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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.
a/b - c/d ----> ad/bd - bc/bd -----> (ad - bc)/bd
Since ad, bc, and bd are integers since integers are closed under the operation of multiplication and ad-bc is an integer since integers are closed under the operation of subtraction, then (ad-bc)/bd is a rational number since it is in the form of 1 integer divided by another and the denominator is not eqaul to 0 since b and d were not equal to 0. Thus a/b - c/d is a rational number.