Class 10 Maths for Real Numbers of Chapter 1 Revision Notes

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.