Proper Fraction
Fractions & DecimalsA 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 a fraction where the top number (numerator) is smaller than the bottom number (denominator). Proper fractions are always less than $1$ whole. Examples: $1/2$, $3/4$, $5/8$.
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, so $3/5$ is "proper" - it fits inside one whole.
Key Insight
The word "proper" hints that these fractions stay in their place between $0$ and $1$. They represent parts of a single whole, not more than a whole.
Definition
A proper fraction is a fraction $a/b$ where $0 \le a < b$ and $b > 0$. Its absolute value is strictly less than $1$, so it lies between $-1$ and $1$ on the number line (excluding $-1$ and $1$). All proper fractions are rational numbers in the interval $(-1, 1)$.
Example
$7/9$ is proper ($7 < 9$), and equals approximately $0.778$, which is between $0$ and $1$. Contrast with $9/7$, which is improper ($9 > 7$) and equals approximately $1.286$, lying outside the interval $(0, 1)$.
Key Insight
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 a proper fraction.
Definition
A proper fraction $a/b$ satisfies $|a| < |b|$. In the context of partial fraction decomposition, a proper rational function $P(x)/Q(x)$ satisfies $\deg(P) < \deg(Q)$. Improper rational functions must first be divided to extract a polynomial part before partial fractions can be applied - exactly mirroring the integer/proper-fraction split.
Example
$(x^3+2x)/(x^2+1)$ is an improper rational function. 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 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 $|\text{remainder}| < |\text{divisor}|$.