1011 stars. The Sun's mass is 2x1033 grams. Using this measurement as an average star mass, we can work out that the mass (visible) of our Galaxy is therefore approximately 1045 grams. Within the Universe, there are some hundred billion galaxies, a number comparable to the number of stars in our Galaxy. Together, we have therefore gathered some 1056-1057 grams of matter. This matter is essentially formed of baryons (protons and neutrons) connected to the nucleus of atoms, which, ranging from hydrogen to uranium (amongst others), form part of our Universe. When calculating mass, the electrons, which weigh approximately one two thousandth of the nucleons, may easily be disregarded. Bearing in mind that the mass of a proton (and neutron) is 1.7 x 10-24 g, we can work out that the number of baryons present in the Universe is approximately 1080. A large number, but substantially smaller than a googol; to be accurate, a hundredth of a billionth of a billion of a googol. There are more neutrinos and photons, but even their numbers are substantially smaller than a googol. To exceed a googol, we must turn to the largest container we know and its smallest relative part. The smallest length, in terms of physics, that we are know of, is the Planck length. It equals 1.6 x 10-33 centimeters. In a cubic centimeter, there are 2.5 x 1098 cubes, the side of which measures a Planck length. Not even a tenth of a googol. Within the entire Universe, which has a radius of approximately 1028 cm, there are therefore approximately 10184 Planck cubes. This number - the number of Planck cubes in the Universe – is probably the largest number we can give to an entity within the physical world. If we put aside physical size and remain in the sphere of mathematical abstraction, we know of some Mersenne prime numbers which exceed the googol, beginning with 2521 – 1 (which includes 157 figures) and ending with 243.112,609 – 1 which includes 13 million figures and which is believed to be the largest amongst the Mersenne prime numbers known (however we will undoubtedly discover more in the future). We have managed to exceed a googol, but we are still dealing with small numbers when compared to a googolplex.
A googolplex, is, in fact, equal to 10googol and can only be written in exponential format. A googol, which is equal to 10100, may also be written as 1010^2; the Planck cube number containable within the Universe can also be written as 1010^2,27, however a googolplex is 1010^100! Not only would the paper or ink not be enough, but there is not enough space or time to be able to write a googolplex in its full format. Even if you wrote every figure using miniature characters, so small they could fit in a Planck cube, there would not be enough space in the entire Universe, which, as we have seen, contains, at maximum, enough space to write the first 10184 figures. However, we need many more figures! In turn, a googolplex, although equal to 1098 greater than a googol, can be considered a smaller number, to quote an example, when compared to that which is considered the largest number ever used in a mathematical context, known as G, Graham's number. This number, the relative number of figures being unknown, cannot be easily written not even when using the exponential format and it is necessary to resort to new formats such as tetration and subsequent exponential loops which enable its development: addition – multiplication – exponentiation – tetration and so forth. Tetration is indicated by two arrows facing upwards, in between the factors. Subsequent calculations are indicated by an increasing number of arrows. Hence 3↑3 = 33 = 3x3x3. Subsequently 3↑↑3 = 3↑(3↑3) (and hence 33^3) and 3↑↑↑3 = 3↑↑(3↑↑3) – which equates to (33^3)^(33^3)^(33^3). Finally 3↑↑↑↑3 = 3↑↑↑(3↑↑↑3). This represents the starting point in order to reach Graham's number which we will refer to as g1. Step 2 will be g2 = 3↑↑…↑↑3 where the number of arrows is equal to g1. The subsequent step will be g3 = 3↑↑↑↑…↑↑↑↑3 where, this time, the number of arrows is equal to a g2. And so forth. This continues up until the 64th level where we reach g64 = G, Graham's number, an evidently, unimaginable number. It is possible, even easy, to devise operations which would lead us to larger numbers: it is possible to go from the simple +1 to an exponential calculation with terms even greater than that which defines Graham's number (g65), or which include higher factors (using 4 instead of 3 for example).
This is not however the point. The point is to find numbers that serve a purpose, hold specific significance, numbers which result from operations, from mental logic problems, or which cannot be, like prime numbers, expressed in terms of smaller numbers. In this light, both the googol and googolplex are simply two powers of 10 which have been named, are well-known and appear in dictionaries, encyclopedias, documentation and extracts. Graham's number, quite differently, represents an upper bound (but not necessarily the smallest) of the “smallest number of necessary dimensions” to work out the properties of a hypercube (a geometric shape with four or more spatial dimensions). This is why it was labeled the largest number amongst those with the same meaning. I will finish off with some odd information on Graham's number. Its prime numbers remain unknown, and we have good reason to think that they will never be discovered, seeing as they are calculated from the very bottom (however, I do not want to make the same arrogant error which I wrote about recently, therefore it is best "never to say never”). The latest figures are known (at the last count 500 were calculated and this number is on the rise). With regards, G, which is simply an exterminate sequence of multiplications of the number 3, there are 7! Lastly, to put into perspective Graham's number, let's not forget that all the figures are a pittance when compared to the infinite ones, which describe the relationship between the length and diameter of a circumference, or even the relationship between the length of the diagonal of the square and its side (even if we now speak of figures “after the decimal”, which will not significantly change the size of the number). But even when we speak of infinites, we must remember that both larger and smaller ones exist…
Extracted from: Le Stelle no. 107, June 2012
