# week08 - Irrational and Algebraic Numbers, IVT, Upper and...

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Irrational and Algebraic Numbers, IVT, Upper and Lower Bounds Original Notes adopted from October 30, 2001 (Week 8) c ± P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics Is 3 4 irrational? 3 4 = m/n 4 = m 3 /n 3 4 n 3 = m 3 If m = p α 1 1 p α 2 2 ··· p α s s ,n = q β 1 1 q β 2 2 ··· q β t t , then m 3 = p 3 α 1 1 p 3 α 2 2 ··· p 3 α s s , n 3 = q 3 β 1 1 q 3 β 2 2 ··· q 3 β t t , 2 2 q 3 β 1 1 q 3 β 2 2 ··· q 3 β t t = p 3 α 1 1 ··· p 3 α s s . 2 2 needs to occur on the right side. On the right side, the power of 2 occurring is a multiple of 3. On the left side, it is not a multiple of 3 (the power of 2 is 2 (mod 3)). But this is a contradiction, since prime factorizations are unique. Theorem. k L is rational only if k L is an integer. Proof : Suppose k L = m/n . Then L = m k /n k , so Ln k = m k . If m = p α 1 1 p α 2 2 ··· p α s s and n = q β 1 1 q β 2 2 ··· q β t t , then m k = p 1 1 p 2 2 ··· p s s , n k = q 1 1 q 2 2 ··· q t t , so Lq 1 1 ··· q kβt t = p 1 1 ··· p s s . Write L as a product of primes:

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## This note was uploaded on 04/26/2011 for the course MATH 246 taught by Professor Applebaugh during the Spring '10 term at University of Toronto.

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week08 - Irrational and Algebraic Numbers, IVT, Upper and...

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