## CBSE Class 10 Maths Chapter 1 – Real Numbers – Free PDF Download

Free PDF download of Class 10 Maths Chapter 1 – Real Numbers Revision Notes & Short Key-notes prepared by expert Mathematics teachers from latest edition of CBSE(NCERT) books.

You can also Download __Maths NCERT Solutions Class 10__ to help you to revise complete Syllabus and score more marks in your examinations.

**CBSE Class 10 Maths Revision Notes Chapter 1 Real Numbers**

**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.

1. **Euclid’s division lemma :**

For given positive integers ‘a’ and ‘b’ there exist unique whole numbers ‘q’ and ‘r’ satisfying the relation .

Theorem: If and are non-zero integers, the least positive integer which is expressible as a linear combination of and is the HCF of and , i.e., if is the HCF of and , then these exist integers and , such that and is the smallest positive integer which is expressible in this form.

The HCF of and is denoted by HCF.

2. **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

b = Divisor

q = Quotient

r = Remainder

**Step II**: If r = 0, HCF (a,b)=b if , 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.

3. **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.

4. Let 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.

5. Let be a rational number, such that the prime factorization of 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.

6. 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

8. If a and b are two positive integers, then HCF(a, b) x LCM(a, b) = a x b

i.e., (HCF x LCM) of two intergers = Product of intergers.

9. 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 where m and n are natural numbers, then has a terminating decimal expansion.

10. We conclude that every rational number can be represented in the form of terminating or non-terminating recurring decimal.