X mathematics concept chapter -1

Euclid’s Division Lemma

For three positive integers a, b  there exists a unique integer q and r such that a = bq + r  and here value of  r will always less then b

That means if we divide number a by b and  q is our quotient and r is remainder then value of remainder will always less then deviser b .

For example 5 and 19

If we divide 22 by 5 we get 4 as quotient and 2 is remainder

So we can write it as  22= 5*4+ 2

And you can see that value of remainder is less then deviser

Composite number

Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

The Fundamental Theorem of Arithmetic

If we ignore the order, any number which is more than 1, is either a prime number or can be written as unique product of prime number  .

Or

Every composite number can be written in term of product of unique set of prime numbers ignoring their

For example

26 is a composite number and it can be written as 2*13 according to fundamental theorem you cannot get another set of value which product is 26.

So   if we ignore the order of factors, prime factorization of all natural number is always unique.

Theorem of rational number: - for any prime number p which is divides a^2 then it will divide a also.

For example  5 divides 100 then it will divide root of 100( 10) also.

Theorem of terminator:-  for any rational number x which is written as p/q   and if q can be written in form of 2^n*p^m  and value of m and n are positive integer or equal to 0  then  decimal expansion of x will terminate.

For exampletake  x = 7/50

And 50 = 2*5*5 = 2^1*5^2

And powers 1 and 2 both are positive integer So value of x will terminate

Theorem of non-terminating repeating (recurring) :-  for any rational number x which is written as p/q   and if q ca not be  written in form of 2^n*p^m  and value of m and n are positive integer or equal to 0  then  decimal expansion of x will non-terminating repeating (recurring).

For example

X = 7/15

Here denominator 15 = 3*5

And it cannot be written in form of 2^n*5^m

x =0.4666……………

So x will non-terminating repeating (recurring) .