सिद्ध कीजिए कि 5 एक अपरिमेय संख्या है Prove That 5 is Irrational Number Class 10th NCERT Maths

Understanding Why √5 Is an Irrational Number

In CBSE Class 10 Mathematics, the chapter on Real Numbers introduces students to the fundamental classification of numbers into rational and irrational categories. One of the most important and frequently asked proofs in board examinations is establishing that √5 is an irrational number. This proof relies on the method of contradiction, a powerful logical technique where we assume the opposite of what we want to prove, and then demonstrate that this assumption leads to an impossible or contradictory result. Understanding this proof thoroughly helps students build strong logical reasoning skills and prepares them for similar proofs involving other irrational numbers.

The proof begins by assuming the opposite — that √5 is a rational number. If √5 were rational, it could be expressed in the form p/q, where p and q are coprime integers (having no common factors other than 1) and q is not equal to zero. So, we write √5 = p/q. Squaring both sides gives us 5 = p²/q², which means p² = 5q². Since p² is divisible by 5, and 5 is a prime number, it follows that p itself must be divisible by 5 (this is based on the theorem that if a prime number divides a perfect square, it must also divide the original number). Therefore, we can write p = 5a for some integer a. Substituting this back into the equation p² = 5q² gives us (5a)² = 5q², which simplifies to 25a² = 5q², and further to q² = 5a². This means q² is also divisible by 5, and by the same logic, q must be divisible by 5.

Here lies the contradiction: we have now shown that both p and q are divisible by 5, meaning they share a common factor of 5. However, our initial assumption stated that p and q are coprime — they should have no common factors other than 1. This contradiction means our original assumption that √5 is rational must be false. Therefore, we conclude that √5 is indeed an irrational number. This elegant proof beautifully demonstrates how contradiction can be used to establish mathematical truths with absolute certainty.

• √5 cannot be expressed as a fraction p/q where p and q are integers with no common factors, proving it is irrational.
• The proof uses the contradiction method — assume the opposite and show it leads to a logical impossibility.
• A key theorem used: if a prime number divides a perfect square (like p²), then that prime must also divide the base number (p).
• The contradiction arises when both p and q turn out to be divisible by 5, violating the condition that they are coprime.
• This same method can be applied to prove that √2, √3, √7, and other square roots of non-perfect squares are irrational.
• Remember: the rationality or irrationality of numbers is a key topic in the Real Numbers chapter and frequently appears in CBSE board exams as a 2-mark or 3-mark question.

This proof is part of the Real Numbers chapter in the NCERT Class 10 Mathematics textbook and is directly connected to the Fundamental Theorem of Arithmetic, which states that every composite number can be uniquely expressed as a product of prime numbers. Mastering this proof not only helps you score well in examinations but also strengthens your understanding of number theory concepts that form the foundation for higher mathematics, including algebra and real analysis.