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For this problem, let's say that a pair of real numbers (a, b) is funky iff a and b are both irrational but a" is rational. Do any funky pairs exist? We'll show the answer is Yes in two different ways. (a) a Define r = √22 and show that a funky pair exists with a proof by cases: r is either rational or irrational. Hint: What is r2? What funky pair (a, b) can you find in each case? (b) Let a √2, and choose b so that a = 3, i.e., b = log√(3) = log2 (9). Use contradiction to prove that b is irrational. (This proves that (a, b) = (√2, log2 9) is funky!)

For this problem, let's say that a pair of real numbers (a, b) is funky iff a and b are both irrational but a" is rational. Do any funky pairs exist? We'll show the answer is Yes in two different ways. (a) a Define r = √22 and show that a funky pair exists with a proof by cases: r is either rational or irrational. Hint: What is r2? What funky pair (a, b) can you find in each case? (b) Let a √2, and choose b so that a = 3, i.e., b = log√(3) = log2 (9). Use contradiction to prove that b is irrational. (This proves that (a, b) = (√2, log2 9) is funky!)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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For this problem, let's say that a pair of real numbers (a, b) is funky iff a and b are both irrational but a' is rational. Do any funky pairs exist? We'll show the answer is Yes in two different ways. (a) 5√2 Define r = √2 either rational or irrational. and show that a funky pair exists with a proof by cases: r is Hint: What is rv2? What funky pair (a, b) can you find in each case? (b) Let a = √2, and choose b so that a = 3, i.e., b = log2 (3) = log₂ (9). Use contradiction to prove that b is irrational. (This proves that (a, b) = (√2, log29) is funky!)
Transcribed Image Text:For this problem, let's say that a pair of real numbers (a, b) is funky iff a and b are both irrational but a' is rational. Do any funky pairs exist? We'll show the answer is Yes in two different ways. (a) 5√2 Define r = √2 either rational or irrational. and show that a funky pair exists with a proof by cases: r is Hint: What is rv2? What funky pair (a, b) can you find in each case? (b) Let a = √2, and choose b so that a = 3, i.e., b = log2 (3) = log₂ (9). Use contradiction to prove that b is irrational. (This proves that (a, b) = (√2, log29) is funky!)
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