Learn how to show that √2 is an irrational number with a simple and complete proof. In this explanation, we use the contradiction method to prove that root 2 is not a rational number. According to this method, if √2 were rational, it could be expressed as a fraction p/q, where p and q have no common factors. However, through logical steps, we find a contradiction, proving that √2 is indeed irrational.
This topic is important in number theory and frequently appears in Class 9 and Class 10 maths. Our easy-to-understand approach is ideal for all students — whether you’re preparing for board exams or entrance tests. Learn the proper steps to solve the root 2 irrational number proof and build a strong foundation in mathematics.
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FAQ: Show That √2 Is an Irrational Number
Q1. What does it mean that √2 is an irrational number?
A: It means that √2 cannot be expressed as a simple fraction like p/q, where p and q are integers and q ≠ 0. Its decimal form goes on forever without repeating.
Q2. How do you prove that √2 is irrational?
A: To show that √2 is an irrational number, we use the contradiction method. We assume √2 is rational (can be written as p/q), and then prove this leads to a contradiction — hence, √2 is irrational.
Q3. Why is the contradiction method used in the proof of √2?
A: It’s used because it helps logically disprove the assumption that √2 is rational, proving instead that √2 is not a rational number.
Q4. What are some other irrational numbers like √2?
A: Examples include √3, √5, π (pi), and e. These numbers cannot be written as exact fractions.
Q5. Where is this proof used in academics?
A: This topic is covered in Class 9 and Class 10 Maths under real numbers and is commonly asked in board exams and competitive tests.
Q6. Is √2 a recurring decimal?
A: No. √2 = 1.4142135… is a non-terminating, non-repeating decimal, which is a key trait of irrational numbers.
Q7. Is root 2 the only irrational square root?
A: No. The square roots of non-perfect squares (like √3, √7, √11) are all irrational.
Q8. Can √2 ever be written exactly?
A: No. It can only be approximated as a decimal. The exact value of √2 is irrational and infinite in length.
Q9. Why is √2 important in mathematics?
A: √2 is a historical example of irrational numbers and is used in geometry, algebra, and trigonometry.
Q10. Who provides the best tutoring for this topic in Delhi?
A: Amit Home Tuition is Delhi’s leading home tutor and private tuition provider, offering clear explanations and personalized learning in maths and more.
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