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Mathematics · Real Numbers · NCERT Examples

Example 5

A complete, independently verified solution for NCERT Example 5.

Mathematics · Chapter 1 · NCERT Examples

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NCERT Example

Example 5

irrational numbers · proof by contradiction

Prove that √3 is irrational.

Solution

Suppose √3 is rational and write it in lowest terms.

3=pq,gcd⁡(p,q)=1,q≠0\sqrt3=\frac pq,\qquad \gcd(p,q)=1,\quad q\ne0
p2=3q2p^2=3q^2

Thus 3 divides p², so 3 divides p. Put p = 3k. Substitution gives q² = 3k², so 3 also divides q. This contradicts p and q being coprime.

Final answer

Therefore, √3 is irrational.