# The difference between two rational numbers always is a rational number [duplicate]

Claim: The difference between two rational numbers always is a rational number

Proof: You have a/b - c/d with a,b,c,d being integers and b,d not equal to 0.

Then:

Another definition of rational number is a real number $x$ such that there exists a non-zero integer $n$ for which $nx$ is an integer.
Since $bd(a/b-c/d) =ad-bc$ is an integer, $(a/b-c/d)$ is rational.