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obsolescent English
system. The name of the internet search engine Google™ is derived from googol.
    Kasner introduced the googol to the world in his book Mathematics and the Imagination, written with James Newman, and they tell us that a group of children in a kindergarten worked out that the number of raindrops falling on New York in a century is much less than a googol. They contrast this with the claim (in a ‘very distinguished scientific publication’) that the number of snowflakes needed to form an ice age is a billion to the billionth power. This is 10 9000000000 , and you could just about write it down if you covered every page of every book in a very large library with fine print - all but one symbol being the digit 0. A more reasonable estimate is 10 30 . This makes the point that it is easy to get confused about big numbers, even when a systematic notation is available.
    All of this pales into insignificance when compared with Skewes’ number, which is the magnificent
When considering these repeated exponentials, the rule is to start at the top and work backwards. Form the 34th power of 10, then raise 10 to that power, and finally raise 10 to the resulting power. A South African mathematician, Stanley Skewes, came across this number in his work on prime numbers. Specifically, there is a well-known estimate for the number of primes π(x) less than or equal to any given number x, given by the logarithmic integral

    In all cases where π(x) can be computed exactly, it is less than Li(x), and mathematicians wondered whether this might always be true. Skewes proved that it is not, giving an indirect argument that it must be false for some x less than his gigantic number, provided that the so-called Riemann hypothesis is true (Cabinet, page 215).

    To avoid complicated typesetting, and in computer programs, exponentials a b are often written as a^b. Now Skewes’ number becomes
    10^10^10^34
In 1955 Skewes introduced a second number, the corresponding one without assuming the Riemann hypothesis, and it is
    10^10^10^963
    All this has mainly historical interest, since it is now known that without assuming the Riemann hypothesis, π(x) is larger than Li(x) for some x < 1.397 × 10 316 . Which is still pretty big.
    In our book The Science of Discworld III: Darwin’s Watch, Terry Pratchett, Jack Cohen and I suggested a simple way to name really big numbers, inspired by the way googol becomes googolplex: If ‘umpty’ is any number, 8 then ‘umptyplex’ will mean 10 umpty , which is 1 followed by umpty zeros. So 2plex is a hundred, 6plex is a million, 9plex is a billion. A googol is 100plex or 2plexplex, and a googolplex is 100plexplex or 2plexplexplex. Skewes’ number is 34plexplexplex.
    We decided to introduce this type of name to talk about some of the big numbers appearing in modern physics without putting everyone off. For instance, there are about 118plex protons in the known universe. The physicist Max Tegmark has argued that the universe repeats itself over and over again (including all possible variations) if you go far enough, and estimates that there should be a perfect copy of you no more than 118plexplex metres away. And string theory, the best known attempt to unify relativity and quantum theory, is bedevilled by the existence of 500plex variants on the theory, making it hard to decide which one, if any, is correct.
    As far as big numbers go, this is small beer. In my 1969 PhD thesis, in an esoteric and very abstract branch of algebra, I proved that every Lie algebra with a certain property that depends on an
integer n has another, rather more desirable, property 9 in which n is replaced by 5plexplexplex . . . plex with n plexes. I strongly suspected that this could be replaced by 2 n , if not n + 1, but as far as I know no one has proved or disproved that, and I’ve changed my research subject anyway. This tale makes an important point: the usual reason for finding gigantic numbers in mathematics is