The ceil function (short for "ceiling") returns the smallest integer that is greater than or equal to a given number. It effectively rounds a number up to the nearest integer.

Key Details:

• Mathematical Definition: For any real number $ x $, $ \lceil x \rceil $ is the least integer not less than $ x $.

  • Example: $ \lceil 2.3 \rceil = 3 $, $ \lceil -2.3 \rceil = -2 $.

• In Programming Languages:

  • C/C++: ceil() is in <cmath> or <math.h>. Returns double.

    #include <cmath>
    std::cout << std::ceil(2.3); // Output: 3

  • Java: Math.ceil() returns double.

    Math.ceil(2.3); // Returns 3.0

  • JavaScript: Math.ceil() returns the smallest integer ≥ input.

    Math.ceil(2.3); // Returns 3

  • Python: Use math.ceil() (from math module).

    import math
    math.ceil(2.3) # Returns 3

  • SQL (Oracle, Redshift, etc.): CEIL(n) or CEILING(n) returns smallest integer ≥ $ n $.

    SELECT CEIL(2.3) FROM DUAL; -- Returns 3

  • Excel: CEILING(number, significance) rounds up to the nearest multiple of significance (not just to the next integer).

    =CEILING(4.42, 0.05)  // Returns 4.45 (nearest nickel)

Special Cases:

• If the input is already an integer, ceil returns the same value.

• For negative numbers, it rounds toward zero (e.g., ceil(-2.3) is -2).

• If the input is NaN, ±∞, or ±0, the function returns the same value.

Note: In most languages, ceil() returns a floating-point number (e.g., double or float) even when the result is an integer.