Euclid’s division algorithm as the name suggest, has to do with divisibility of integers. It says that any positive integer ‘a’ can be divided by another positive integer ‘b’ in such a way that it leaves a remainder ‘r’ that is smaller than b. This result has many applications related to the divisibility properties of integers. It is used mainly to complete the HCF of two positive integers.
Fundamental theorem of arithmetic
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 says that every composite number can be factorized as a product of primes. It says that given any composite number it can factorized as a product of prime numbers in a ‘unique’ way, except for the order in which the primes occur.
Irrational numbers : The numbers which cannot be expressed in decimal form either in terminating or in repeating decimals, are known as irrational numbers. An irrational number cannot be written in the form of p/q , where ‘p’ and ‘q’ are both integers and q is not equal to 0.
Let ‘P’ be a prime number. If ‘P’ divides a2, then ‘P’ divides a, where a is a positive integers.
Real numbers : Rational numbers and irrational numbers taken together are known as real numbers. Hence we can say that any real number is either rational number or an irrational number.
1. Let x be a rational number then x can be expressed in the form p/q , where p and q are co-prime and the prime factorization of q is in the form of 2n 5m, where n and m are non-negative integers.
2. Any rational number of the form a/b, where b is a power of 10, will have a terminating decimal rational number of the form p/q, where q is of the form 2n 5m to an equivalent rational number of the form a/b where b is a power of 10.
3. x = p/q be a rational number, such that the prime factorization of q is of the form 2n 5m, where n and m are non-negative integer. Then x has a decimal expression which terminates.
4. x = p/q be a rational number, such that the prime factorization of q is not of the form 2n 5m where n and m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring)