If a/b and c/d are two rational numbers with b and d positive integers then a/b<c/d if *1 pointa>cb<da×d>c×ba×d <c×b About the author Jade
If a=c, then a+ b =c+ d ⇒ b = d ⇒b=d So, let a =c. Then, there exists a positive rational number x such that a=c+x. Now, ⇒a+ b =c+ d ⇒c+x+ b =c+ d [∵a=c+x] ⇒x+ b = d ⇒(x+ b ) 2 =( d ) 2 ⇒x 2 +2 b x+b=d ⇒ b = 2x d−x 2 −b ⇒ b is rational [∴d,x,b are rationals∴ 2x d−x 2 −b 2 is rational] ⇒ b is the square of a rational number. From(i), we have d =x+ b ⇒ d is rational ⇒ d is the square of a rational number. Hence, either a=c and b=d or b and d are the squares of rationals. Reply
If a=c, then a+
b
=c+
d
⇒
b
=
d
⇒b=d
So, let a
=c. Then, there exists a positive rational number x such that a=c+x.
Now,
⇒a+
b
=c+
d
⇒c+x+
b
=c+
d
[∵a=c+x]
⇒x+
b
=
d
⇒(x+
b
)
2
=(
d
)
2
⇒x
2
+2
b
x+b=d
⇒
b
=
2x
d−x
2
−b
⇒
b
is rational [∴d,x,b are rationals∴
2x
d−x
2
−b
2
is rational]
⇒ b is the square of a rational number.
From(i), we have
d
=x+
b
⇒
d
is rational
⇒ d is the square of a rational number.
Hence, either a=c and b=d or b and d are the squares of rationals.