### Theory:

Proper fraction:
1. A fraction in which the absolute value of numerator (top number) is smaller than the absolute value of denominator (bottom number).

2. It is represented as, $$p / q$$; $$q ≠ 0$$ and $$|p|< |q|$$, where $$p$$ is the numerator, $$q$$ is the denominator.

3. Proper fractions are greater than $$0$$ but lesser than $$1$$.
1. Negative proper fraction:
A negative proper fraction will have a negative sign either in numerator or denominator.

It can be represented as $\phantom{\rule{0.147em}{0ex}}\frac{-p}{q}\phantom{\rule{0.147em}{0ex}}\mathit{or}\phantom{\rule{0.147em}{0ex}}\frac{p}{-q}\phantom{\rule{0.147em}{0ex}}\mathit{where}\phantom{\rule{0.147em}{0ex}}q\ne 0\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}|p|<|q|$
Example:
$\frac{-9}{16}$; where $$p = -9$$, $$q = 16 ≠ 0$$ and $$|-9| < |16|$$
2. Positive proper fraction:
A positive proper fraction will have a positive sign in both numerator and denominator.

It can be represented as $\phantom{\rule{0.147em}{0ex}}\frac{p}{q}\phantom{\rule{0.147em}{0ex}}\mathit{or}\phantom{\rule{0.147em}{0ex}}\frac{p}{q}\phantom{\rule{0.147em}{0ex}}\mathit{where}\phantom{\rule{0.147em}{0ex}}q\ne 0\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}|p|<|q|$
Example:
$\frac{4}{5}$; $$p = 4$$, $$q = 5 ≠ 0$$ and $$|4| < |5|$$
Improper fractions:
1. Fractions where the numerator (the top number) is greater than or equal to the denominator (the bottom number).

2. It is represented as $\frac{p}{q}$; $$q ≠ 0$$ and $$|p| ≥ |q|$$; where $$p$$ is the numerator, $$q$$ is the denominator.

3. Improper fractions will be always $$1$$ or greater than $$1$$.
Example:
$\frac{3}{1}, \frac{4}{2},\phantom{\rule{0.147em}{0ex}}\frac{13}{2}$
1. Positive improper fraction:
A positive improper fraction will have a positive sign in both numerator and denominator and the value of a positive improper fraction will always be equal or greater than $$1$$.

It can be represented as $\phantom{\rule{0.147em}{0ex}}\frac{p}{q};\phantom{\rule{0.147em}{0ex}}\mathit{where}\phantom{\rule{0.147em}{0ex}}q\ne 0\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}|p|\ge |q|$
Example:
$\begin{array}{l}1/1=1\\ 5/3=1.6>1\\ 9/8=1.125>1\end{array}$
2. Negative improper fraction:
A negative improper fraction will have a negative sign in either numerator or denominator and the value of a negative improper fraction will always be equal or greater than $$-1$$.

It can be represented as, $\phantom{\rule{0.147em}{0ex}}\frac{-p}{q} \mathit{or}\phantom{\rule{0.147em}{0ex}}\frac{p}{-q};\phantom{\rule{0.147em}{0ex}}\mathit{where}\phantom{\rule{0.147em}{0ex}}q\ne 0\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}|p|\ge |q|$
Example:
$\begin{array}{l}-6/3=-2<-1\\ -9/5=-1.8<-1\\ -2/1=-2<-1\end{array}$
Important!
All the mixed fractions ($2\frac{3}{4},\phantom{\rule{0.147em}{0ex}}6\frac{3}{8}$) and integers ($$-1$$, $$3$$, $$7$$) are improper fractions.