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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.
 Fall '11
 Aluffi

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