$\lceil m/n \rceil$ for naturals $m$ and $n \ge 1$: the least $q \in \mathbb{N}$ with $m \le n q$

Let $m, n \in \mathbb{N}$ with $n \ge 1$ (The natural numbers $\mathbb{N}$ (von Neumann), Order on the natural numbers, On $\mathbb{N}$ the order is membership: $m < n \iff m \in n$), and put

$Q(m,n) \;:=\; \{\, q \in \mathbb{N} \ :\ m \le n\,q \,\} ,$

the multiplication being that of $\mathbb{N}$ (Multiplication of natural numbers).

$Q(m,n)$ is nonempty, so the definition below has something to pick from. Since $n \ne 0$, Every nonzero natural number is a successor gives $n = \sigma(d)$ for some $d \in \mathbb{N}$, and then $n \cdot m = \sigma(d)\cdot m = d\cdot m + m = m + d\cdot m$ by the successor-left law of Distributivity and the successor law for multiplication and the commutativity of addition (Addition is commutative), so $m \le n\cdot m$ by the definition of the order (Order on the natural numbers), which asks for a natural $j$ with $m + j = n m$ and is met by $j = d\cdot m$. Hence $m \in Q(m,n)$.

Definition. $\big\lceil m/n \big\rceil$ is the least element of $Q(m,n)$, which exists by the well-ordering principle (The well-ordering principle) applied to the nonempty subset $Q(m,n)$ of $\mathbb{N}$. It is a natural number, and it is defined for $n \ge 1$ only.

Four clauses, recorded here because they are what the notation is used for.

(a) $m \le n\big\lceil m/n\big\rceil$. This is membership of $\big\lceil m/n\big\rceil$ in $Q(m,n)$.

(b) Minimality. If $q \in \mathbb{N}$ satisfies $m \le n q$ then $\big\lceil m/n\big\rceil \le q$; equivalently, every $q$ with $q < \big\lceil m/n\big\rceil$ satisfies $n q < m$, by trichotomy (Trichotomy of the order on $\mathbb{N}$).

(c) Two values read off directly. $\big\lceil 0/n\big\rceil = 0$, since $0 \le n\cdot 0$ makes $0$ the least element of $\mathbb{N}$ that qualifies; and $\big\lceil m/1\big\rceil = m$, since $m \le 1\cdot m = m$ while $m \le 1\cdot q$ forces $m \le q$ (Zero and one under multiplication, Multiplication is commutative).

(d) The reading in $\mathbb{R}$. With $\iota : \mathbb{N} \to \mathbb{R}$ the canonical natural (The canonical natural $\iota(n) = n \cdot 1_F$ of a field), clause (a) gives $\iota(m) \le \iota(n)\,\iota\big(\lceil m/n\rceil\big)$ by the multiplicativity of $\iota$ (clause 0 of Laws of finite sums and products in $\mathbb{N}$, and $\iota\big(\sum_{k<n} a_k\big) = \sum_{k<n} \iota(a_k)$) and its strict monotonicity (clause 7); and $\iota(n) > 0$ because $n \ge 1$. Since $\mathbb{R}$ is an ordered field (Ordered field, Field), dividing by $\iota(n)$ gives

$\frac{\iota(m)}{\iota(n)} \ \le\ \iota\Big(\big\lceil m/n\big\rceil\Big) .$

• This is not a floor and it is not a ceiling function. It is defined for a pair of natural numbers with $n \ge 1$, its value is a natural number, and it is fixed by one order property and one minimality property. It is not defined for a real argument, it does not extend to negative numbers, and it carries no division: the symbol $m/n$ inside the brackets is part of the notation and not an operation performed anywhere above.

• Why it is introduced at all. The strong form of the pigeonhole principle says that some fibre has at least "the average, rounded up" elements, and that phrase needs a name for the rounding. The least $q$ with $m \le nq$ is exactly what the proof produces, and the well-ordering principle is exactly what makes it exist, so nothing stronger is required.

• Nothing among this page's declared prerequisites supplies a division with remainder, and the definition above deliberately does not attempt one: no $r$ with $m = nq + r$ and $r < n$ is produced or claimed here.