Graphics Reference
In-Depth Information
A set of numbers containing an element in every interval on the
number line is said to be
dense
on the line. The argument of question 7
can be used to show that there is an
infinity
of elements of a dense set
in every interval of the number line. We have shown in question 6 that
therational numbrs are
dense
on the number line, even though the
proof only used rationals with denominator a power of 2. Beware!
When drawing a figure, there is no difference to be seen between a
dense set of points and a continuous line segment.
8 Let
T
m
/2
m
Z,
n
Z
.Is
T
dense on the number line? Is
there a smallest positive number in
T
? Is there a smallest positive
rational number?
9 Let
D
m
/10
m
Z,
n
Z
. Prove that every element of
T
(in qn
8) is an element of
D
. Deduce that
D
is dense on the number line.
Show that
D
, and use the Fundamental Theorem of Arithmetic
to show that
.
The elements of
D
arecalled
terminating decimals
.
Is thest
T
T
closed under
(i) addition,
(ii) subtraction,
(iii) multiplication and
(iv) division, except by 0?
Would your answers have been the same if the questions had been
asked about the set
D
?
10 Check that every term of the sequence with terms
a
,
a
(
)
,
a
(
)
(
)
,...,
a
((
)
(
)
(
)
...
(
)
), ...
belongs to
T
, as defined in qn 8.
ยท
1
(
)
Useqn 1.3(vi) to show that
a
)
,
1
(
and with the help of qn 3.42, the difference rule (3.54(v)) and the
scalar rule(3.54(i)), provethat (
a
)
. Thus a sequence in the
dense set
T
may converge to a point outside
T
.
Infinite decimals
11 Use the Fundamental Theorem of Arithmetic to show that
D
as
defined in question 9. That is to say, there is no terminating
decimal equal to
.
