Question

Must the difference between two rational numbers be a rational number? Explain

Answer 1

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:

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.

Answer 2

there is no difference. but there is a difference between a rational and irrational number which is rational can be formed as a decimal and fraction but irrational can only be formed as a decimal and not a fraction. hope this helps!