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Unformatted text preview: The Real Numbers as Decimal Expansions Even though defining the real numbers as infinite decimal expansions makes proving that they are a complete ordered field diﬃcult, most people think of them that way. So let’s define a real number to be + or an integer, fol lowed by a decimal point, followed by an unending string of numbers chosen from { , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } which is not unbroken string of 9’s after some point. We also need to agree that all of these expansions are different real numbers with the exception that +0 . = . . If you think about constructing rules for adding and multiplying these expansions so that the result is a complete ordered field, you will see why mathematicians wanted a different definition for the real numbers. How ever, one can give rules which reﬂect the way that one manipulates explicit real numbers. I think that it worthwhile to do this. For reals beginning with a plus sign (“positive reals”) I will define x > y to mean that in the first decimal place where the entry for x is different from the entry for y , the entry for x is greater than the entry for y . This definition works only because....
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This note was uploaded on 01/30/2012 for the course MATH 131a taught by Professor Hitrik during the Spring '08 term at UCLA.
 Spring '08
 hitrik
 Math, Real Numbers

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