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A set of all rational number & irretional number is called real number.

All the arithmetic operations can be performed on real numbers.

It can be represented in the number line.

It can be both positive or negative and are denoted by the symbol “R”.

Set of real numbers consists of:

  1. Natural Numbers: All numbers such as 1, 2, 3, 4, 5, 6,…..…
  2. Whole Numbers: All natural numbers including 0.
  3. Integers : All whole numbers and negative of all natural numbers.
  4. Rational Numbers :Numbers which can be written in the form of p/q, where q≠0 such as 1,3/5 etc
  5. Irrational Numbers : Numbers that are not rational and cannot be written in the form of p/q like √2,√3

Euclid’s Division Lemma

What is Lemma?

A lemma is a proven statement used for proving another statement.

What is Algorithm?

An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.

In Mathematics, lemma represent Dividend.

We know that , Dividend = (Divisor × Quotient) + Remainder.

For example, for two positive numbers 58 and 8, Euclid's division lemma holds true in the form of 58 = (7 × 8) + 2.

Euclid's Division Lemma says, if we have two positive integers a and b, then there would be whole numbers q and r that satisfy the equation: a = bq + r, (where 0 ≤ r < b)


a is called as dividend.

b is called as divisor.

q is called as quotient and r is the remainder

How to find HCF using Euclid’s Algorithm?

Follow the simple and easy procedures to find the HCF using Euclid’s Algorithm of positive integers a, b where a>b.

Step 1: On applying Euclid’s division lemma to integers a and b, we get other two whole numbers q and r such that, a = bq+r ; 0 r < b.

Step 2: If r =0, then b is the HCF of a, b. If r ≠ 0 then apply Euclid’s Division Lemma to b and r.

Step 3: Continue the steps until the remainder is zero (r =0).

Step 4: When r =0, the divisor at this stage is called the HCF of given numbers a and b.

The Fundamental Theorem of Arithmetic

The fundamental theorem of arithmetic states that, every composite number can be expressed (factorised) as the product of primes and this factorisation is unique, apart from the order in which the prime factors occur.

For example, the number 45 can be written in the form of its prime factors as:

45 = 3×3×5 =3 ²X5

Here, 3 and 5 are the prime factors of 45

Revisiting Rational Numbers and Their Decimal Expansions

Any number x is said to be an irrational number if it cannot be expressed in the form of pq where q≠0.

√5, √7, √2 etc. are irrational numbers.

Irrational numbers have non-terminating and non-repeating decimal representation.

Properties of irrestional numbers

The sum or difference of any two irrational numbers is also rational or an irrational number.

The sum or difference of any rational and any irrational number is also an irrational number.

Product of any rational and an irrational number is also an irrational number.

Product of any two irrational numbers is also rational or an irrational number.


If p be a prime number. If p divides a², then p also divides a, where a is a positive integer.

Decimal Representation of Rational Numbers

Any rational number can have two types of decimal representations :

1. Terminating

2. Non-terminating but repeating

How to find a fraction is terminating or repeating without long devision method.

Let we have a the rational number p/q, where q ≠ 0.

Take the prime factorization of denominator (q).

If we can write the denominator (q) in terms of 2^m x 5^n or 2^m or 5^n only , then the rational number p/q will be in the form as shown below and have terminating else repeating .

fraction is terminating or repeating

How to find a position of decimal in terminating decimal number form of any rational number(p/q, whereq≠0 ) without long devision method.


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