Give and examples each to show that that the rational numbers are closed under addition and subtraction
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a/b + c/d = (ad+bc)/bd, so closed under addition.
a/b - c/d = (ad-bc)/bd, so closed under subtraction.
a/b * c/d = (ac)/(bd), so closed under multiplication.
a/b / c/d = (ad)/(bc), so closed under division.
Wait!! but what if bc = 0?
b cannot be zero, because a/b is rational. But, c can be 0, so bc might be zero, and if so, then (ad)/(bc) is not rational, so
NOT closed under division, unless you add "except for division by zero
a/b - c/d = (ad-bc)/bd, so closed under subtraction.
a/b * c/d = (ac)/(bd), so closed under multiplication.
a/b / c/d = (ad)/(bc), so closed under division.
Wait!! but what if bc = 0?
b cannot be zero, because a/b is rational. But, c can be 0, so bc might be zero, and if so, then (ad)/(bc) is not rational, so
NOT closed under division, unless you add "except for division by zero
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