Fraction

A fraction is an expression representing division, conventionally written as $\frac{a}{b}$ or $a/b$, where $b$ is nonzero. In elementary arithmetic, its terms are integers, and its value is a rational number. Fractions describe equal parts of a whole, but also express quantities greater than one, negative quantities, and division results. More broadly, fractional notation can contain algebraic expressions rather than integers. (openstax.org)

Notation and interpretation

The upper term $a$ is the numerator, and the lower term $b$ is the denominator. The separating bar indicates division and groups the entire numerator and denominator. Thus $\frac{a+c}{b+d}$ means that the whole expression $a+c$ is divided by the whole expression $b+d$. (openstax.org)

For positive integers, a fraction has an equal-parts interpretation: $\frac{3}{4}$ represents three parts when a unit is divided into four equal parts. Equal size is essential; three pieces selected from four unequal pieces need not constitute three-quarters of the whole. Fractions can also be located on a number line, where $\frac{3}{4}$ lies three quarter-unit intervals to the right of zero. (openstax.org)

A denominator of zero is excluded because division by zero is undefined in ordinary arithmetic. A zero numerator is permitted: $\frac{0}{b}=0$ whenever $b\ne0$. Signs obey the usual division rules:

Consequently, a negative fraction is ordinarily written with its minus sign before the expression or in the numerator. (openstax.org)

Forms of fractions

For nonnegative integer numerators and positive integer denominators, a proper fraction has numerator smaller than denominator, such as $\frac{2}{5}$. An improper fraction has numerator at least as large as denominator, such as $\frac{7}{5}$ or $\frac{5}{5}$. Improper fractions are valid expressions, not errors. (openstax.org)

A mixed number combines a whole-number part and a proper fraction:

In this notation, juxtaposition means addition rather than multiplication. A unit fraction has numerator one, such as $\frac{1}{6}$. A complex fraction contains a fraction within its numerator, denominator, or both; for example, $\frac{1/2}{3/4}$ represents a division of two fractions. (openstax.org)

Equivalence and simplification

Different fractional expressions can represent the same value. Multiplying numerator and denominator by the same nonzero number preserves the quotient:

Thus $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{50}{100}$ are equivalent fractions. Conversely, a common nonzero factor can be divided out of both terms. (openstax.org)

An integer fraction is in lowest terms when numerator and denominator have no common positive divisor other than one. Dividing both by their greatest common divisor produces this form. For example,

With a positive denominator, every rational number has a unique lowest-terms representation; zero is represented by $\frac{0}{1}$. (openstax.org)

For nonzero denominators, equality can be tested by

For positive denominators, comparison similarly reduces to comparing $ad$ and $bc$. This distinguishes numerical size from the apparent size of the written terms: $\frac{3}{5}<\frac{2}{3}$, because $9<10$. (math.utah.edu)

Arithmetic operations

Addition and subtraction require expressions with a common denominator:

For instance, $\frac{1}{2}+\frac{1}{3}=\frac{5}{6}$, not $\frac{2}{5}$. The least common multiple of positive denominators supplies the least common denominator and often reduces the amount of calculation. (openstax.org)

Multiplication combines numerators and denominators directly:

It can represent taking a fraction of a quantity: half of three-quarters is $\frac{1}{2}\times\frac{3}{4}=\frac{3}{8}$. Common factors may be cancelled before multiplication. (openstax.org)

Division uses the reciprocal of the divisor:

For example, $\frac{3}{4}\div\frac{1}{2}=\frac{3}{2}$. Multiplication or division by a fraction therefore need not make a positive quantity larger or smaller, respectively. (openstax.org)

Decimals and percentages

A fraction can be converted to a decimal by dividing its numerator by its denominator. Integer fractions yield terminating or eventually repeating decimal expansions: $\frac{1}{8}=0.125$, whereas $\frac{1}{3}=0.\overline{3}$. Conversely, every terminating or eventually repeating decimal represents a rational number. Irrational numbers cannot be expressed as fractions of integers. Both classes belong to the real numbers. (openstax.org)

A percentage expresses a quantity in hundredths, so $25\%=\frac{25}{100}=\frac{1}{4}$. A terminating decimal likewise gives an exact fraction with a power-of-ten denominator. A rounded decimal, however, may only approximate the original fraction; $0.33$ is not exactly $\frac{1}{3}$. (openstax.org)

Algebraic and formal extensions

In algebra, fractional notation extends to expressions involving variables. A quotient of polynomials is a rational expression. The familiar arithmetic rules still apply, but denominator restrictions must be preserved. For example,

Cancellation removes common factors, not arbitrary terms in sums. A function defined by the original quotient remains undefined at $x=1$, even though the simplified expression can be evaluated there. (openstax.org)

In set theory, rational numbers can be constructed from integer pairs $(a,b)$, with $b\ne0$. The rule $(a,b)\sim(c,d)$ when $ad=bc$ defines an equivalence relation. Each equivalence class collects all fractional representations of one rational number. Under the induced arithmetic operations, these numbers form a field, providing an abstract foundation for fraction arithmetic. (math.utah.edu)