Rational number ($\mathbb{Q}$)

A rational number is a quotient of two integers with nonzero denominator.

A binary tree enumerating every positive reduced fraction exactly once. Begin with boundary fractions $0/1$ and $1/0$; between neighboring fractions $a/b<c/d$, insert their mediant $(a+c)/(b+d)$ and recurse into the two new intervals. The determinant relation $bc-ad=1$ is preserved, and the new denominator is $b+d$.

A fraction $a/b$ represents the quotient of its numerator $a$ by its nonzero denominator $b$. For integers $a,b$ this is a rational number. Different fractions can represent the same number: $252/198=14/11$. Dividing both entries by their greatest common divisor gives a reduced fraction, with the sign convention that the denominator is positive.

A nonnegative fraction with positive denominator and numerator smaller than its denominator, so its value lies in $[0,1)$. The zero value has the reduced fraction representation $0/1$. The term is sometimes extended to negative fractions by comparing absolute values; the nonnegative convention is the one used here.

A fraction of the form $1/n$ for a positive integer $n$. Its numerator is 1 and its denominator determines it uniquely.

An Egyptian fraction expresses a nonnegative rational number as a finite sum of positive unit fractions. A computational representation may permit repeated denominators, while the classical distinct-denominator convention adds a further requirement. The empty sum represents zero. For example, $3/7=1/3+1/11+1/231$; each denominator is a positive integer. A representation that permits repetition also has the simple expansion $n/d={\sum }_{j=1}^{n}1/d$ for nonnegative integers $n$ and positive $d$.

For a positive proper residual fraction $n/d$, with $0<n<d$, choose $q=\lceil d/n\rceil$, append the unit fraction $1/q$, and replace the residual by $(nq-d)/(dq)$ in lowest terms. This is a greedy algorithm: the chosen unit fraction is the largest one not exceeding the residual.

The inequalities $q-1<d/n\le q$ imply $0\le nq-d<n$. Thus the numerator of the reduced residual strictly decreases while positive, proving algorithm termination by a nonnegative-integer loop variant. The loop invariant is that the accumulated sum plus the residual is the original rational number. Since $q\ge 2$ and $n/d<1/(q-1)$, the new residual is less than $1/(q(q-1))\le 1/q$. If it remains positive, its next chosen denominator is therefore strictly greater than $q$; this expansion also satisfies the distinct-denominator convention. The proof does not bound the denominators by the input's fixed-width range. Arbitrary-precision arithmetic, checked conversion, or a different representable expansion can therefore be necessary in code.

Exact arithmetic on rational numbers represented by integer numerators and nonzero integer denominators. Common factors can be removed with the greatest common divisor. Exact rational arithmetic avoids rounding error when deciding equality or integrality.

For nonzero denominators, $a/b=c/d$ exactly when $ad=bc$. For positive denominators, $a/b<c/d$ exactly when $ad<bc$. This compares fractions with exact integer arithmetic, avoiding rounding from floating-point arithmetic.

The numerator is the upper entry $a$ in a fraction $a/b$.

The denominator is the nonzero lower entry $b$ in a fraction $a/b$.

The increasing list of reduced rational numbers in $[0,1]$ with denominators at most its level $Q$, including zero and one.

A member of a Farey sequence. Two distinct level-$Q$ fractions have ordinary distance at least $Q^{-2}$. On the quotient circle identify zero and one before forming a set; the shorter wrapped distance of two remaining distinct fractions obeys the same bound.

A reduced fraction $a/b$ has coprime integers $a,b$ and positive denominator $b$. Distinct reduced fractions with denominators at most $Q$ differ by at least $Q^{-2}$, since their difference has a nonzero integer numerator over a denominator at most $Q^2$.

Between any two distinct real numbers lies a rational number.

A real number is rational exactly when its expansion in any integer base is eventually periodic.

The shortest prefix before the repeating tail of a decimal representation. For a reduced fraction with denominator $2^u{5}^{v}m$, where $m>1$ is coprime to ten, its length is $\max (u,v)$; the tail length is given by decimal period from multiplicative order.

A positional expansion in an integer base represents a rational number exactly when its digit sequence is eventually periodic. Long division proves the forward implication because only finitely many remainders occur; summing a periodic tail as a geometric series proves the converse.