## NCERT Solutions For Class 10 Maths Real Numbers

### Real Numbers CBSE Class 10 Maths Chapter 1 Solutions

NCERT Solutions For Class 10 Maths Real Numbers Exercise 1.1    NCERT Solutions For Class 10 Maths Real Numbers Exercise 1.2  “>      NCERT Solutions For Class 10 Maths Real Numbers Exercise 1.3    NCERT Solutions For Class 10 Maths Real Numbers Exercise 1.4     Real Numbers:
Rational numbers and irrational numbers taken together form the set of real numbers. The set of real numbers is denoted by R. Thus every real number is either a rational number or an irrational number. In either case, it has a nonâ€“terminating decimal representation. In case of rational numbers, the decimal representation is repeating (including repeating zeroes) and if the decimal representation is nonâ€“repeating, it is an irrational number. For every real number, there corresponds a unique point on the number line â€˜l’ or we may say that every point on the line â€˜l’ corresponds to a real number (rational or irrational).

From the above discussion we may conclude that:
To every real number there corresponds a unique point on the number line and conversely, to every point on the number line there corresponds a real number. Thus we see that there is oneâ€“toâ€“one correspondence between the real numbers and points on the number line â€˜l’, that is why the number line is called the â€˜real number line’.

Objectives:
The students will be able to ;
prove Euclid’s Division Lemma
state fundamental theorem of arithmetic
find HCF and LCM using prime factorisation
establish the given number as an irrational number
conclude the decimal expansion of a rational number is either terminating or non-terminating repeating.

Summary:

We have studied the following points:
1. Euclid’s Division Lemma : Given positive integers a and b, there exist whole numbers q and r satisfying a = bq + r where 0 = r = b.
2. Euclid’s Division Algorithm: According to this, which is based on Euclid’s division lemma, the HCF of any two positive integers a and b with a > b is obtained as follows:
Step 1 Apply the division lemma to find q and r where a = bq + r, O = r < b.
Step 2 If r = 0, the HCF is b . If r ? 0 apply Euclid Lemma to b and r
Step 3 Continue the process till the remainder is zero. The divisor at this stage will be HCF (a, b). Also HCF (a, b) = HCF (b, r)
3. The Fundamental Theorem of Arithmetic: Every composite number can be expressed (factorised) as a product of primes and this factorisation is unique, apart from the order in which the prime factors occur.