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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.
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