Open App

Class 10 Exam > Class 10 Videos > Mathematics (Maths) Class 10 > How to prove an irrational number?

How to prove an irrational number? Video Lecture | Mathematics (Maths) Class 10

FAQs on How to prove an irrational number? Video Lecture - Mathematics (Maths) Class 10

1. What is an irrational number?

Ans. An irrational number is a type of real number that cannot be expressed as a fraction of two integers. This means that their decimal representation is non-terminating and non-repeating. Common examples include numbers like √2, π (pi), and e (the base of natural logarithms).

2. How can I prove that √2 is irrational?

Ans. To prove that √2 is irrational, we can use a proof by contradiction. Assume that √2 is rational, meaning it can be expressed as a fraction a/b, where a and b are integers with no common factors (in simplest form). Then, squaring both sides gives us 2 = a²/b², or a² = 2b². This implies that a² is even, thus a must be even (since the square of an odd number is odd). If a is even, we can express it as a = 2k for some integer k. Substituting this back gives 2k² = b², which means b² is also even, thus b must be even. This contradicts our assumption that a and b have no common factors, proving that √2 is indeed irrational.

3. Are all square roots of positive integers irrational?

Ans. No, not all square roots of positive integers are irrational. The square root of a perfect square, such as 1, 4, or 9, is an integer and thus rational. However, the square roots of non-perfect squares, such as 2, 3, or 5, are irrational.

4. Can you provide more examples of irrational numbers?

Ans. Yes, besides √2, other common examples of irrational numbers include π (approximately 3.14159), e (approximately 2.71828), and the golden ratio (approximately 1.61803). All these numbers cannot be expressed as a simple fraction and have non-terminating, non-repeating decimal expansions.

5. Why is it important to understand irrational numbers?

Ans. Understanding irrational numbers is crucial because they are fundamental in various fields of mathematics, science, and engineering. They help in modeling real-world situations where measurements cannot always be expressed as simple fractions. Additionally, they play a vital role in advanced mathematics, including calculus and number theory, contributing to a deeper understanding of the number system.

About this Video

4.75/5 Rating

Oct 02, 2025 Last updated

Related Exams

Class 10 Grade 10

Video Description: How to prove an irrational number? for Class 10 2025 is part of Mathematics (Maths) Class 10 preparation. The notes and questions for How to prove an irrational number? have been prepared according to the Class 10 exam syllabus. Information about How to prove an irrational number? covers all important topics for Class 10 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for How to prove an irrational number?.

Introduction of How to prove an irrational number? in English is available as part of our Mathematics (Maths) Class 10 for Class 10 & How to prove an irrational number? in Hindi for Mathematics (Maths) Class 10 course. Download more important topics related with notes, lectures and mock test series for Class 10 Exam by signing up for free.

Description

Video Lecture & Questions for How to prove an irrational number? Video Lecture | Mathematics (Maths) Class 10 - Class 10 full syllabus preparation | Free video for Class 10 exam to prepare for Mathematics (Maths) Class 10.

Information about How to prove an irrational number?

Here you can find the meaning of How to prove an irrational number? defined & explained in the simplest way possible. Besides explaining types of How to prove an irrational number? theory, EduRev gives you an ample number of questions to practice How to prove an irrational number? tests, examples and also practice Class 10 tests.

	Mathematics (Maths) Class 10 127 videos\|584 docs\|79 tests

Mathematics (Maths) Class 10

127 videos|584 docs|79 tests

Join Course for Free

Explore Courses for Class 10 exam

video lectures