Properties of numbers : Number Theory #Properties #of #numbers : #Number #Theory
So far we have seen about the historical background to Number Theory.
Now we enter into subject starting with the basic definitions & properties of Numbers.
Contents :
* The Principle of Induction
* The well - Ordering Principle
* Divisibility
* Properties of Divisibility
* Greatest Common Divisor (gcd)
* properties of gcd
The Principle of Induction :
If Z is the set of all integers such that
i) 1 𝜖 Z
ii) n 𝜖 Z implies n+1 𝜖 Z
then
iii) all integers ≥ 1 belong to Z.
In another manner
The principle of induction is useful to define a statement p(n) is exists for all integers n
which are to be proved in the following steps.
i) We have to prove P(1) is true i.e. the statement is true for n=1
ii) Assume P(k) is true i.e. the statement is true for n=k
iii) Again we have to prove the statement is true for n-k+1.
If all the above conditions are exists for the statement P(n) , then we say the statement P(n) is true for all n 𝜖 Z.
For Example :
Let the statement be P(n) is 1 + 2 +3 + ... + n = n (n+1 ) /2.
i) Put n=1.
Then R.H.S. = n(n+1) /2
= 1 ( 1+1)/2
= 2/2
=1
= L.H.S.
∴ The statement is true for n=1.
ii) Assume the statement P(n) is true for n=k for k 𝜖 N.
i.e. 1 + 2 + 3 +... + k = k ( k+1 ) / 2.
iii) Now L.H.S. = 1 + 2 + 3 + ... + (k+1) = 1 + 2 + 3 + ... + k + (k+1)
= k ( k+1 ) / 2 + ( k + 1 ) | ∵ from ii) |
= [k ( k+1 )+ 2 ( k+1 ) ] / 2 | By taking l.c.m |
= (k+1 ) ( k+2 ) /2 | by taking (k+1) common |
= ( k+1 ) (( k+1)+1) /2
= R.H.S.
∴ The statement P(n) is true for n = k+1.
By mathematical induction on n the statement P(n) is true forall n 𝜖 N.
i.e. 1 + 2 + 3 + ... + n = n(n+1)/2 ∀ n 𝜖 N.
The Well- Ordering Principle :
If A is a non-empty set of positive integers, then A contains a smallest member.
For example : The set A = { odd positive integers }
= { 1,3,5,7,9,11,...} has the smallest member 1.
Divisibility :
We say d divides n and we write d/n whenever n = cd for some c. We also say that n is a multiple of d, that d is a divisor of n, or that d is a factor of n. If d does not divide n we write d ∤ n.
Divisibility establishes a relation between any two integers with the follwoing elementary properties.
Properties of Divisibility :
1. n/n ( reflexive property )
2. d/n and n/m implies d/m ( transitive property 0
3. d/n and d/m implies d/(an+bm) ( linearity property )
4. d/n implies ad/an ( multiplication property )
5. ad/an and a ≠ 0 implies d/n ( cancellation law )
6. 1/n ( 1 divides every integer )
7. n/0 ( every integer divides 0 )
8. 0/n implies n=0 ( 0 divides only 0 )
9. d/n and n ≠ 0 implies |d| ≤ |n| ( comparison property )
10. d/n and n/d implies |d| = |n|
11. d/n and d ≠ 0 implies (n/d)/n
Note : If d/n then n/d is called the divisor conjugate to d.
Greatest Common Divisor :
if d divides two integers a and b , then d is called a common divisor of a and b.
Thus , i is a common divisor of every pair of integers a and b.
Result 1. Given any two integers a and b, there is a common divisor d of aand b of the form
d = ax + by
i.e. every pair of integers a and b has a common divisor which can be expressed as a linear
combination of a and b.
Result 2 : Given integers aand b, there is one and only one number d with the following properties.
i) d ≥ 0 ( d is non negative)
ii) d/a and d/b ( d is a common divisor of a and b )
iii) e/a and e/b implies e/d ( every common divisor divides d )
Note : 1. Here the number d is called the greatest common divisor (gcd) of a and b and is deonoted by (a,b) or aDb.
2. if (a,b) = 1 then a and b are called relatively prime integers.
properties of gcd :
i) (a,b) = (b,a) ( commutative law )
ii) (a,(b,c)) = ((a,b),c) ( associative law )
iii) (ac,bc) = |c|(a,b) ( distributive law )
iv) (a,1) = (1,a) = 1
(a,0) = (0,a) = |a|
#Properties #of #numbers : #Number #Theory
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