RE: Why x*0 = x
December 14, 2015 at 9:54 am
(This post was last modified: December 14, 2015 at 9:55 am by Alex K.)
(December 14, 2015 at 9:03 am)LastPoet Wrote: Just like rational numbers can be represented by many fractions, real numbers can also have more than one decimal representation. 0.9999... is one of the representations of the real number 1.
Exactly. The construction of the real numbers from the rational numbers which I alluded to above is really quite fascinating, and there being several representations is almost an automatic prediction in this approach. So the idea is that in the rational numbers, there are infinite sequences a_n which seem to converge because following elements are closer and closer together, but some of them don't seem to converge to any element of the rational numbers. Such sequences are called Cauchy. The idea, which I find stunning, is to say the following: We now look at the set of all those Cauchy-sequences, and do one more thing: any pair of sequences the difference of which converges to zero we call equivalent. One then gets classes of equivalent Cauchy-sequences. This set of equivalence classes is the real numbers. Let that sink in
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition
