# week07 - Rational and Irrational Numbers Original Notes...

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Rational and Irrational Numbers Original Notes adopted from October 23, 2001 (Week 7) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong Lemma: If x 2 is even then x is even. Theorem: ¥ 2 is irrational We can’t have ¥ 2 = m / n or 2 = (m / n) 2 for m, n integers Proof : If 2 = (m/n) 2 , we can divide out all common factors of m & n , getting m 0 /n 0 in lowest terms. (m 0 /n 0 ) 2 =2 m 0 2 = 2n 0 2 m 0 2 is even m 0 is even (from Lemma) m 0 = 2k, some k. (m 0 is even) (m 0 /n 0 ) 2 =2 m 0 2 = 2n 0 2 , substitute m 0 = 2k, some k (from above) (2k) 2 = 2n 0 2 4k 2 = 2n 0 2 2k 2 = n 0 2 n 0 2 is even n 0 is even (from Lemma) Now since 2|n 0 , 2|m 0 m 0 /n 0 is not in lowest terms so ¥ 2 is NOT rational. Theorem: ¥ 3 is irrational Proof: Assume rational ¥ 3 = m/n means m/n are in lowest terms ( We will show there is a contradiction) 3 = m 2 /n 2 (square both sides) 3n 2 = m 2 (Note since 3 is prime theorem states p|ab p|a or p|b) Note : 3 | m * m 3|m or 3 |m.

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