Historical Introduction to Number Theory - Part 1
Historical Introduction to Number Theory : Part 1
The theory of numbers is that branch of mathematics which deals with properties of the whole numbers 1,2,3,... also called the counting numbers or positive integers.
The positive integers are undoubteldly man's first mathematical creation. It is hardly possible to imagine human beings without the ability to count, at least within a limited range. Historical record shows that as early as 5700 BC the ancient sumerians kept a calender, so they must have developed some form of arithmetic.
By 2500 BC of Sumerians had developed a number system using 60 as a base. This was passed on to Babylonians, who became highly skilled calculators. Babylonians clay tablets containing elaborate mathematical tablets have been found, dating back to 2000 BC.
When ancient civilizations reached a level which provided leisure time to ponder about things , some people began to speculate about the nature and properties of numbers. This curiosity developed into a sort of number mysticism or numerology, and even today numbers such as 3,7,11,and 13 are considered omens of good or bad luck.
Numbers were used for keeping records and for commercial transactions for over 5000 years before anyone thought of studying numbers themselves in a systematic way. The first scientific approach to the study of integers, that is, the true origin of the theory of numbers, is generally attributed to the Greeks. Aroound 600 BC Pythagoras and his disciples made rather thorough studies of the integers. They were the first to classify integers in various ways.
Even numbers : 2,4,6,8,10,...
Odd numbers : 1,3,5,7,9,11,...
Prime numbers : 2,3,5,7,11,13,...
Composite numbers : 4,6,8,9,10,12,14,...
A prime number is number greater than 1whose divisors are 1 and the number itself. Numbers that are not prime numbers are called the composite numbers except 1 is considered neither prime nor composite.
The Pythogoreans also linked numbers with geometry. They introduced the idea of polygonal numbers : triangular numbers, square numbers, pentagonal numbers , etc.
Another link with geometry came from the famous theorem of pythagoras wihch states that in any right angled triangle the sum of the squares lenghts of the two sides is equal to the square of lenght of hypotenuse. The Pythagoreans were interested in right triangle s whose sides are integers. Such triangles are also called Pythagorean triangles. The corresponding triple of numbers { a,b,c } representing the lengths of the sides is called the pythagorean triple.
A Babylonian tablet has been found , dating from about 1700 BC, which contains an extensive list of pythogorean triples, some of the numbers being quite large. The Pythogoreans were the first to give a method for determining infinitely many triples. In modern notation it can be described as follows. Let n be any odd number greater than 1, and let x=n, y= 1/2 ( n2 -1) , z= 1/2 ( n2 +1).
The resulting triple {a,b,c } will always be a Pythagorean triple with z=y+1.
**Historical Introduction to Number Theory : Part 2 **
Nice research
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