if a/b and c/d are rational numbers then what is a×b/b×d
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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 −b2
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.
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