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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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:
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online