September 20, 2009

The Book of Numbers - Peter J. Bently

The book of numbers
The secret of numbers and how they changed the world - by peter . J. Bently.

This is a good book on how mathematics - its history and Gurus behind the scene; including interesting story behind numbers.

1000 years ago, when there was no difference between science and religion, numbers seemed to hold the key to understanding the universe. Einstein once said, " There are 2 ways to live your life. One is as though nothing is a miracle. The other is a as though everything is a miracle".

Even though Romans would have counted 2000 years ago, around 4000 years ago, several tribes developed shorter spoken words for the numbers. Amazingly the words that these farmers/tribes and hunters created from the basisi for the words used throughout Europe today: including Punjab, Hindi, Persian, Afgan, Lycian, Greek, Latin, German, Armenian, Italian, Spanish, Portuguees, French, Romanian, Sardirian, Dalmatian, Welsh, Cornish, Erse, Manx, Scotts, Gaelic, Dutch, Friesian, Anglo-Saxon & English. These people are now known as the Indo-Europeans. They helped to create one of the largest family of languages that we know today.

The Invention of Nothing:

The problem with Roman numbers is that addition/subtraction is difficult. However the Arabic numerals seems so much easier. The reason for calculation seems simpler in Arabic numerals is because we use the position of numerals to give extra meaning. The number furthest to the right always means, " a value less than 10'. The number left o the left of that always means, " a number of tens less than 100." This advantage made Arabic numerals a standard over Roman numerals.

Nothing was invented about 1800 years ago in India. The first representation of zero was made in 628 AD by a 30-year old Indian Mathematician named Brahmagupta. He wrote, Brahmasphutasiddhanta which explained the movement of the planets and how thier precise paths could be calculated. He said, 'Zero is the result of subtracting a number from itself". To prove it, he wrote down a series of mathamatical rules to show what you do with zero.

" When zero is added to a number or subtracted from a number, the number remains unchanged and a number multiplied by zero becomes zero". In the history of Mathamatics, that was very remarkable & important invention.

However he thought a number divided by zero was zero which was a terribly wrong. It was corrected by Bhaskara after 468 years which Baskara said, it should be infinite. But Baskara's version of the answer was wrong too. It took 1000 years after Bramagupta, French mathematician names l'Hopital received the credit for the idea of shrinking series to zero over over zero and it is known today as l"Hopital rule. With his argument, it is simple to show that zero divided by zero could be anything at all. So the answer is indeterminate - meaning it could be any number.

Rational Number:

Pythagoras was born in 569BC to 475BC - same period when Buddha lived. HIs students has to follow strict rules; they had to give up their possessions become vegetarians and follow his beliefs which are the following.
1. That at its deepest level, reality is mathematical in nature
2. That philosophy can be used for spiritual purification
3. That the soul can rise to union with the divine
4. That certain symbols have a mystical significance
5. That all brothers of the order should observe strict loyalty and secrecy.

[Some of the dates associated with famous Greek people)

Pythagoras (569 to 475BC)
Plato (427-347 B.C.)
Aristotle (384-322 B.C.)
Euclid (300 B.C.)
Archimedes (287?-212 B.C.)
Eratosthenes (276-194 B.C.)
Aristarchus (200s B.C.)
Hipparchus (180?-125 B.C.)]

The first Pythagoras theorem - sum of the square of the two sides of a right-angle triangle equals the square of the remaining side - was found on a Babylonian tablet, which dates back from 1900-1600 BC (1000 years before Pythagoras lived).

Pythagoras was perhaps the first professional explorer of the world of numbers, but he wasn't very good at fractions. The fractional notation that we know today was listed in Brahmagupta's book. Fractional numbers have transformed our ability to think small and understand the dimensions of things like atoms. While Pythagoras was learning about numbers, Buddha had already mastered a seemingly impossible trick with tiny numbers.

Perfect Numbers:
These numbers are that can be formed by adding up all the smaller numbers that make up their divisors.

There are only 4 such perfect numbers in the first 10 million natural numbers. (6,28,496,8128)
6= 1+2+3; 6 can be divided by 1,2,&3
28= 1+2+4+7

These first 4 numbers have been known for over 2000 years. Their significance has been debated since thier discovery.
As per St. Augustine, 6 is a prefect number because God created all things in 6 days.

Likewise 28 was thought to be chosen by God as a perfect number of days to take the Moon to orbit the Earth

Amicable Numbers:
These numbers are whose divisions add up to their 'friend'. The best 2 examples of amicables numbers are 220
& 284.
220's divisions are 1+2+4+5+10+11+20+22+44+55+110=284
284's divisions are 1+2+4+71+142=220

Prime Numbers:
These are numbers that can be divided by 1 and itself. The first prime numbers are 1,2,3,5,7,11,13,17,19,23,29.

Euclid proved the following:
All natural numbers greater than 1 are made from products of prime numbers.
Prime numbers can be made by adding a sequence of numbers that double each time, then the4 prime number multiplied by the largest factor will be a perfect number.
(1+2+4=7; 7*4=28 which is a perfect number
1+2+4+8+16=31; 31*16= 496 which is a perfect number)

Secure Numbers (Strong Prime numbers):
Strong prime numbers are used for secure transaction over internet (PKI security).

A prime number is strong if the average value of the 2 primes on each side of the prime is less than the value of the prime.
(e.g 17 is the 7th prime. 6th & 8th prime numbers is 13 & 19. If we add them up (32) and half the summation is 16 which is less than 17 and hence 17 is a strong prime.

Most successful and important work in Geometry was performed by Euclid who invented the Fundamental Theorem of Arithmetic in his 13 volume book
His famous wording - Eureka is an interesting story.
His famous sentence was " Those who claim to discover everything, but produce no proofs of the same, may be confused as having pretended to discover the impossible.

Algebra - When is a number not a number:
Geometry was clearly an important tool for those wanting to draw and design with accuracy. But it took a 1000 years before it became the flexible tool we know today. It was a man named Abu Ja'far Muhammad bin Musa al-Khwarizmi born in Baghdad around 780 who invented Algebra.

Golden Phi:
Many philosphers and mathamaticians believe in a magical number for hundres of years; becasue they think they know the one of these magical irrational numbers(today we call Phi - 1.618033988...); an irrational numbers, it goes on forever, never forming a repeating pattern). Today this number is called golden section, golden ratio or simply golden number.

Few claim that Leonardo DA Vinci used the golden ratio to help him find the perfect propositions for his famous painting 'MONO LISA'. The division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons.

At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.

e is one of the mysterious irrational numbers that lie at the heart of matahmatics. its value (to the first 20 decimal places) is 2.71828182845904523536

Its existence was more or less implied in the work of John Napier, the inventor of logarithms, in 1614. Euler was also the first to use the letter e for it in 1727 (the fact that it is the first letter of his surname is coincidental). As a result, sometimes e is called the Euler Number, the Eulerian Number, or Napier's Constant (but not Euler's Constant).

An effective way to calculate the value of e is not to use the defining equation above, but to use the following infinite sum:

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...

Without e, it would have been impossible to make the same progress in science and technology over the last few centuries.

Jacob Bernoulli was the first to discover e. He was looking for compound interest on money. He knew that if you added the interest to the sum more frequently then the sum would increase faster. But happen when you calculate in shorter intervals (6 months, 1 month or 1 week or 1 day..). He quickly disocvered that if you deposited $1 at 100 % interest APR:
if compounded annually, it becomes $2.0
if compounded biannually, it becomes $2.25
if compounded quarterly, it becomes $2.44
if compounded monthly, it becomes $2.61
if compounded weekly, it becomes $2.69
if compounded daily, it becomes $2.71
if compounded continuously, it becomes $2.718

Some people believe today that pi=22/7. But this is a rational number as its start to show repeating patterns, but actual value of pi is irrational number.
By definition, pi is the ratio of the circumference of a circle to its diameter. Pi is always the same number, no matter which circle you use to compute it. Pi is an infinite decimal.

Pi continued to fascinate mathematician for centuries and slowly came to know more accurately. = 3.14159265... For many years, the power of super-computers are determined by showing how many digits of pi, it can calculate.

i: the imaginary number:

What is i? It's the square root of -1 (see footnote below). And it's NOT a real number. i was invented because people wanted to be able to take square roots of negative numbers, and you can't do that if you limit yourself to real numbers. So we can make an imaginary number by taking a real number like 5 and multiplying it by i. That gives us 5i.

There are many superstitions about the number one - some are listed below.
If you break one egg, you will break a log.
It is unlucky to walk around the house in one slipper
Only keep money in one pocket or you will lose it.
One-eyed person is a witch
Seeing one magpie bodes a death in yuor family
If you wash your hair on the first day of the month you will have a shorter life
It is unlucky to get married on Aug1 or Jan 1.

In many religion, the number 1 is associated with the unity of God.In China, 1 is linked to growth and prosperity. Number 1 is single, unequivocal, exclusive oneness. The oneness has led to intolerance and centuries of bitter, bloddy battle.

Mobius is remembered for his work in topology; esp. on simple shape that bears his name. Mobius is not the first one to notice the shape, but Listing. Mobisu stripe has only one side and one edge.
(Make a Mobius strip and cut into thirds. Again carefully make a hole and cut parallel to the edge, about a third cross. You will quickly discover that although you are trying to cut it into 3, you only make one cut)

To be even has always meant more than a dull mathematical definition. Early Christains believe 2 represented devil or the division of soul and God. Zorostraints beleive that the 2 is symbolic of an eternal, evenly balanced battle between good and evil. In Russia, even number of flowers are for funerals.

In computer world, there is no 2; but only 1 or 0. Because it uses base of 2 (binary). The origins of binary are ancient, but perhaps the first person to study binary in detail was Gottfried Leibnitz (was born in 1646 in Germany).

Euler noticed that if you add the number of corners(vertices) to the number of faces and subtract the number of edges, teh answer is always 2.

Bernard Russel worked with Einstein and together they released Russell-Einstein Manifesto in 1955. While working with logic and mathematics, he found a paradox. Something that are both true and not true at the same time. His paradox seems to imply that the whole of mathematics was faulty.

This cannot be explained in mathematics, but here is an example.

"There is a barber who shaves precisely those people who don't shave themselves. Does he shave himself?"
If he does not shave himself, then he must shave himself. But if he does shave himself, then he will not shave himself. The only way this makes sense is if he shaves himself and does not shave himself at the same time - but that is logically impossible. That is why it is paradox.

There are lots of 3s. Traffic lights - red amber & green lights. Tom, Dick & Harry. Tremendously Triumphant Trio. Alphabet ABC. 3 meals- breakfast, lunch & supper.
3 is central to many religions; Holy Trinity - Father, Son & Holy Spirit. Hindhu - Bhrama, Vishnu & Shiva.

One of the most famous trinagular numbers is the so-called 'number of beast':666
(There is now some question about 666 being the number of the beast -- it turns out that it may have been a mistake made centuries ago in copying of the scriptures. In the oldest surviving copy of the New Testament-some 1,500 years old- the true 'evil' number is 616)

The magic relationship between the first 4 numbers and 10 led them to create a whole philosophy based on 10 sets of 4. (1+2+3+4=10)
Numbers- 1,2,34
Magnitudes - point, line surface, solid
Elements - fire,air, water, earth
Figures - pyramids, octahedron,icosahedron,cube
Living things - seed, growth in length, breadth, in thickness
Societies - man, village, city, nation
Faculties - reason, knowledge, opinion, sensation,
Seasons - Spring, summer, fall, winter
Ages of a person - infancy, youth, adult,m old age.
Parts of living things - body, three parts of soul.

When first 4 numbers are written one blow the other using dots, then they form a perfect triangle.

famous five' equation connecting the five most important numbers in mathematics, 0, 1, e, pi, and i:
e^(i*pi) + 1 = 0.

No comments: