[I : 14]
REAL VARIABLES
24
14.
Some theorems concerning quadratic surds.
Two pure
quadratic surds are said to be
similar
if they can be expressed as rational
multiples of the same surd, and otherwise to be
dissimilar
. Thus
√
8 = 2
√
2
,
q
25
2
=
5
2
√
2
,
and so
√
8,
q
25
2
are similar surds.
On the other hand, if
M
and
N
are
integers which have no common factor, and neither of which is a perfect
square,
√
M
and
√
N
are dissimilar surds. For suppose, if possible,
√
M
=
p
q
r
t
u
,
√
N
=
r
s
r
t
u
,
where all the letters denote integers.
Then
√
MN
is evidently rational, and therefore (
Ex.
ii
. 3) integral.
Thus
MN
=
P
2
, where
P
is an integer.
Let
a
,
b
,
c, . . .
be the prime
factors of
P
, so that
MN
=
a
2
α
b
2
β
c
2
γ
. . . ,
where
α
,
β
,
γ, . . .
are positive integers. Then
MN
is divisible by
a
2
α
, and
therefore either (1)
M
is divisible by
a
2
α
, or (2)
N
is divisible by
a
2
α
, or
(3)
M
and
N
are both divisible by
a
.
The last case may be ruled out,
since
M
and
N
have no common factor. This argument may be applied to
each of the factors
a
2
α
,
b
2
β
,
c
2
γ
, . . .
, so that
M
must be divisible by some
of these factors and
N
by the remainder. Thus
M
=
P
2
1
,
N
=
P
2
2
,
where
P
2
1
denotes the product of some of the factors
a
2
α
,
b
2
β
,
c
2
γ
, . . .
and
P
2
2
the product of the rest.