10.27 - Puzzle from Monday's lecture: Does there exist an...

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irrational number r such that r sqrt(2) is rational? Hint: The answer is in Wednesday’s lecture (sort of!). Solution: We can easily prove that either r = sqrt(2) or r = sqrt(2) sqrt(2) works, but the proof won’t tell us which! Proof: If sqrt(2) sqrt(2) is rational, then r = sqrt(2) works. If sqrt(2) sqrt(2) is irrational, then r = sqrt(2) sqrt(2) works, because then r sqrt(2) = (sqrt(2) sqrt(2) ) sqrt(2) = sqrt(2) sqrt(2)sqrt(2) = sqrt(2) 2 = 2. As it happens, it’s been proved that sqrt(2) sqrt(2) is irrational. Last time we found empirically (i.e. with a calculator) that lim x 0+ (1+ x ) 1/ x and lim x 0– (1+ x ) 1/ x both are about 2.7. In fact, both limits are equal to the famous irrational number e = 2.718… That is: e = lim x 0 (1+ x ) 1/ x (a two-sided limit). We can also write e as lim n →∞ (1+1/ n ) n and lim n →∞ (1–1/ n ) n (Why? ..?. . Put
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10.27 - Puzzle from Monday's lecture: Does there exist an...

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