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Rudin_1

# Rudin_1 - The Real and Complex Number Systems Written by...

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The Real and Complex Number Systems Written by Men-Gen Tsai email: [email protected] 1. 2. 3. 4. 5. 6. 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 ) where 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 . 1

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Proof: For (a): mq = np since m/n = p/q . Thus b mq = b np . By Theorem 1.21 we know that ( b mq ) 1 / ( mn ) = ( b np ) 1 / ( mn ) , that is, ( b m ) 1 /n = ( b p ) 1 /q , that is, b r is well-defined. For (b): Let r = m/n and s = p/q where m, n, p, q are integers, and n > 0 , q > 0. Hence ( b r + s ) nq = ( b m/n + p/q ) nq = ( b ( mq + np ) / ( nq ) ) nq = b mq + np = b mq b np = ( b m/n ) nq ( b p/q ) nq = ( b m/n b p/q ) nq . By Theorem 1.21 we know that (( b r + s ) nq ) 1 / ( nq ) = (( b m/n b p/q ) nq ) 1 / ( nq ) , that is b r + s = b m/n b p/q = b r b s .
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