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December 1st, 2012, 04:52 PM  #1 
Newbie Joined: Nov 2012 Posts: 25 Thanks: 0  Real Numbers  Distributive law of Multiplication
Hi All... I am stuck with this problem. As we know that the real numbers are defined to be the set of equivalence classes of pairs of rational sequences( ai, bi) where (1) {ai} is increasing, (2) {bi} is decreasing, (3) for each i = 1, 2, .... (bi  ai) > 0 and (4) limit (i > 0) (bi  ai) = 0 (a) How can I prove the distributive law of multiplication for numbers. (b) Can we represent ? in this sense. Can anyone please help me with these. I will appreciate any help in this regard. Thank You. 
December 14th, 2012, 05:51 PM  #2 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Real Numbers  Distributive law of Multiplication
How have you defined multiplication itself? I would be inclined to define xy for x and y positive as the equivalence class containing sequences of the form (a_xa_y) and (b_xb_y). Then define (x)(y) as (xy), (x)(y) as (xy), (x)(y) as xy.


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