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December 1st, 2012, 04:52 PM   #1
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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.
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December 14th, 2012, 05:51 PM   #2
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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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