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What's the difference between the following fractions:

$ \frac{-4}{-5}$

$ \frac{4}{-5}$

$ \frac{-4}{5}$

$ - \frac{4}{5}$

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up vote 9 down vote accepted

The first one is $4/5$. The other three are all $-4/5$.

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thank you! Good to have that cleared up. – bot_bot Mar 9 '12 at 11:54

Another way to think about this would be in terms of equivalence classes. If you are not familiar with this, it is pretty much how mathematicians say in the rationals that

$$\frac{1}{2} = \frac{2}{4} = \frac{4}{8}.$$

In fact one says that given two fractions

$$x = \frac{m}{n} \hspace{2mm} \text{and} \hspace{2mm} y= \frac{a}{b},$$

they are equal iff $mb - na = 0$. So in your case for example $\frac{-4}{5}$ and $-\frac{4}{5}$ are equal because

$$\frac{-4}{5} - \bigg((-1)\frac{4}{5}\bigg) = \frac{-4}{5} + \frac{4}{5} = \frac{-4 + 4}{5} = 0$$

recalling that $-\frac{4}{5} = (-1)\frac{4}{5}$. You can go on like this for all of them, e.g. for example the first and second are not equal because

$$\frac{-4}{-5} - \frac{4}{-5} = \frac{-4 - 4}{5} \neq 0.$$

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thanks, I'll have a go at working those out for myself. Can you fix your last example please? – bot_bot Mar 9 '12 at 12:40
sorry, the mark-up didn't work on the last example. Looks good now. – bot_bot Mar 9 '12 at 12:45

The first one is positive, while the rest are all negative. That is just the difference. But observe that there modulus is the same.

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Hint $\ $ Use $\rm\ \dfrac{a}b \dfrac{c}d\: =\: \dfrac{ac}{bd}\ $ for $\rm\ a,b\: =\: \pm 1\ \:$ (hence $\:\rm\dfrac{a}b\: =\: \pm 1\:$)

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