Sample

Chapter 1

Numbers and Constants

 

Numbers in some form have been with us in one way or another for several thousand years or more.  One of the earliest documented recording was discovered in ancient Babylon now modern day Iraq and is thought to date back over 5,000 years.

From these times many developments have occurred and in particular various symbols have been produced: Sumerian, Chinese, Egyptian, Roman, Mayan, Arabic and in the computer digital age, the language of machines, binary numbers.

The most widely used today is the denary base 10 numbers and we owe a great debt to the Hindu-Arab numerals originating from India.  This is so well expressed by the famous French mathematician Pierre Laplace (1749 -1827):

 “ It is India that gave us the ingenious method of expressing all numbers by means of ten symbols each receiving a value of position, as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity”

As the great interests in astronomy and science developed as a serious disciplines, accompanied by the brilliance and ingenuity of mathematicians, philosophers and scientists so special numbers and in particular physical constants have emerged to crystallise concise statements of results.

Constants that influence our lives, our world and even the universe: the gravitational force that permeates the universe and is characterised by the gravitational constant G, which in turn allows us to calculate the mass of the Earth, planets, moon and sun; the velocity of light c which is fundamental to aspects of astronomy, communications, radar and global positioning GPS systems; Avogadro’s number dealing with atoms and matter, (can we produce nano-sized machines and devices?) and not least the “six numbers” describing the universe and its possible fate.                      

Numbers over the ages have not just been used for “counting”. They are the essential counters for the development of mathematics and mathematics in turn has become the language of science.  One surprising diversion is that numbers have been studied not only for theoretical properties but also as an insight for mystical properties.

The Greeks, particularly Pythagoras (569 – 475 BC) and his followers, were interested in positive numbers.  They formed a cult believing many things in nature could be explained in terms of natural numbers.  One of their many interests was geometry and numbers that could be associated with shapes.
For example, triangular numbers:

    *                  *                 *                      *
                    *      *         *      *              *      *
                                    *     *     *        *   *   *   *
                                                          *   *   *   *    *
      1                 3                6                       10

The triangular numbers have the property that the difference between successive numbers follows the natural sequence
    1  2  3  4  5  6  7  8  …
The sum of successive triangular numbers gives the square numbers:
    1 4 9 16 25 36  49 ….

Square numbers:
     *              *    *        *   *   *        *  *  *  *
                     *    *        *   *   *        *  *  *  *
                                     *   *   *        *  *  *  *
     1                 4                9                  16
The difference between successive square numbers follows the series of odd numbers:
      1 3  5  7  9  11  13  15 ……

Hexagonal numbers:
     *                *   *                      *   *
                     *   *   *               *   *   *   *
                       *    *               *  *   *   *   *
                                                *   *   *   *
                                                     *   *
      1                 7                          19

Pythagoras made many important contributions to the theory of numbers and mathematics.  He also looked at numbers in a metaphysical and abstract way and had a particular interest in demonstrating the harmony of numbers and how they could relate to music.  He claimed that the whole universe was filled with music and even argued that the movement of the sun and planets generated musical notes for the universe to be in harmony.
He initiated a number of schools in Italy, Crete and his home town Samos. He formed the Pythagoran Society, a highly secretative society which lasted 200 years. 

The Pythagorans were also immensely interested in primes and the concept of perfect and amicable numbers.  Prime numbers are numbers which are only divisible by 1 or themselves,
For example, the first 14 prime numbers are:
      1 2 3 5 7 11 13 17 19 23 29 31 37 41…
Rather surprisingly as we go to higher number ranges there appear to be fewer primes:
  Number range    Number of primes    Percentage
                              in the range
    1 to 10                        4                           40 %
    1 to 100                     25                          25 %
    1 to 1000                  168                         17 %
    1 to 10,000              1,229                        12 %
    1 to 100,000             9,592                        9.6 %
Prime numbers are of special interest in modern day encryption systems and the search for the higher order primes still goes on.  The largest prime found to date has 12,978,189 digits. The encryption technique using prime numbers is a very powerful one. A pass key number, N say, is created by multiplying two primes, p and q, together, i.e. N = p x q.  N can be known to everyone but the prime numbers p and q only to the user. Finding p and q is extremely difficult and time consuming, a real deterrent to hackers.  For example try finding the two primes making up 49,264,099   (answer 7919 and 6221)
Banks sending highly sensitive data N can be billions and billions and billions… in size.
There is an interesting property where using the number 29 prime numbers may be generated: prime = 29 + 2 x n x n where n takes the whole numbers 1, 2, 3, 4…
So we generate 31, 37, 61, 79, 101, 127, 157, 191, 229…

A perfect number is one in which the sum of its divisors equals the number itself.  The definition is made much clearer by showing the first three perfect numbers:

       6 = 1+2+3    (its divisors 1,2 and 3 add up to 6)
     28 = 1+2+4+7+14
   496 = 1+2+4+8+16+31+62+124+248

Amicable numbers relate to pair of numbers where the divisors of one sum to the other and vice-versa.
For example,
     220 has the divisors  1 2 4 5 10 11 20 22 44 55  110
     which sum to 284
     284 has the divisors  1 2 4 71 142 which sum to 220
Thus  220 and 284 are amicable numbers

We can find perfect numbers by writing a series of powers of 2 starting with 1 and sum the series until the sum equals a prime number, the perfect number is then found by multiplying the last term in the series by the sum.  As usual this is best illustrated by example:
        1+2+4+8+16   … sum at this point is a prime 31
       So the perfect number in this case is 16 x 31 = 496
       1+2+4+8+16+32+64….sum is 127, which is prime
       So the fourth perfect number is  64 x 127 = 8128

Pythagoras and his followers studied these numbers for their mystical rather than the elegance of their theoretical properties.  6 may be regarded as the number of days taken by God to create the world.  It was believed the number was chosen because it was perfect.  28 the next perfect number is the number of days for the Moon to circle the Earth.  Did God choose 28 for this reason?
The Pythagoreans were particularly fascinated by “magic squares”, arranging the numbers 1 to 9 in a square so the sum in any row, column or diagonal is always the same whichever line is added.  The 3x3 magic square is shown below:
and is unique, all lines sum to 15. There are 8 possible permutations found by rotating the lines in sequence.

View Synopsis View Information Purchase Options