Positive and Negative Fractions
A fraction whose numerator and denominator are integers $$ \frac{m}{n} \ \text{with} \ m,n \in \mathbb{Z}, \ n \neq 0 $$ can be either positive (+) or negative (-).
Fractions fall into two simple categories
- Positive fractions
Fractions with a positive sign are called positive fractions $$ + \frac{m}{n} $$ A fraction is positive when the numerator and the denominator have the same sign. $$ + \frac{m}{n} \sim \frac{+m}{+n} \sim \frac{-m}{-n} $$ In standard notation, the leading plus sign is usually omitted, so we simply write the fraction without it.Example. The following are positive fractions $$ \frac{2}{3} \ , \ \frac{6}{4} \ , \ \frac{7}{5} \ , \ ... $$
- Negative fractions
Fractions with a negative sign are called negative fractions $$ - \frac{m}{n} $$ A fraction is negative when the numerator and the denominator have opposite signs. $$ - \frac{m}{n} \sim \frac{-m}{+n} \sim \frac{+m}{-n} $$Example. The following are negative fractions $$ - \frac{2}{3} \ , \ - \frac{6}{4} \ , \ - \frac{7}{5} \ , \ ... $$
A practical example
Consider the negative fraction
$$ - \frac{2}{3} $$
This fraction can be written in different but equivalent ways, depending on where the negative sign is placed.
$$ - \frac{2}{3} \sim \frac{-2}{3} $$
It can also be written as
$$ - \frac{2}{3} \sim \frac{2}{-3} $$
By the transitive property of equality, these forms are equivalent to each other
$$ \frac{-2}{3} \sim \frac{2}{-3} $$
Verification. To check that the fractions $$ \frac{-2}{3} \sim \frac{2}{-3} $$ are equivalent, compute the cross products by multiplying the numerator of each fraction by the denominator of the other. $$ -2 \cdot (-3) = 3 \cdot 2 $$ $$ 6 = 6 $$ Since the products are equal, the fractions are equivalent.
This reasoning applies in general to all fractions.
