Graphics Reference
In-Depth Information
1 Find non-negative integers
a
,
b
,
c
,
d
,
e
,
f
such that
200 772
2
· 3
· 5
· 7
· 11
· 13
.
The
Fundamental Theorem of Arithmetic
states that each natural
number greater than 1 may be expressed in a unique way as a product
of prime numbers. The proof of this theorem can be found in most
undergraduate textbooks on number theory. See for example the first
chapter of Burn (1997) or Hardy and Wright. For the simple
enunciation of this theorem it is important that the number 1 is
not
called prime.
2 Find integers
a
,
b
,
c
,
d
,
e
,
f
such that
2775/999 999
2
· 3
· 5
· 7
· 11
· 13
.
3 Use the Fundamental Theorem of Arithmetic to enunciate a unique
factorisation theorem for
positive rational
numbers. (The set of
rational numbers,
Q
, was defined at the start of chapter 2.)
Dense sets of rational numbers on the number line
4 Illustrate on a number line those portions of the sets
,
m
/2
m
Z
,
m
/4
m
Z
,
m
/8
m
Z
which lie between
3. Is
each set contained in the set which follows in this list?
What would an illustration of thest
m
m
Z
m
/2
m
Z
look likefor
some large positive integer
n
?
5 Do you believe that if only
n
were large enough there would be a
rational number of the form
m
/2
lying between 57/65 and 64/73?
The difference between these two numbers is 1/(65· 73). Try to find
such a number where
m
and
n
are positive integers.
6 To show that there is always a rational number of the form
m
/2
between two numbers
a
and
b
, where
a
)is
a null sequence by qn 3.39, since it is a geometric progression with
common ratio less than 1. So, for suMciently large
n
,(
b
, remember that ((
)
)
b
a
.
Only when
n
is that big can we expect to find an
m
/2
between
a
and
b
. Now let
k
be a positive integer such that (
)
b
a
.Use
the integer function [ ] defined just before qn 3.19, to locate two
integers close to
a
· 2
, with [
a
· 2
]
a
· 2
[
a
· 2
]
1. Now show
that
m
[
a
· 2
]
1 and
n
k
satisfy the required conditions.
7 By repeatedly bisecting the interval
x
a
x
b
, show that
between any two distinct numbers on the number line, there is an
infinity of numbers of the form
m
/2
, where
m
is an integer and
n
is
a non-negative integer.
