# Why is $\frac{p}{-q}$ written as $\frac{-p}{q}$ and not as $\frac{p}{-q}$?

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.

• 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. Commented Aug 11 at 9:02
• Which book is it by G.H. Hardy? Commented Aug 11 at 9:05
• Pure Mathematics centenary edition, by gh hardy
– user1372724
Commented Aug 11 at 9:08
• 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. Commented Aug 11 at 9:10
• 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. Commented Aug 11 at 9:14