This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 521  Advanced Calculus I Instructor: J. Metcalfe Due: February 1, 2010 Assignment 5 1. Fix b > 1. (a) If m,n,p,q are integers, n > 0, q > 0, and r = m/n = p/q , prove that ( b m ) 1 /n = ( b p ) 1 /q . Hence it makes sense to define b r = ( b m ) 1 /n . (b) Prove that b r + s = b r b s if r and s are rational. (c) If x is real, define B ( x ) to be the set of all numbers b t , where t is rational and t x . Prove that b r = sup B ( r ) when r is rational. Hence it makes sense to define b x = sup B ( x ) for every real x . (d) Prove that b x + y = b x b y for all real x and y . (a) If m/n = p/q , we have mq = np . By definition, we have y = ( b m ) 1 /n satisfies y n = b m . Similarly, we have z = ( b p ) 1 /q satisfies z q = b p . Then, y np = b mp = z qm . Since np = qm N , we can take this root of both sides. By uniqueness of n th roots, we have that y = z . (b) Let r,s Q . By finding a common denominator, we can write r = m/n and s = l/n . Then, b r + s = b ( m + l ) /n = ( b ( m + l ) ) 1 /n = ( b m b l ) 1 /n ....
View Full
Document
 Spring '10
 MetCalfe
 Calculus, Integers

Click to edit the document details