My Math Forum Real Numbers - Distributive law of Multiplication

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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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### proof of distributive law for real numbers

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