Here you will find NCERT Solution for class 10 Maths Chapter 1 Real Numbers Exercise 1.3
students who are in class 10 and studing NCERT Books can refer these solutions.
In this exercise you will learn how to prove the irrationality of given numbers.
A number is irrational if it cannot be written in the form of p/q where p and q are integers and q ≠ 0 or we can say that the number which is not a rational number then it is a irrational number.
to prove this irrationality we use this theorem-
Let p be a prime number. if p divides a² then p divides a where a is a positive integer.
All the proofs are based on a technique called " Proof by contradiction".
some rules which you have studied in class 9th :
The sum or difference of a rational and an irrational number is irrational number.
The product and quotient of a non zero rational and irrational number is irrational.
Here you will find NCERT Solution for class 10 Maths chapter 1 Real Numbers Exercise 1.2.
Fundamental Theorem of Arithmetic:Every composite number can be expressed (factorised) a product of primes and this factorisation is unique, apart from the order in which the prime factors occur.
In previous classes you have studied that natural number can be written as a product of its prime factors.
Fundamental Theorem of Arithmetic says that every composite number can be factorized as a product of Prime Numbers in a unique way except the order in which the prime number occurs.
In this exercise we also find HCF and LCM which you have learnt in earlier classes.
HCF = product of the smallest power of each common prime factor in the number.
LCM= product of the greater power of each prime factor involved in the number.
We also find LCM and HCF by using property
LCM × HCF = Product of two numbers
Here you will find NCERT Solution for class 10 Maths Chapter 1 Real Numbers Exercise 1.1
students who are in class 10 and studing NCERT Books can refer these solutions.
To do exercise 1.1 you have to understand some concept
1. Real Number- Real number includes all rational and irrational numbers. and we can plot any real number on a number line.
2. Euclid Division Lemma- Lemma means a proper statement used for proving another statement.
and Euclid division Lemma says that For each pair of given integers a and b, there exist unique whole number q and r which satisfies the relation
a = bq + r where 0 ≤ r < b where q and r can also be 0.
Here a is dividend and b is divisor, q is quotient and r is remainder.
So, dividend = (divisor ✕ quotient) + remainder
3. Euclid Division Algorithm- An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.
as the name suggest has to do with divisibility of integers. which means that any positive integer a can divided by another integer b in such a way that it leaves a remainder r that is smaller than b.
Many people usually recognise it as a long division method but we use Euclid Division Algorithm to compute HCF of given two numbers.
to obtain the HCF of two positive integers say a and b with a > b follow these steps
Step 1: Apply Euclid's division lemma to a and b. so, we find whole numbers q and r such that a = bq + r where 0 ≤ r < b.
Step 2: If r = 0, then b is the HCF of a and b. If r ≠ 0, apply the division lemma to d and r.
Step 3 : Continue the process till the remainder is zero. the divisor at this stage will be the required HCF.
Now you can easily understand the exercise 1.1 (real numbers)
Question 1:
Use Euclid's division algorithm to find the HCF of
1. 135 and 225
2. 196 and 38220
3. 867 and 255
Solution:
