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Transcript
Few more elegant proofs Irrational numbers1 R. Inkulu http://www.iitg.ac.in/rinkulu/ 1 objective of this lecture is not to understand the properties of rational and irrational numbers; instead the focus was on learning how to prove few propositions together with learning few summation techniques (Irrational Numbers) 1 / 11 Definition A real number that can be expressed as ba with b 6= 0 is termd as a rational number. A real number that is not rational is said to be an irrational. (Irrational Numbers) 2 / 11 Few properties • Every rational number can be expressed as a finite continued fraction (a0 + 1 a1 + 1 ...+ a1 n ), whose integer coefficients ai can be determined by applying the Euclidean algorithm • The decimal expansion of an irrational number never repeats nor terminates, unlike a rational number • Almost all real numbers are irrational: rational numbers are countable whereas the real numbers are uncountable — proved later • Rational numbers form a field — not proved in class (Irrational Numbers) 3 / 11 √ 2 is irrational We have seen various proofs of this stmt with • proof by contradiction • proof by infinite descent (Irrational Numbers) 4 / 11 √ k is irrational, if it is not integral We have seen various proofs of this stmt with • proof by contradiction • proof by infinite descent • proof using fundamental theorem of arithmatic (Irrational Numbers) 5 / 11 Golden ratio is irrational golden ratio φ is a constant: √ 1+ 5 2 if φ is rational then 2φ − 1 = (Irrational Numbers) √ 5 is rational — reaching contradiction 6 / 11 lg2 3 is irrational proof by contradiction (Irrational Numbers) 7 / 11 √ √ 2 2 is irrational non-constructive existence proof (Irrational Numbers) 8 / 11 e is irrational if e = a b for integers a and b > 0, then n!be = n!a for every n ≥ 0; but bn!e = bn!((1 + 1 1! + 1 2! + ... + 1 n! ) 1 + + ( (n+1)! 1 (n+2)! + . . .)) = (integral) +n!(non-integral) for sufficiently large n (Irrational Numbers) 9 / 11 e2 is irrational if e2 = a b for integers a and b > 0, then n!be = n!ae−1 for every n ≥ 0; but n!ae is just a bit smaller than an integer and n!be is a bit larger than an integer (Irrational Numbers) 10 / 11 e4 is irrational — additional reading (Irrational Numbers) 11 / 11