From 0 to Infinity in 26 Centuries (19 page)

In the last 100 years mathematics has diversified into a vast number of fields, each of which requires years of work in order to excel at it – gone are the days of the
generalist mathematician with equal facility across all fields. While a significant proportion of these mathematical developments are beyond the scope of this book – and, if I’m being
perfectly honest, beyond the scope of this author, too – there are some interesting areas where, initially at least, we can appreciate the mathematics and the mathematicians that contributed
to them.

Pick a prize

Named in honour of the original host of the American game show
Let’s Make a Deal
, the
Monty Hall problem
is a good example of maths at its most
counter-intuitive. Problems such as this one have been around for some time. Parisian mathematician
Joseph Bertrand (1822–1900) posed a very similar problem called
Bertrand’s Box way back in 1889.

On
Let’s Make a Deal
, the winner of the game show was offered the opportunity to choose one box out of three. One box contained the grand prize – the keys to a brand-new car.
The other two boxes contained consolation prizes of much less worth.

The contestant picked a box but was forbidden from opening it. The host, Monty Hall, then revealed that one of the two remaining boxes was a consolation prize. He gave the contestant the option
to swap their box for the remaining closed box. The question was:

On the surface it seems that it should not matter. After all, the contestant is choosing between his or her own box and the other box, so it’s a 50:50 call, right?

Wrong.

There are several ways of thinking about this problem and perhaps the most obvious way is to look at all possible outcomes of these events. Let’s call the boxes A, B and
C and say that A contains the prize.

If we look at the outcome column, we can see that there are three ways to win and three ways to lose. So it was 50:50 after all!

Wrong.

If you look, one of the wins happens when you stick, and two of them happen when you swap. This means that the chance of winning if you swap is 2/3, and only 1/3 if you
stick.

Another, less laborious way of thinking about it is this: when you pick initially, you have a 1 in 3 chance of choosing the correct box, which means there is a 2 in 3 chance that the prize box
is one of the other two boxes. When the consolation box is revealed, it does not change the fact that there is a 2 in 3 chance that the remaining box is the prize.

A D
IFFERENT
L
ANGUAGE

We saw earlier how Alan Turing and other code breakers helped the Allies during the Second World War, but the discipline of
cryptography
precedes their efforts. People
have been using codes and ciphers to communicate privately for thousands of years.

A
code
is, in effect, a language, and they work on the principle that A and B can communicate privately in a language that C does not know. The disadvantages here are that C can either
find someone else who speaks the language, or they can find a dictionary or codebook and work out the information themselves.

Ciphers
, on the other hand, involve altering the original message using a sort of algorithm; the art of the cryptologist is to break the cipher, often using mathematical
means.

A new alphabet

The Roman army used Caesar’s cipher to encrypt orders. This simple cipher worked by taking the letter in the original or
plaintext
and shifting each letter in it
along the alphabet by a predetermined amount to create the
ciphertext
. For example, ATTACK would become BUUBDL if each letter is shifted along one place.

Although, admittedly, it is not a terribly cunning device, it is nevertheless effective if the orders were very short term. The problem here is that if you can work out the shift, you can
translate the whole message easily.

More effective would be to rewrite the alphabet in a more random fashion, assigning each letter a new cipher letter without a constant shift. I could write A as Q, T as G, C as X and K as Z, so
ATTACK would become QGGQXZ. This cipher is a bit more secure, but maybe vulnerable to attack if I set my computer the task of trying out different combinations of the alphabet. Let’s see:

For A, I have 26 choices of letter (it might help to confuse things if sometimes the plaintext letter and the ciphertext are the same).

For B I have 25 choices, 24 for C, 23 for D, etc. This means there are 26 × 25 × 24 × 23 × … × 3 × 2 × 1 possible alphabets for the computer to
check. The shorthand for this is written 26! or 26 factorial.

This is a large number, but computers these days are so fast they can deal with a number of this size quite easily. Let’s say my computer can check
100 alphabets per second to see if they make sense, so it would take my computer:

to check all the possible alphabets a message could be in. Clearly such a brute force approach would not work, but people have been able to break such ciphers for hundreds of
years. How?

Language patterns

Arab mathematician Al-Kindi (801–73) developed something we now call
frequency analysis
. Al-Kindi discovered that every language uses its letters in unique
proportions. For example, in English we use the letter e most often – 13% of the time, followed by t (9%), a (8%) and o (7.5%) and so on. Different languages have different percentages of
letter distribution.

This means that, if you know an encrypted message is in English, you can count how many times each letter appears in the ciphertext and match it up with the real letters. For example, if the
letter n appears 13% of the time it is most likely representing e. There are also idiosyncrasies in languages that help – such as q, which in English is nearly always followed by the letter
u.

There are many other ciphers, each of which has become increasingly elaborate and difficult to break. However, in today’s
online world we have started transmitting
far more of our private information, to the point where sending personal details such as bank account details, dates of birth and passwords is commonplace. So how do we protect this
information?

Keys to the lock

In the nineteenth century the Dutch cryptographer Auguste Kerckhoff (1835–1903) summed up what makes an ideal cipher: even if someone knows the full workings of your
cipher, it should still be impossible for them to decipher your message. This relies on having a good cipher and something called a
key
, which is the crucial information without which you
are unable to break the cipher. In the example of Caesar’s cipher, the key is how many letters along the alphabet you should shift.

Central to the workings of modern ciphers that control e-commerce, and also for people that like to keep their emails protected, are
one-way functions
, so-called because it’s a
calculation that is easy to do one way, but very difficult to do in reverse. British economist William Jevons (1835–82) recognized that it is fairly easy to multiply two prime numbers
together; much harder is to work backwards from the result to find out which two prime numbers had been multiplied together, especially if the numbers are large.

For example, in a matter of moments you can multiply 23 and 19 using a calculator to get 437. However, to work out which prime numbers multiply to make 437 you have to go through the rather
laborious process of checking whether each prime number goes into 437. In this case, I would have to perform a
check for 2, 3, 5, 7, 11, 13, 17 before getting to 19. Checking
seven times is not a big deal, but imagine checking prime numbers with thousands of digits. The largest primes known at the moment have over 12
million
digits. This will take you a very long
time, even using a computer.

In effect we now have two keys: the product of the two primes, which can be used to encrypt information, and the two prime numbers themselves, which can be used to decipher anything encrypted
with the product of the prime numbers in the first place. This pair of keys is referred to as
public and private keys
. They allow information to be transmitted safely, even though people can
get access to our encrypted data and they know what method we used to encrypt it.

So, when you buy something online, the website’s computer lets your computer know its public key – a whopping great big number that is the sum of two big
primes multiplied together. Your computer uses that number to encrypt all your details (which, if they weren’t already numbers, have been converted to numbers) and then sends it to the
website. The website’s computer can then use its private key – the two prime numbers used in the first place – to decipher the message and remove the money from your account.
Clever, eh?

G
OING
L
ARGE

When it comes to big numbers we have adopted the American system. Starting from 1 million, which originally derives from Italian, we then use Latin numbers as prefixes to
indicate a number a thousand times larger. For example, 1 billion is 1,000 million, ‘bi’ being a prefix meaning two; 1 trillion is 1,000 billion, and 1 quadrillion is 1,000 trillion,
and so forth.

The big and the small

The practical use for these numbers is limited to fields that deal with very large things (such as astronomy and cosmology) or very small things (such as chemistry and physics).
For example,
Avogadro’s constant
tells us how many atoms or molecules there are in a standard measure of a particular element, and that it is roughly equal to 600 sextillion. The mass
of the earth is approximately equal to 6 septillion kilograms.

Mathematicians have another system with large and small numbers called
standard form
or
scientific notation
.
This system exploits the fact that a
number with n zeros after it is the same as the number multiplied by 10
ⁿ
. Hence, Avogadro’s constant is often written 6 × 10
²³
, which is much more convenient to
use in calculations, and to enter into a calculator too. Standard form can also be convenient for very small numbers, because multiplying a number by a negative power of ten is the same as putting
zeros and a decimal point in front of the number. For example, an electron’s mass is approximately 9 × 10
^-31
kg, e.g. a 9 preceded by 31 zeros with a decimal point between
the first two: 0.0000000000000000000000000000009.

P
ARTY
T
IME

In 1939 Austrian engineering scientist Richard von Mises (1883–1953) posed the
Birthday problem
: how many people need to be in a room for there to be a 50% chance
of two of them sharing a birthday? The answer to this puzzle is surprising.