Proof of irrationality of sqrt(2)
Assume sqrt(2) = p/q in lowest terms. After squaring, 2q^2 = p^2, so p is even. That leads to q also being even, which contradicts the fraction being in lowest terms.
Practice This ConceptMain explanation
Teacher explanation
This is a proof by contradiction. If both p and q become divisible by 2, then the fraction was not in lowest terms after all. That contradiction shows sqrt(2) cannot be rational.
Example
If p is even, write p = 2k. Then 4k^2 = 2q^2, so q^2 = 2k^2 and q is also even.
Simple analogy
Even numerator leads to even denominator; lowest terms cannot survive.
Common confusion
Students stop after proving p is even and forget to show that q is even too.
Exam tip
In a proof by contradiction, always end with the words 'this contradicts lowest terms'.
Answer writing and exam use
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Define proof of irrationality of sqrt(2) and add one example or condition.
3-mark use
Explain proof of irrationality of sqrt(2), show the method or example, and mention the common mistake.
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