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I have seen in a pure mathematics book by G.H Hardy that $\frac{-p}{q}$ is equal to $\frac{p}{-q}$, but why it is taken as that why not substitute the whole equation as $\frac{p}{-q}$, also if a equation comes as $\frac{-p}{q}$ why don't we write $\frac{p}{-q}$ it might be a basic problem but it's quite important to know.

For example $\frac{2}{-3}$ is written as $\frac{-2}{3}$ and same goes for $\frac{7}{-8}$ is written as $\frac{-7}{8}$ why are they not taken as their original form, $\frac{p}{-q}$ = $\frac{-p}{q}$, but we don't take $\frac{-p}{q}$ = $\frac{p}{-q}$, we don't write $\frac{-9}{5}$ as $\frac{9}{-5}$ and $\frac{-7}{2}$ as $\frac{7}{-2}$ why don't we do that with rational numbers.

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  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Jonathan Xu
    Commented Aug 11 at 9:02
  • $\begingroup$ Which book is it by G.H. Hardy? $\endgroup$
    – Gribouillis
    Commented Aug 11 at 9:05
  • $\begingroup$ Pure Mathematics centenary edition, by gh hardy $\endgroup$
    – user1372724
    Commented Aug 11 at 9:08
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    $\begingroup$ I'm pretty sure it's just because they want the denominator to be as simply as possible, and a positive integer is "simpler" than a negative one. Similar to how we would rather not have a surd in the denominator, and we thus multiply it by the conjugate. $\endgroup$
    – La-on
    Commented Aug 11 at 9:10
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    $\begingroup$ It is a general convention that denominators do not carry a sign, so one would prefer $-\frac mn$ or $\frac{-m}n$ over $\frac m{-n}$. By the way, an equation is something of the form $P=0$ where $P$ is an expression that may contain one or more variables (sometimes dubbed unknowns). If there is no equal sign it is merely an expression. $\endgroup$
    – Andrea Mori
    Commented Aug 11 at 9:14

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Short answer: convention.

We have collectively agreed that having a sign in the denominator is a bit uglier. Both are correct.

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