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week9 - The Real Numbers(Theorems Original Notes adopted...

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The Real Numbers (Theorems) Original Notes adopted from November 6, 2001 (Week9) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong Prove 2 + 3 is irrational Suppose 2 + 3 rational 2 + 3 = m/n 2 = m/n - 3 2 = m 2 /n 2 – 2 m/n 3 + 3 2m/n 3 = m 2 /n 2 + 1 3 = n/2m (m 2 /n 2 + 1) But 3 irrational. Contradiction. Extra problem : Is 2 + 3 + 5 rational? Several ways of constructing real numbers from the rationals. We'll use "Dedekind cuts". Q = set of rational numbers Each real number will be a set of rational numbers. -It's like the set of rationals less than the number. 0 2 Eg. 2 = { p Q: p < 0 or p 2 < 2 } Definition: A real number x is a subset of Q such that: 1) x , x Q (set of all Q) 2) if p x & s Q with s < p, then s x. 3) There is no largest number x. Eg. 3 = { p Q: p < 3 } To each rational number m/n, associate the set {p Q: p < m/n } = m/n x y if x y x < y if x y & x y. 2 = { p Q: p < 0 or p 2 < 2 } Is 2 a real number? 2 (eg. 1 2 ) 2 Q (eg. 7 2 ) Show p 2 & q < p q 2 Two Cases: 1) if p 0 if q < p, then q < 0, so q 2 2) if p > 0 & p

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week9 - The Real Numbers(Theorems Original Notes adopted...

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