Graphics Reference
In-Depth Information
Since the sequence is monotonic increasing, the limit of the sequence is
thelimit of (
a
).
1
10
14
1
1
10
1
999
·
1
10
1/10
1/10
...
1/10
·
1
.
1
10
1
a
12.45,
b
6.78.
With a recurring block of length
l
, theinfinitedcimal will bethelimit
of a sequence with
n
th term
1
10
1
1
10
1
10
1
10
b
10
a
b
...
a
·
1
10
1
b
1
10
a
1
.
10
1
If
a
and
b
are terminating decimals, the sequence with this as
n
th term
has a rational limit.
15 If
A
and
B
are integers and
A
/
B
is not equal to a terminating decimal,
thelong division of
A
by
B
will not terminate. The remainder at each
stageof thedivision is oneof thenumbrs 1, 2, . . .,
1. Sincethe
process is endless, the remainders must recur and thus the dividends
will recur.
B
16 If thefirst digits to diffr are
d
and
e
, then we suppose
d
e
and so
d
.
Now 0.00 . . . 0
d
1
e
d
...
d
(1/10
)(1
1/10
), with equality only
when each
d
9, so
x
d
.
d
d
...(
d
1). But also
e
.
e
e
...
e
x
since an infinite decimal sequence is monotonic increasing. Thus
e
.
e
e
...
e
x
d
.
d
d
...(
d
1). So
9 for all
i
.
17 Every rational number is equal to a terminating or recurring decimal.
The infinite decimal given here is neither terminating nor recurring, so
if it converges it does not converge to a rational number.
x
e
.
e
e
...
e
d
.
d
d
...(
d
1) and
e
0,
d
18 In theprimefactorisation of both
p
, all primes occur to an
even power. So 2 appears to an even power in
p
and to an odd power
in 2
q
,so
p
2
q
contradicts the fundamental theorem of arithmetic.
Likewise
3,
6 and
2 cannot berational numbrs.
and
q
19 If
a
b
2
x
were rational, then (
x
a
)/
b
2 would also haveto
berational. Contradiction.
