For example we have:
$$ \frac{-1}{2} $$
Does this mean that only the numerator of the fraction is negative?
Can we put it like this?
$$ -\frac{1}{2} $$ Does this means that the whole fraction is negative?
And then can we put it like this?
$$ \frac{1}{-2} $$
Does this mean that only the denominator of the fraction is negative?
Do the above suggestions [look] right? Or how should I understand when in some examples the minus sign is moving from numerator to the whole fraction?
Best Answer
You might want to ask yourself what is meant when you put a minus sign in front of something. The answer could be, for example, that $-\frac ab$ is the solution to the equation $$x+\frac ab = 0$$ where the unknown is $x$. Now, you want to know whether $\frac {-a}b$ is the same as $-\frac ab$. Just check if it solves the equation! $$\frac {-a}b + \frac ab = \frac{-ab+ab}{b^2} = \frac 0{b^2} = 0$$ But lo and behold, the same goes for $\frac a{-b}$ $$\frac a{-b} + \frac{a}{b} = \frac{ab + a(-b)}{(-b)(b)} = \frac{ab-ab}{-b^2} = \frac 0{-b^2} = 0$$
We conclude that both $\frac {-a}b$ and $\frac a{-b}$ are opposites of the number $\frac ab$, i.e. they are both $-\frac ab$. This works because of the way that fractions and their operations are defined. Note that I have used a primitive way of adding fractions, by taking as the common denominator the product of denominators (to avoid using known tricks).