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terminating decimal is the limit of a recurring decimal sequence.
(
Hint
. Analyse the remainders on dividing the integer
A
by the
integer
B
by long division.)
16 Supposetwo infinitedcimals havethesamelimit, say
x
d
. . .. If they are different as
infinite decimals then they must differ at some digit. Suppose
d
.
d
d
d
...
d
...
e
.
e
e
e
...
e
e
, and in fact
d
e
. Then
d
1
e
sincethenumbrs
are integers.
From qn 12, 0.
d
. Sincean infinitedcimal
sequence is monotonic increasing,
x
d
d
d
...
d
1
1/10
1, by theinequality
rule, qn 3.76. Again, since an infinite decimal sequence is monotonic
increasing,
e
x
.
x
d
e
,so
x
e
d
Now
e
1
1, and
e
0,
d
9
when 1
i
.
Revise this argument for the case when the first digits to differ are
d
, to show that of two equal infinite decimals, one has
recurring 9s and the other terminates.
For example, 0.5
0.49
.
and
e
17 If the infinite decimal sequence for the infinite decimal
0.101 001 000 100 001 . . ., with 1s in the
n
(
n
1)th positions and 0s
elsewhere is convergent, could its limit be a rational number?
Irrational numbers
18 If
p
and
q
are non-zero integers, prove that an equation of the form
p
2
q
would contradict the Fundamental Theorem of
Arithmetic. So there can be no rational number equal to
2. List
some other square roots and cube roots which cannot equal
rational number for similar reasons.
19 If
a
and
b
arerational numbrs, with
b
0, is it possiblefor
a
b
2 to bea rational numbr?
20 Let
a
and
b
berational numbrs with
a
b
. Show that
(
a
b
2)/(1
2) is not rational. Show also that
a
b
. Deduce that there is an irrational
number between any two distinct rationals. This, with qn 7, shows
that the irrationals are dense on the number line.
(
a
b
2)/(1
2)
21 (Optional) Show that there can be no rational number equal to
log
2.
