From 0 to Infinity in 26 Centuries Read Online Free PDF Page B

withdefinitions of words to help make it clear what Euclid means when he refers to words such as point, line,
straight, surface, etc. Euclid then sets out a list of axioms or statements that are evidently true, such as ‘all right angles are equal to each other’ and ‘if A=B and A=C, then
B=C’.
    The next section of the
Elements
is called ‘Propositions’, in which Euclid proposes a method of how to carry out a mathematical task. For example, in Proposition 1 of Book 1
Euclid shows how to draw an equilateral triangle (all the sides are the same length and all the angles are equal to 60°), and he then goes on to prove that the triangle is, in fact,
equilateral.
    E RATOSTHENES (276–195 BC )
    It would be wrong to talk too much about prime numbers without mentioning multi-disciplined mathematician Eratosthenes, who hailed from a Greek city in modern-day Libya. He was
responsible for many great intellectual endeavours, including calculating the earth’s circumference to a surprising degree of accuracy and coining the word ‘geography’, which
means ‘drawing the earth’ in Ancient Greek. Mathematically, Eratosthenes’ greatest contribution is the Sieve of Eratosthenes .
    In their prime
    Before we look at the sieve let us first contemplate prime numbers : numbers that have only two factors – themselves and 1. Hence 13 is a prime number because 1 and
13 are the only numbers that divide into it without leaving a remainder. 9 is not prime, because it can be divided by 1, 3 and 9, which means it has three factors. 1 is also not a prime number
because it has only one factor.
    Prime numbers are important for two reasons:
    1. Any whole number or integer greater than 1 can be written as a chain of multiplied prime numbers. For example, the numbers between 20 and 30 can be written as
     follows:
    20 = 2 × 2 × 5
    21 = 3 × 7
    22 = 2 × 11
    23 = 23 (prime)
    24 = 2 × 2 × 2 × 3
    25 = 5 × 5
    26 = 2 × 13
    27 = 3 × 3 × 3
    28 = 2 × 2 × 7
    29 = 29 (prime)
    30 = 2 × 3 × 5
    There is only one way of doing this for each number so it seems to me, at least, that primes are the equivalent of DNA for numbers.
    Fundamentals
    The idea that any whole number greater than 1 can be expressed as the unique product of a chain of multiplied prime numbers is called the fundamental
     theorem of arithmetic .
    2. They are very mysterious – there is no pattern to prime numbers, and there is no formula that will produce them. To this day the nature of prime numbers is
     still under intensive study by mathematicians.
    Eratosthenes’ sieve works using a very simple principle to help find prime numbers up to a certain limit. 2 is the first prime number. Anything that 2 goes into cannot be
prime, because it would then have 2 as a factor as well as itself and 1.
    If we set ourselves a limit of 100, we could highlight 2 as a prime and then eliminate all the numbers that have 2 as a factor: 4, 6, 8, etc. up to 100. If we use a grid we can shade them in to
generate a pattern:

    You don’t even need to be brilliant at your two-times table to do this – you could just count on 2 each time and shade in each square you land on.
    After you’ve shaded in all the multiples of 2 you move on to the next unshaded number, which also happens to be the next prime number: 3. We highlight 3 as a prime and then eliminate all
the multiples of 3, some of which have already been eliminated in the first round. The next unshaded number is 5, which again is also prime. As before, highlight that and then eliminate the
multiples of 5.
    As you move along, the next unshaded number must be prime because none of the prime numbers that went before it could go into it. If you keep on repeating this process eventually you’ll
have a completed sieve. Turn to page 55 to see what this looks like.
    To Infinity and Beyond
    Euclid’s theorem demonstrated that there are infinitely many prime numbers. We know that any number can be made by multiplying a chain of