We prove that √5 is irrational using proof by contradiction.
Proof by Contradiction:
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Assume the opposite: Let's assume that √5 is rational.
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Definition of a rational number: If √5 is rational, then it can be expressed as a fraction p/q, where p and q are integers, and q ≠ 0. Furthermore, we can assume that the fraction is in its simplest form (i.e., p and q have no common factors other than 1).
√5 = p/q
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Manipulate the equation: Square both sides of the equation:
(√5)² = (p/q)²
5 = p²/q²Multiply both sides by q²:
5q² = p²
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Deduction: This equation (5q² = p²) tells us that p² is divisible by 5.
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Implication: If p² is divisible by 5, then p itself must also be divisible by 5. This is because if 5 doesn't divide p, then it won't divide p².
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Express p in terms of 5: Since p is divisible by 5, we can write p as 5k, where k is some integer.
p = 5k
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Substitute into the original equation: Substitute p = 5k back into the equation 5q² = p²:
5q² = (5k)²
5q² = 25k² -
Simplify: Divide both sides by 5:
q² = 5k²
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Another deduction: This equation (q² = 5k²) tells us that q² is divisible by 5.
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Another implication: Similar to step 5, if q² is divisible by 5, then q itself must also be divisible by 5.
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Contradiction: We have now shown that both p and q are divisible by 5. This contradicts our initial assumption that p/q was in its simplest form, meaning p and q have no common factors other than 1.
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Conclusion: Since our initial assumption leads to a contradiction, it must be false. Therefore, √5 is not rational, and must be irrational.
In summary, we assumed √5 was rational, which led to the conclusion that both the numerator and denominator of its simplest fraction form are divisible by 5. This contradicts the definition of a simplified fraction, proving our initial assumption false. Hence, √5 is irrational.
