In view of the coronavirus pandemic, we are making LIVE CLASSES and VIDEO CLASSES completely FREE to prevent interruption in studies

Let us learn some important definitions before we get into the concept of real numbers Class 10:

  • A natural number is a counting number. Hence a set of natural number can be shown as N = {1, 2, 3, . .}
  • A whole number is all the natural numbers including 0. Therefore a set of whole numbers is W = {0, 1, 2, ..}
  • An integer is a set of all positive and negative whole numbers. The set of integers can be written as Z = { -4, -3, -2, -1, 0, 1, 2, . . }. The natural numbers (excluding zero) are known as positive integers. When a number gives zero on being added to its corresponding positive value is known as a negative integer. When we consider zero along with the natural numbers, we call it non-negative integers.
  • A rational number is any number that can be expressed in p / q form where both the numerator and denominator are integers and the value of q is positive. This includes all integers, natural numbers and rational number. Any two rational numbers has infinite rational numbers between them. The rational number can either be terminating decimal or non-terminating decimal, which can again be recurring or non-recurring in nature. There are certain operations of rational numbers that need to be accounted for. The sum of two rational numbers is a rational number. The same hold for difference and product too. This may or may not be true for division.
  • An irrational number is a number that cannot be expressed in p / q format. We can represent the set of irrational numbers on a number line with the help of Pythagoras theorem.

What are real numbers?

A collection of rational numbers and irrational numbers make up the set of real number. A real number can be expressed on the number line and has some specific properties. They satisfy:

  • The commutative law of addition. That is, when a and b are two real numbers then a + b = b + a. For example 1 + 3 = 3 + 1 = 4
  • The commutative law of multiplication. That is, when a and b are two real numbers then a x b = b x a. For example 1 x 3 = 3 x 1 = 3
  • The associative law of addition. That is, when a, b and c are three real numbers then a + (b + c) = (a + b) + c. For example 1 + (3 + 4) = (1 + 3) + 4 = 8
  • The associative law of multiplication. That is, when a, b and c are three real numbers then a x (b x c) = (a x b) x c. For example, 1 x (3 x 4) = (1 x 3) x 4 = 12
  • The law of distribution. That is, when a, b, c are three real numbers then

a x (b +c) = (a x b) + (a x c). For example 1 x (3 + 4) = (1 x 3) + (1 x 4)

= 7

There are some laws of exponents as well that is demonstrated by real numbers. They are:

  1. ap x aq = a (p+q)
  2. ap / aq = a(p-q)
  3. (ap)q = apq
  4. ap x bp = (ab)p

Here a and b, both are real numbers.

What is Euclid’s division algorithm?

There is a value q and r for every set of positive integer such that

a = bq +r, where 0 ≤ r < b.

There are specific names for each term, a is dividend, b is divisor, q is quotient and r is remainder. It is a method used to find the highest common factor.

For example, when we need to use the Euclid’s algorithm to find the HCF of 135 and 225.

Since 225 > 135, we can write 225 = 135 x 1 +90

Since 90>0, the remainder is 45. We apply the division rule again to get
135 = 90 x 1 +45

We then consider 90 to be the new divisor with the remainder as 45 and on applying the algorithm again we get, 90 = 2 x 45 + 0. Here, since the remainder is 0, the HCF = 45, which is the divisor at this particular stage.

Some solved questions

  1. Prove that √2 is irrational number

Answer: Let us assume that it is rational number

Then, √2=p/q

Where, p and q are co-primes.

Or, q√2=p

Squaring both sides, we get


Hence, 2 divide p2.

From theorem we know that,

2 will divide p also. p = 2c, where c is any constant


or q2=2c2

So q is divisible by 2 also

So 2 divide both p and q, which contradicts our assumption.

So √2 is an irrational number.

  1. Show that a positive odd integer is in either of the following forms: 4q+1 or 4q+3 where q is an integer.

Answer: let a be any positive off integer. On applying the division algorithm with a, b = 4, we get that the remainder can be 0, 1, 2, or 3.

So, a can be 4q, 4q+1, 4q+2 or 4q+3 where the quotient is q.

Since a is odd, it cannot be 4q or 4q+2 (since both are divisible by 2)
hence, an odd integer is always of form 4q+1 or 4q+3.

  1. Express 140 and 5005 as a product of its prime factor:


140 = 2 x 2 x 5 x 7 = 22 x 5 x 7

5005 = 5 x 7 x 11 x 13

  1. Find the LCM and HCF of 26 and 91

Answer: 26 = 2 x 13

91 = 7 x 13

HCF = 13

LCM = 2 x 7 x 13 = 182

Product of 2 numbers = 2366

HCF X LCM = 2366

Hence, proved

To know how to improve your Maths Scores visit here!

Tired of hunting for solutions? We have it all

Access 300,000+ questions with solutions curated by top rankers.

No thanks.

Request a Free 60 minute counselling session at your home

Please enter a valid phone number
  • Happy Students


    Happy Students
  • Questions Attempted


    Questions Attempted
  • Tests


    Tests Taken
  • Doubts Answered


    Doubts Answered