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The difference of two rational numbers is rational. Let the two rational numbers be a/b and c/d, where a, b, c, and d are integers. Any rational number can be represented this way. Their difference is

a/b-c/d

= ad/bd-cb/bd

= (ad-cb)/bd.

Products and differences of integers are always integers. This means that ad-cb is an integer, and so is bd.

Thus, (ad-cb)/bd is a rational number (since it is the ratio of two integers). This is equivalent to the difference of the original two rational numbers.

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Q: Would the difference of a rational number and a rational number be rational?

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