If $x$ is a rational number, then $1/x$ is a rational number

Question

Why is this statement false?

If $x$ is a rational number, i.e. $\frac{p}{q}$, then shouldn't it be obvious that $\frac{q}{p}$ is also a rational number, by definition of rational numbers?

Answer 1

$x = 0$ is a rational number but $\dfrac{1}{x} = \dfrac{1}{0}$ isn't defined.

Answer 2

HINT: What happens in the case that $p=0$?

Answer 3

Strictly speaking the statement has undefined meaning (because "$1/x$ is rational" has undefined meaning when $x$ can be$~0$) rather than that it is false. To make the statement meaningful, you need to ensure it is not talking about the undefined quantity $1/0$, and depending on how you do that this can make the statement false or true.

Since in mathematics we are more interested in truths than in falsehoods, when we assert a statement, one usually takes it to also implicitly affirm that everything it talks about is well defined. Applying this to your case would transform the statement into

(For any real number $x$,) if $x$ is a rational number then $1/x$ is defined and it is a rational number.

which statement is false (it fails for $x=0$).

However an other way to repair the statement is to recognise from the outset that $x=0$ is going to cause an undefined expression to turn up in the formulation, and therefore exclude it, This leads to

For any nonzero real number $x$, if $x$ is a rational number then $1/x$ is a rational number.

which statement is true (since $\Bbb Q$ is a sub-field of $\Bbb R$).

The example shows one must always be careful in handling statements that contain potentially undefined expressions.