**Googol**. It is
a large number, unimaginably
large. It is easy to write in exponential format: 10^{100}, 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 10^{11} stars. The Sun's mass
is 2x10^{33} grams. Using this measurement as
an average star mass, we can work out that the mass (visible) of our Galaxy is
therefore approximately 10^{45} 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 10^{56}-10^{57} 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 10^{80}. 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
10^{98} 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
10^{28} cm, there are therefore approximately 10^{184} 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 2^{521} – 1 (which includes 157 figures) and ending with 2^{43.112,609} – 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
10^{googol} and can only be written in
exponential format. A googol, which is equal to 10^{100}, may also be written as 10^{10^2}; the Planck cube number
containable within the Universe can also be written as 10^{10^2,27}, however a googolplex is 10^{10^100}! 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 10^{184} figures. However, we need many
more figures! In turn, a googolplex, although
equal to 10^{98} 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 = 3^{3} =
3x3x3. Subsequently 3↑↑3
= 3↑(3↑3) (and hence 3^{3^3})
and 3↑↑↑3
= 3↑↑(3↑↑3) – which equates to (3^{3^3})^(3^{3^3})^(3^{3^3}). 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