Fraction
Fractions & DecimalsA 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. It is written as two numbers with a line between them, like $3/4$. The bottom number tells how many equal parts the whole is split into, and the top number tells how many of those parts you have.
Example
If you cut a pizza into $8$ equal slices and eat $3$ of them, you ate $3/8$ of the pizza. The pizza was split into $8$ parts and you had $3$ of them.
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.
Definition
A fraction $a/b$ represents the quotient of two integers where the denominator $b$ is not zero. It expresses a ratio of parts to a whole, a division operation, or a point on the number line. Fractions can be proper ($a < b$), improper ($a \ge b$), or written as mixed numbers.
Example
$5/6$ means $5$ divided by $6$, or $5$ parts out of $6$ equal parts. On a number line, $5/6$ sits between $0$ and $1$, closer to $1$. The fraction $7/4$ is improper and equals $1$ and $3/4$ as a mixed number.
Key Insight
Fractions are not just parts of shapes - they are numbers on the number line. Every fraction $a/b$ names an exact location, making fractions just as precise as whole numbers.
Definition
A fraction $a/b$ 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 equivalence relation $(a, b) \sim (c, d)$ iff $ad = bc$. The set $\mathbb{Q}$ is the field of fractions of the integers $\mathbb{Z}$, the smallest field containing $\mathbb{Z}$.
Example
The fractions $2/3$, $4/6$, and $8/12$ all belong to the same equivalence class and name the same rational number. In abstract algebra, 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
The construction of $\mathbb{Q}$ from $\mathbb{Z}$ mirrors how real numbers are built from $\mathbb{Q}$ via Cauchy sequences or Dedekind cuts. The field-of-fractions construction shows that any integral domain embeds into a field - fractions are the algebraic mechanism that guarantees division.