Number theory : The set of it integers and its properties are at the root of all mathematical disciplines. In fact , it is impossible to do mathematics without making use of integers in some form or another. Number theory which involves the study of integers itself , is a rich and fascinating branch o mathematics .many volume have been written on this subject and some of the best mathematician in history have devoted much of their time to the study of number theory.
We can subdivide the number theory as follow :
(i) Combinatorial Number Theory
(ii) Algebraic Number Theory
(iii) Analytic Number Theory
(iv) Transcendental number theory
(v) Geometric number theory
(vi) Computational number theory
What is a Natural Number?
Introduction To natural number : Since our childhood we are using numbers 1 , 2, 3, 4,………………………..t count and calculate. For example 3 banana , 5 apples , 7 mangoes , 2 books etc. here banana , apples , mangoes are objects whereas three , five , seven , two etc indicates about the quantities of theses objects .
To define natural numbers we might put these in this way, as when we count objects in groups of objects , we start counting from one and then go on to two, three , four etc . there is a natural way of counting objects .Hence 1 , 2 , 3 , 4 , …………………are called natural numbers .In fact number from 1 to 1crore are all natural numbers.Let us see what is a natural numbers ?We start counting from1 , so 1 is the first natural number , if we add1 to the first natural number , then we get 2 the second natural number/.by adding 1 to any natural number , then we get 2 , the second natural number. In fact adding 1 to any natural number , we get the next natural number let us take few examples of natural numbers , 1000 is the natural number next to 999 , 10001 is the natural number, next to 10000 and so on .Thus if we think of any natural number , there is always a natural number next to it .Consequently there is no last or greatest natural number consequently there is no last or greatest natural number. now in simple word , let us define natural numbers: natural numbers are number from 1 onwards, ie , 1 , 2 3 4 , 5 , 6 ……………………..and are used for counting
Properties of natural numbers :Following are some properties of natural number (i) The first and smallest natural number is 1.(ii) Every natural number (except 1) can be obtained by adding to 1 to the previous natural number(iii)For the natural number 1, there is no previous natural number (iv)There is no last or greatest natural number (v)We cannot complete the counting of all natural number .
