Is a rational number divided by an irrational number always irrational
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Is a rational number divided by an irrational number always irrational
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No. If we let x be irrational, then 0/x = 0 is a
counterexample.
However, if we consider nonzero rational numbers, then our
conjecture is true. We shall prove this by contradiction.
Suppose we have nonzero rational numbers x and y, and an
irrational number z, such that x/z = y. Since z is not equal to 0,
x = yz. Since y is not equal to 0, x/y = z. Since x/y is a quotient
of rational numbers, x/y is rational. Therefore, z is rational,
contradicting our assumption that z was irrational. QED.