Proper Fraction
A proper fraction is a fraction where the numerator is less than the denominator, representing a quantity less than one whole.
Definition
A proper fraction is one where the numerator is smaller than the denominator, so its value is always less than $1$ whole: examples include $1/2$, $3/4$, $5/8$. More precisely, a proper fraction $a/b$ satisfies $0 \le a < b$ with $b > 0$, so its absolute value is strictly less than $1$ and it lies between $-1$ and $1$ on the number line. The idea extends to rational functions: in partial fraction decomposition, a proper rational function $P(x)/Q(x)$ satisfies $\deg(P) < \deg(Q)$, and an improper rational function must first be divided to extract a polynomial part before partial fractions can be applied, exactly mirroring the integer/proper-fraction split.
Example
$3/5$ is a proper fraction because $3$ is less than $5$: if you eat $3$ slices of a $5$-slice pizza, you ate less than the whole pizza. Contrast $7/9$ (proper, $\approx 0.778$) with $9/7$ (improper, $\approx 1.286$, lying outside the interval $(0, 1)$). For the improper rational function $(x^3+2x)/(x^2+1)$, polynomial long division gives $x + x/(x^2+1)$; the remainder $x/(x^2+1)$ is now proper (degree $1$ < degree $2$) and can be integrated directly as $\frac{1}{2}\ln(x^2+1)$.
Key Insight
The word "proper" hints that these fractions stay in their place between $0$ and $1$, representing parts of a single whole rather than more than a whole, and every rational number in $(0, 1)$ is a proper fraction: when reducing an improper fraction or simplifying a mixed number, the fractional part left over is always proper. The proper/improper distinction for rational functions is the algebraic analogue of the integer/fractional-part split for real numbers via the floor function; both cases reflect the division algorithm, $\text{dividend} = (\text{quotient})(\text{divisor}) + \text{remainder}$, with $|\text{remainder}| < |\text{divisor}|$.