Fraction
A fraction represents a part of a whole, written as one number over another separated by a line.
Formula
\frac{a}{b} \text{ (where } b \neq 0 \text{)}
Definition
A fraction is a way to show part of a whole thing or part of a group, written as two numbers with a line between them, like $3/4$: the bottom number (denominator) tells how many equal parts the whole is split into, and the top number (numerator) tells how many of those parts you have. More precisely, a fraction $a/b$ is the quotient of two integers where $b$ is not zero, expressing a ratio of parts to a whole, a division operation, or a point on the number line; it can be proper ($a < b$), improper ($a \ge b$), or written as a mixed number. Formally, $a/b$ is an element of the rational numbers $\mathbb{Q}$, defined as the equivalence class of ordered pairs $(a, b)$ with $b \neq 0$ under the relation $(a, b) \sim (c, d)$ iff $ad = bc$; $\mathbb{Q}$ itself is the field of fractions of the integers $\mathbb{Z}$, the smallest field containing $\mathbb{Z}$.
Example
Cut a pizza into $8$ equal slices and eat $3$ of them: you ate $3/8$ of the pizza, since the whole was split into $8$ parts and you had $3$. The same idea extends further: $5/6$ means $5$ divided by $6$, and on a number line it sits between $0$ and $1$, closer to $1$; the improper fraction $7/4$ equals $1$ and $3/4$ as a mixed number. Fractions like $2/3$, $4/6$, and $8/12$ all name the same rational number, belonging to the same equivalence class, and this construction generalizes: for any integral domain $R$, the field of fractions $\text{Frac}(R)$ is built from equivalence classes of pairs $(a, b)$ with $b \neq 0$.
Key Insight
Fractions only make sense when the pieces are equal, half of a pizza cut unevenly is not really $1/2$; equal parts are what make fractions fair and useful. They are not just parts of shapes, though: every fraction $a/b$ names an exact location on the number line, making fractions just as precise as whole numbers. The construction of $\mathbb{Q}$ from $\mathbb{Z}$ mirrors how real numbers are built from $\mathbb{Q}$ via Cauchy sequences or Dedekind cuts, and it shows that any integral domain embeds into a field: fractions are the algebraic mechanism that guarantees division.