 Ceiling function

The ceiling function gives the least integer large enough to meet a real lower bound. For a real number $x$,Thus $\lceil 2.7\rceil =3$, $\lceil -2.7\rceil =-2$, and $\lceil 2\rceil =2$. The operation rounds upward in the total order on the real numbers, which can move a negative input towards zero. Its difference from the input obeys $0\le \lceil x\rceil -x<1$. Python's math.ceil implements this least-integer definition. ^[ref].

The ceiling count for fixed-size batches is a direct application. Suppose each box holds at most $k$ items, with positive integer $k$, and there are $N\ge 0$ items. The smallest number of boxes with enough capacity is $\lceil N/k\rceil$: a candidate count $b$ must satisfy $bk\ge N$. For $N=101$, $k=20$, six boxes are needed, leaving $120-101=19$ unused places. Exactly 100 items need only five boxes, and zero items need zero boxes. Adding one to the floor function result unconditionally would therefore overcount at exact multiples of $k$.

For nonnegative integer $N$ and positive integer $k$, Euclidean division gives the exact identityWrite $N=qk+r$, $0\le r<k$. The count is $q$ when the remainder is zero and $q+1$ otherwise, and the right-hand side encodes these two cases. This argument specifies the domain of the identity; arbitrary negative denominators require their own sign conventions.

The ceiling function is a nondecreasing function, constant on each of the half-open intervals $(m-1,m]$, and has a discontinuity of a function at every integer. It is tied to the floor function by $\lceil x\rceil =-\lfloor -x\rfloor$. Adding an integer $j$ shifts its output by the same amount: $\lceil x+j\rceil =\lceil x\rceil +j$. At an integer the floor and ceiling agree; at every other input they differ by one. These statements refer to exact real numbers; an input already perturbed by floating-point arithmetic can land on the other side of a threshold.