 # NCERT Solutions for Class 10 Maths Exercise 1.3 Chapter 1 Real Numbers  View Notes

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

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

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