fractions with negative numerators and denominators

Question

I've been having this doubt in my mind.. I really don't get the concept behind fractions with negative numerators and denominators being equal to there positive form. For eg.: $\frac{-2}{-3} = \frac{2}{3}$. I tried to google it but there were no relevant answers. Can anyone help, please?

Answer 1

On a basic level, when you consider $\frac xy$, you ask how many $y$ go into $x$. E.g. if you do $\frac 42$ you think: how many $2$s go into $4$ and deduce the answer to be $2$.

What about with negative numbers. How many $-2$s go into $-4$? The answer is also $2$ as $2*-2=-4$. This can be represented as $\frac{-4}{-2}=2$, and explains why a negative divided by a negative is positive.

Answer 2

Using the fact that $\frac{a}{a}=1$ for all $a\neq 0$, you get $$\frac{-2}{-3} = \frac{(-1)\cdot 2}{(-1)\cdot 3} = \frac{-1}{-1}\cdot\frac23 = 1\cdot\frac23=\frac23$$

Answer 3

Quite simple with the rules of simplification: $$\frac{\not ka}{\not kb}=\frac ab, \quad\text{so}\enspace\frac{-2}{-3}=\frac{(-1)2}{(-1)3}=\frac 23.$$

Variant: $$\frac ab=\frac cd\iff ad=bc,\quad\text{and observe }\;(-2)\,3=(-3)\,2.$$