Proof of irrationality of sqrt(5)
Assume sqrt(5) = p/q in lowest terms. After squaring, 5q^2 = p^2, so p is divisible by 5. That then forces q to be divisible by 5, which contradicts lowest terms.
Practice This ConceptMain explanation
Teacher explanation
This proof follows the same contradiction pattern as the proofs for sqrt(2) and sqrt(3). The only change is the prime number 5. Once both p and q are shown to share factor 5, the assumption of lowest terms collapses.
Example
If p is divisible by 5, write p = 5k and continue until q is also shown to be divisible by 5.
Simple analogy
Prime 5 locks both ends of the fraction.
Common confusion
Students copy the earlier proofs but forget to replace the prime with 5 in every divisibility step.
Exam tip
Keep the same proof steps and only change the prime factor to 5.
Answer writing and exam use
1-mark use
Write the exact meaning of proof of irrationality of sqrt(5) in one clean line.
2-mark use
Define proof of irrationality of sqrt(5) and add one example or condition.
3-mark use
Explain proof of irrationality of sqrt(5), show the method or example, and mention the common mistake.
Practice this concept with focused MCQs
Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.
Help improve this page
Found something confusing, incorrect, or missing?