# NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers (Ex 1.3) Exercise 1.3

## NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers (Ex 1.3) Exercise 1.3

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Q1: Which aspect of Real number does Class 10 Maths Chapter 1 exercise 1.3 cover?

A1: Exercise 1.3 of NCERT solutions for class 10 maths chapter 1 Real Number is the third exercise of Chapter 1 of class 10 Maths. Real Numbers is introduced in class 9 and this is discussed more in details in class 10. It is crucial to have a fair knowledge of the topic – irrational numbers to understand these solutions. The exercise discusses how to prove that root p is irrational.

Revisiting Irrational Numbers – It includes 3 questions based on the theorem where question no – 3 has 3 roots to be proved as irrational.

Q2: What do you mean by irrational numbers?

A2: An irrational number is a real number that cannot be expressed as a ratio of integers, for example, √ 2 is an irrational number. Again, the decimal expansion of an irrational number is neither terminating nor recurring. How do you know a number is irrational? The real numbers which cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0 are known as irrational numbers. For Example √ 2 and √ 3 etc. Whereas any number which can be represented in the form of p/q, such that, p and q are integers and q ≠ 0 is known as a rational number.

Q3: What are the properties of an irrational number?

A3: The following are the properties of rational numbers:

The addition of an irrational number and a rational number gives an irrational number. For example, let us assume that x is an irrational number, y is a rational number, and the addition of both the numbers x +y gives a rational number z.

While Multiplying any irrational number with any nonzero rational number results in an irrational number. Let us assume that if xy=z is rational, then x =z/y is rational, contradicting the assumption that x is irrational. Thus, the product xy must be irrational.

The least common multiple (LCM) of any two irrational numbers may or may not exist.

The addition or the multiplication of two irrational numbers may be rational; for example, √2. √2 = 2. Here, √2 is an irrational number. If it is multiplied twice, then the final product obtained is a rational number. (i.e) 2.

The set of irrational numbers is not closed under the multiplication process, unlike the set of rational numbers.

Q4: Mention a few popular irrational numbers that are extensively used in Maths.

A4: Few popular irrational numbers used extensively in Maths is as follows:

Pi, π - 3.14159265358979…

Euler’s Number, e - 2.71828182845904…

Golden ratio, φ - 1.61803398874989….

√3 - 1.7320508075688772935274463415059

√99 - 9.9498743710661995473447982100121