Graphics Reference
In-Depth Information
20 If (
a
b
x
, then (
a
x
)/(
x
b
)
2)/(1
2)
2, so
x
cannot be
rational.
21 Suppose that there are integers
p
and
q
such that log
2
p
/
q
, then
10
2, so 10
2
, and 5 divides the left-hand side but not the
right-hand side, contradicting the fundamental theorem of arithmetic.
22 The new decimal will be different from the others above it if each entry
differs from the diagonal entry above it.
23 Constructed sequence of infinite decimals
0.
a
a
a
a
a
a
...
0.
a
a
a
a
a
a
...
0.
a
a
a
a
a
a
...
0.
a
a
a
a
a
a
...
0.
a
a
a
a
a
a
...
...
...
...
...
...
...
...
...
0.
a
a
a
a
a
a
...
...
...
...
...
...
...
...
...
new infinite decimal
0.
b
b
b
b
b
b
...
Choose
b
different from 0, 9 and
a
.
Choose
b
different from 0, 9 and
a
.
Choose
b
different from 0, 9 and
a
.
...
Choose
b
.
By avoiding 0 and 9, unwitting duplication is avoided. By avoiding
a
different from 0, 9 and
a
a new infinite decimal is guaranteed.
24 0,
1,
1,
2,
2,
3,
3, . . .
0, 1, 2, 3, 4, . . .
1, 2, 4, 8, 16, . . .
1,
2,
3, ...
6, 12, 24, . . .
So sequence built from increasing natural numbers is 0, 1, 2,
1, 3,
2, 4, . . .
25 positive
a
/
b
b
a
1
2
3
4
1
6
12
24
48
2
18
36
72
144
3
54
108
216
432
4
162
324
648
1296
Sequence
,
,
,
,[
],
,
,
, ...
