Professor Stewart's Hoard of Mathematical Treasures (9 page)

How to rotate the ellipse.

Putting this all together, using measurements from modern ostrich eggs and intact ancient ones, led to an average figure of 570 square centimetres for one egg. This seemed quite large, but experiments with a modern egg confirmed it. The sums then indicated that at least six eggs had been deposited in Structure 07, the largest concentration of ostrich eggs in any single Predynastic deposit.

You never know when mathematics will be useful.

For the archaeological details, see:

Many puzzles, indeed most of them, lead to more serious mathematical ideas as soon as you start to ask more general
questions. There is a class of word puzzles in which you have to start with one word and turn it into a different one in such a way that only one letter is changed at each step and that every step is a valid word.
⁷
Both words must have the same number of letters, of course. To avoid confusion, you are not allowed to rearrange the letters. So CATS can legitimately become BATS, but you can’t go from CATS to CAST in one step. You can using more steps, though: CATS-CARS-CART-CAST.

Here are two for you to try:

Even though these puzzles involve words, with all the accidents and irregularities of linguistic history, they lead to some important and intriguing mathematics. But I’ll postpone that until the Answers section, so that I can discuss these two examples there without giving anything away here.

 

Answers on page 283

Big numbers have a definite fascination. The Ancient Egyptian hieroglyph for ‘million’ is a man with arms outstretched - often likened to a fisherman indicating the size of ‘the one that got away’, although it is often found as part of a symbolic representation of eternity, with the two hands holding staffs that represent time. In ancient times, a million was pretty big. The Hindu arithmeticians recognised much bigger numbers, and so did Archimedes in The Sand Reckoner, in which he estimates how many grains of sand there are on the Earth and demonstrates that the number is finite.

The million that got away . . .

In mathematics and science the usual way to represent big numbers is to use powers of 10:

There was a time when an English billion was 10
¹²
, but the American usage now prevails almost universally - if only because a billion is now common in financial transactions and we need a snappy name for it. The obsolete ‘milliard’ doesn’t have the right ring. In this age of collapsing banks, trillions of pounds or dollars are starting to be headline material. Billions are passé.

In mathematics, far bigger numbers arise. Not just for the sake of it, but because they are needed to express significant discoveries. Two relatively well-known examples are:
with one hundred zeros, and
which is 1 followed by a googol of zeros. Don’t try to write it down that way: the universe won’t last long enough and you won’t be able to get a big enough piece of paper. These two names were invented in 1938 by Milton Sirotta, the American mathematician Edward Kasner’s nine-year-old nephew, during an informal discussion of big numbers (Cabinet, page 213). The official name for googol is ten duotrigintillion in the American system and ten thousand sexdecillion in the 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
In 1955 Skewes introduced a second number, the corresponding one without assuming the Riemann hypothesis, and it is

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 that some sort of recursive process has been used in a proof, and this probably leads to a wild overestimate.

In orthodox mathematics, the role played by our ‘plex’ is usually taken over by the exponential function exp x = e
^x
, and 2plexplexplex will look more like exp exp exp 2. However, 10 is replaced by e here, so this statement is a complete lie. However, it’s not hard to complicate it so that it’s true, bearing in mind that e = 10
^0.43
or thereabouts. Theorems about repeated exponentials are often rephrased in terms of repeated logarithms, like log log log x (see page 189 for logarithms). For example, it is known that every positive integer, with finitely many exceptions, is a sum of at most
perfect nth powers - well, ignoring a possible error that is smaller than n. More spectacularly, Carl Pomerance has proved that the number of pairs of amicable numbers (page 110) up to size x is at most
for some constant c.

Several systems for representing big numbers have been worked out, with names like Steinhaus-Moser, Knuth’s up-arrow and Conway’s chained arrow. The topic is much bigger than you
might expect, which is only appropriate, and you can find much more about it at

Which (perhaps unfortunately) reminds me:

Piracy is probably not the first thing that comes to mind in connection with mathematics. Of course, the peak period for piracy, or its state-sanctioned version, privateering, was also the golden age of the mathematics of navigation. Navigators drew geometric diagrams on charts using compasses and protractors; and they ‘shot the Sun’ with sextants and used mathematical tables to calculate the ship’s latitude. But that’s not the connection I’m after here, which is a curious set of historical links between mathematicians and pirates, centred on one of the all-time greats: Leonhard Euler, a Swiss-born mathematician who worked in Germany and Russia. He lived between 1707 and 1783 and produced more new mathematics than anybody who has ever lived. The connections were discovered by Ed Sandifer, and posted on his wonderful ‘How Euler Did It’ website:
www.maa.org/news/howeulerdidit.html