How to prove an irrational number? Video Lecture - Class 10

Ans. An irrational number is a type of real number that cannot be expressed as a simple fraction. This means that it cannot be written in the form of \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \) is not zero. Examples of irrational numbers include \( \sqrt{2} \), \( \pi \), and \( e \).

Ans. To prove that \( \sqrt{2} \) is irrational, we can use a proof by contradiction. We start by assuming that \( \sqrt{2} \) is rational, meaning it can be expressed as \( \frac{a}{b} \) where \( a \) and \( b \) are integers with no common factors. Squaring both sides, we get \( 2 = \frac{a^2}{b^2} \), which leads to \( a^2 = 2b^2 \). This implies that \( a^2 \) is even, so \( a \) must also be even. If \( a \) is even, we can write \( a = 2k \) for some integer \( k \). Substituting back, we get \( (2k)^2 = 2b^2 \), or \( 4k^2 = 2b^2 \), which simplifies to \( b^2 = 2k^2 \). Therefore, \( b^2 \) is also even, meaning \( b \) is even too. Since both \( a \) and \( b \) are even, they have a common factor of 2, which contradicts our assumption that they have no common factors. Hence, \( \sqrt{2} \) is irrational.

Ans. In addition to \( \sqrt{2} \), other examples of irrational numbers include \( \sqrt{3} \), \( \sqrt{5} \), \( \pi \) (the ratio of a circle's circumference to its diameter), and \( e \) (the base of the natural logarithm). All of these cannot be expressed as fractions of integers.

Ans. Proving that a number is irrational is significant because it helps us understand the nature of numbers and their properties. It shows that not all numbers can be neatly categorized as fractions, which is important in mathematics, especially in fields like algebra and calculus. It also has implications in real-world applications such as physics, engineering, and computer science.

Ans. No, not all square roots are irrational. The square root of a perfect square, such as \( \sqrt{4} = 2 \) or \( \sqrt{9} = 3 \), is a rational number because it can be expressed as a fraction (e.g., \( \frac{2}{1} \)). However, the square root of non-perfect squares, like \( \sqrt{2} \) or \( \sqrt{3} \), are irrational numbers.