An irrational number cannot be expressed as p/q where p and q are integers and q ≠ 0. To prove that √3 is irrational, we use the method of proof by contradiction — we assume √3 is rational and show that this leads to a contradiction. This is a standard Class 10 CBSE Maths proof (Chapter 1: Real Numbers).
√3 is irrational — it cannot be expressed as p/q where p and q are coprime integers.
Proof method: Proof by contradiction (assume rational, derive contradiction).
Key step: If 3 divides p², then 3 divides p (prime number property).
The contradiction: both p and q become divisible by 3, violating HCF(p,q) = 1.
√3 = 1.7320508... — a non-terminating, non-repeating decimal.
This proof is the same structure used for √2, √5, √7 — all irrational.
Theorem: √3 is an irrational number. Method: Proof by contradiction
Step 1 — Assume the opposite: Let us assume that √3 is a rational number. Then √3 = p/q, where p and q are integers, q ≠ 0, and p and q have no common factor (i.e., p/q is in its simplest form — HCF(p, q) = 1).
Step 2 — Square both sides: √3 = p/q 3 = p²/q² p² = 3q² ... (i)
Step 3 — Conclude that p is divisible by 3: From (i), p² is divisible by 3. By Theorem: If a prime p divides a², then p divides a. Since 3 divides p², therefore 3 divides p. So let p = 3k for some integer k. ... (ii)
Step 4 — Substitute back: Substituting (ii) into (i): (3k)² = 3q² 9k² = 3q² q² = 3k² ... (iii)
Step 5 — Conclude that q is divisible by 3: From (iii), q² is divisible by 3. Therefore, 3 divides q.
Step 6 — Contradiction: Both p and q are divisible by 3. But this contradicts our assumption that p and q have no common factor (HCF = 1).
Step 7 — Conclusion: Our assumption that √3 is rational is wrong. Therefore, √3 is an irrational number. □
Theorem (used in Step 3): If p is a prime number and p divides a², then p divides a.
This is called the 'Fundamental Theorem of Arithmetic' corollary.
In our proof: • 3 is a prime number • 3 divides p² → 3 divides p
This theorem is the key step that makes the proof work.
A number is irrational if it CANNOT be expressed in the form p/q where: • p and q are integers • q ≠ 0 • HCF(p, q) = 1 (no common factor)
Examples of irrational numbers: • √2, √3, √5, √7 (square roots of non-perfect squares) • π (pi) = 3.14159... • e (Euler's number) = 2.71828... • √3 = 1.7320508... (non-terminating, non-repeating decimal)
Proof by contradiction: Assume √3 = p/q (lowest terms, HCF(p,q)=1). Then 3 = p²/q², so p² = 3q². This means 3 divides p², so 3 divides p. Let p = 3k. Then 9k² = 3q², so q² = 3k², meaning 3 divides q. But then both p and q are divisible by 3, contradicting HCF(p,q) = 1. So √3 is irrational.
√3 is irrational. It cannot be expressed as a fraction p/q of two integers. Its decimal expansion is 1.7320508... which is non-terminating and non-repeating — a characteristic of irrational numbers.
The proof uses the theorem: 'If a prime p divides a², then p divides a.' This is a consequence of the Fundamental Theorem of Arithmetic. In the proof, since 3 is prime and 3 divides p², we conclude 3 divides p.
1 Foot Mein Kitne Centimeter (CM) Hote Hai?
Janiye 1 foot me kitne cm hote hai. Learn the exact value of 1 foot in centimeters and how to convert feet to cm easily.
1 Inch Mein Kitne Centimeter Hote Hai?
Janiye 1 inch me kitne cm hote hai. Accurate mathematical conversion of inches to centimeters (cm) with examples.
How Many Zeros Are in 1 Lakh?
1 lakh = 1,00,000 and has 5 zeros. It equals 100,000 in the international system. Learn the Indian number system with lakhs, crores and zeros.
1 Metre = How Many Feet and Centimetres?
Learn 1 metre in feet and centimetres. 1 m = 3.281 feet and 1 m = 100 cm. Complete length conversion table for Class 6 Maths and everyday use.
1 Metre Mein Kitne Foot Hote Hain?
1 metre = 3.281 feet. 1 foot = 0.3048 metres. Learn the formula to convert metres to feet and feet to metres with examples and a conversion table.
Turn this guide into revision flashcards, a practice exam, or an AI-generated podcast — free, no signup required.