Table of Contents
In the parent chapter on fractions, you saw the basic idea: a fraction represents “how many parts” of a whole, where the whole is divided into equal parts. In this chapter we focus on two special types of fractions: proper and improper fractions.
Proper fractions
A fraction has the form
$$
\frac{a}{b},
$$
where $a$ and $b$ are integers and $b \neq 0$. In that notation:
- $a$ is the numerator,
- $b$ is the denominator.
A proper fraction is a fraction in which the numerator is smaller than the denominator:
$$
\frac{a}{b} \text{ is proper if } 0 \le a < b \text{ and } b > 0.
$$
In everyday terms, a proper fraction represents less than one whole. For example:
- $\dfrac{1}{2}$ is proper, because $1 < 2$.
- $\dfrac{3}{7}$ is proper, because $3 < 7$.
- $\dfrac{9}{10}$ is proper, because $9 < 10$.
If you imagine a whole object (like a pizza) cut into $b$ equal slices, a proper fraction means you are taking fewer than all of those slices.
Recognizing proper fractions
To check if a fraction is proper:
- Look at the numerator and denominator.
- Make sure the denominator is positive.
- If the numerator is a non-negative number smaller than the denominator, the fraction is proper.
Examples:
- $\dfrac{2}{3}$: $2 < 3$, proper.
- $\dfrac{5}{5}$: $5 \not< 5$, not proper.
- $\dfrac{7}{4}$: $7 > 4$, not proper.
- $-\dfrac{3}{5}$: the numerator is negative. This is not called a proper fraction in the usual sense; it is a negative fraction. (Proper vs. improper usually refers to the sizes of positive numerator and denominator.)
You will see more about negative fractions and signs in other chapters; here we focus on the size comparison for positive fractions.
Proper fractions on the number line
If you place numbers on a number line, proper fractions lie between $0$ and $1$:
- $0 \le \dfrac{a}{b} < 1$ when $\dfrac{a}{b}$ is proper.
So if you see a positive fraction between $0$ and $1$ on the number line, it is a proper fraction.
Improper fractions
An improper fraction is a fraction in which the numerator is greater than or equal to the denominator:
$$
\frac{a}{b} \text{ is improper if } a \ge b > 0.
$$
Improper fractions represent one whole or more. For example:
- $\dfrac{4}{3}$ is improper, because $4 > 3$.
- $\dfrac{7}{7}$ is improper, because $7 = 7$.
- $\dfrac{15}{4}$ is improper, because $15 > 4$.
Again thinking of a whole divided into $b$ equal slices, an improper fraction means you have all the slices of one whole and possibly extra slices from more wholes.
Recognizing improper fractions
To check if a fraction is improper (for positive numerator and denominator):
- Look at the numerator and denominator.
- If the numerator is equal to or larger than the denominator, the fraction is improper.
Examples:
- $\dfrac{5}{3}$: $5 > 3$, improper.
- $\dfrac{9}{9}$: $9 = 9$, improper.
- $\dfrac{2}{7}$: $2 < 7$, not improper (this one is proper).
Again, we are not focusing here on negative signs; we are comparing sizes of numerator and denominator for positive values.
Improper fractions on the number line
On the number line, an improper fraction is at least $1$:
- $\dfrac{a}{b} \ge 1$ when $\dfrac{a}{b}$ is improper (with $a, b > 0$).
So fractions like $\dfrac{5}{4}, \dfrac{9}{8}, \dfrac{7}{7}$ are at $1$ or to the right of $1$.
Why distinguish proper and improper fractions?
The difference matters because:
- Proper fractions describe amounts less than one whole.
- Improper fractions describe amounts equal to or more than one whole.
This distinction is useful for:
- Interpreting word problems (e.g., “less than one hour” vs. “more than one meter”),
- Deciding how to rewrite a fraction (for example, as a mixed number, which you will see in the next chapter),
- Understanding where a fraction belongs on the number line (between $0$ and $1$, or beyond $1$).
Comparing proper and improper fractions
Here are paired examples to build intuition:
- $\dfrac{3}{4}$ (proper) vs. $\dfrac{5}{4}$ (improper):
- $\dfrac{3}{4}$ is three out of four equal parts, less than one whole.
- $\dfrac{5}{4}$ is one whole (four quarters) plus one extra quarter.
- $\dfrac{7}{10}$ (proper) vs. $\dfrac{12}{10}$ (improper):
- $\dfrac{7}{10}$ is $0.7$, less than $1$.
- $\dfrac{12}{10}$ is $1.2$, more than $1$.
In each pair, the denominator is the same, but the numerator tells you whether you have less than, equal to, or more than one whole.
Converting between forms: overview
The next chapter will go into detail about mixed numbers and how to work with them. For now, it is enough to understand these ideas at a basic level:
- Every improper fraction can be written either as a whole number or as a mixed number (a whole number plus a proper fraction).
- Example: $\dfrac{7}{3}$ can be seen as $2$ wholes and $\dfrac{1}{3}$ left over.
- Every mixed number can be written as an improper fraction.
- Example: $2 \dfrac{1}{3}$ corresponds to $\dfrac{7}{3}$.
These conversions do not change the amount; they only change how you express it. The key point for this chapter is that the fraction part of a mixed number is always a proper fraction.
Summary
- A proper fraction has a numerator smaller than the denominator ($0 \le a < b$). It represents an amount less than one.
- An improper fraction has a numerator equal to or larger than the denominator ($a \ge b > 0$). It represents one or more wholes.
- On the number line:
- Proper fractions lie between $0$ and $1$.
- Improper fractions are at $1$ or to the right of $1$ (for positive fractions).
- Improper fractions can be connected to mixed numbers, where the fraction part is always proper; the details of that conversion belong to the following chapter.