Real Numbers | Class 10 Mathematics Notes

1.3 Revisiting Irrational Numbers

This section revisits irrational numbers, which were introduced in Class IX, and provides rigorous proofs of the irrationality of certain numbers such as √2, √3, and √p where p is a prime number. It begins by defining irrational numbers as those that cannot be expressed as a ratio of two integers. The section introduces Theorem 1.2, which states that if a prime p divides a², then p divides a. This theorem, based on the Fundamental Theorem of Arithmetic, is critical for the proofs that follow. The irrationality of √2 is proved using proof by contradiction: assuming √2 is rational leads to a contradiction that both numerator and denominator share a factor 2, violating their coprimality. A similar proof is given for √3. The section also discusses properties of irrational numbers, such as the sum or difference of a rational and an irrational number being irrational, and the product or quotient of a non-zero rational and an irrational number being irrational. Examples demonstrate these properties, such as proving 5 - √3 and 3√2 are irrational by contradiction. The section reinforces the importance of the Fundamental Theorem of Arithmetic in understanding the nature of irrational numbers and their behavior under arithmetic operations.

📊 Diagram: No specific diagrams in this section.

🧪 Activity: Exercises include proving irrationality of √5, and proving certain expressions involving irrationals are irrational.

🔗 Connection: Leads to Section 1.4 Summary, consolidating the chapter's key concepts and results.