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# Sols19 - 19 NOVEMBER 10TH DECIMAL EXPANSIONS CARDINALITY I 45 19 November 10th Decimal expansions Cardinality I 19.1 Apply the procedure D R

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19. NOVEMBER 10TH: DECIMAL EXPANSIONS. CARDINALITY I 45 19. November 10th: Decimal expansions. Cardinality I 19.1 . Apply the procedure ‘ D R ’ explained in § 19.1 to the Dedekind cut ( A | B ) with A = { q Q | q< 1 } . What is the corresponding decimal expansion? (Note that this cut is the one associated with the rational number 1.) Answer. The largest integer in A is 0, so b = 0. Among the numbers 0 , 0+ 1 10 , 2 10 , . . . , 8 10 , 9 10 , 1 the largest in A is 9 10 =0 . 9, so the frst digit in the corresponding decimal expansion is 9. Among 9 10 , 9 10 + 1 100 , 9 10 + 2 100 , . . . , 9 10 + 8 100 , 9 10 + 9 100 , 1 the largest in A is 9 10 + 9 100 . 99, so the second digit is 9. Continuing this procedure gives the decimal expansion 0 . 99999 · · · For this Dedekind cut. ± As pointed out, this cut corresponds to the number 1. IF you have never observed the curious Fact that 0 . 9999 · · · = 1, it may seem a little surprising. But indeed, iF r . 9999999 · · · , then 10 r - 9 = 9 . 999 · · · - 9=0 . 999 · · · = r , so 9 r = 9, so r = 1.

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## This note was uploaded on 12/14/2011 for the course MGF 3301 taught by Professor Aluffi during the Fall '11 term at FSU.

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Sols19 - 19 NOVEMBER 10TH DECIMAL EXPANSIONS CARDINALITY I 45 19 November 10th Decimal expansions Cardinality I 19.1 Apply the procedure D R

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