• Natural numbers: Counting numbers are called Natural numbers. These numbers are denoted by N = {1, 2, 3, ………}
• Whole numbers: The collection of natural numbers along with 0 is the collection of Whole number and is denoted by W.
• Integers: The collection of natural numbers, their negatives along with the number zero are called Integers. This collection is denoted by Z.
• Rational number: The numbers, which are obtained by dividing two integers, are called Rational numbers. Division by zero is not defined.
• Coprime: If HCF of two numbers is 1, then the two numbers area called relatively prime or coprime.
- Euclid’s division lemma:
For given positive integers ‘a’ and ‘b’ there exist unique whole numbers ‘q’ and ‘r’ satisfying the relation a = bq + r, 0 < r < b.
Theorem: If a and b are non-zero integers, the least positive integer which is expressible as a linear combination of a and b is the HCF of a and b, i.e., if d is the HCF of a and b, then these exist integers x₁ and y₁, such that d = ax₁ +by₁ and d is the smallest positive integer which is expressible in this form.
The HCF of a and b is denoted by HCF (a, b).
- Euclid’s division algorithms :
HCF of any two positive integers a and b With a > b is obtained as follows:
Step 1: Apply Euclid’s division lemma to a and b to find q and r such that
a = bq + r, 0 < r < b.
b = Divisor
q = Quotient
r= Remainder
Step II: If r = 0, HCF (a,b)=b if r = 0, apply Euclid’s lemma to b and r.
Step III: Continue the process till the remainder is zero. The divisor at this stage will be the required HCF. - The 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.
Ex: 24=2x2x2x3=3×2×2×2 - Let x =p/q , q‡0 to be a rational number, such that the prime factorization of ‘q’ is of the form 2m+5n, where m, n are non-negative integers. Then x has a decimal expansion which is terminating.
- Let x =p/q , q‡0 be a rational number, such that the prime factorizationof q is not of the form 2m+5n, where m, n are non-negative integers. Then x has a decimal expansion which is non-terminating repeating.
- √p is irrational, which p is a prime. A number is called irrational if it cannot be written in the form where p and q are integers and q ‡ 0.
- If a and b are two positive integers, then HCF(a, b) x LCM(a, b) = axb
i.e., (HCF x LCM) of two intergers = Product of intergers. - A rational number which when expressed in the lowest term has factors 2 or 5 in the denominator can be written as terminating decimal otherwise a non-terminating recurring decimal. In other words, if the rational number is, such that the prime factorization of b is of form 2.5″, where m and n are natural numbers, then has a terminating decimal expansion.
- We conclude that every rational number can be represented in the form of terminating or non-terminating recurring decimal.