Proof of irrationality of sqrt(3)
Assume sqrt(3) = p/q in lowest terms. After squaring, 3q^2 = p^2, so p is divisible by 3. That then forces q to be divisible by 3, which contradicts lowest terms.
Practice This ConceptMain explanation
Teacher explanation
The proof uses contradiction just like sqrt(2), but the prime number is 3 here. If both numerator and denominator become divisible by 3, the original fraction cannot have been in lowest terms, so the assumption fails.
Example
Write p = 3k after showing p is divisible by 3, then continue until q is also shown to be divisible by 3.
Simple analogy
For sqrt(3), the prime 3 repeats its way into both sides.
Common confusion
Students repeat the sqrt(2) proof without changing the prime from 2 to 3.
Exam tip
Mention the prime 3 clearly at the divisibility step so the logic stays correct.
Answer writing and exam use
1-mark use
Write the exact meaning of proof of irrationality of sqrt(3) in one clean line.
2-mark use
Define proof of irrationality of sqrt(3) and add one example or condition.
3-mark use
Explain proof of irrationality of sqrt(3), show the method or example, and mention the common mistake.
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