6. The ancient Greeks had the following definition: two lengths (positive numbers) a,b are com-mensurable if there exists a common sublength m of a and b. Otherwise said, a = km and b = Imfor some positive integers k, l.(a) Pythagoras is thought to have believed that all pairs of lengths a, b are commensurable, that isVa, b e R+, 3m E R†, k,l E N such that a = km and b = lmState the negation of Pythagoras' claim using quantifiers.(b) Prove or disprove Pythagoras' claim.(You may use anything seen in the course without proof)

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Description: 6. The ancient Greeks had the following definition: two lengths (positive numbers) a,b are com-mensurable if there exists a common sublength m of a and b. Otherwise said, a = km and b = Imfor some positive integers k, l.(a) Pythagoras is thought to

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Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

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Chapter2: Second-order Linear Odes

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6. The ancient Greeks had the following definition: two lengths (positive numbers) a,b are com-
mensurable if there exists a common sublength m of a and b. Otherwise said, a = km and b = Im
for some positive integers k, l.
(a) Pythagoras is thought to have believed that all pairs of lengths a, b are commensurable, that is
Va, b e R+, 3m E R†, k,l E N such that a = km and b = lm
State the negation of Pythagoras' claim using quantifiers.
(b) Prove or disprove Pythagoras' claim.
(You may use anything seen in the course without proof)

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(a) The negation of Pythagoras’ claim can be stated in simple terms as, “there exist at least one pair of lengths a, b that are not commensurable.”

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