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The ceiling function in mathematics, denoted as ⌈x⌉ or ceil(x), returns the smallest integer greater than or equal to a given real number x. It effectively rounds a number up to the nearest integer.

For positive numbers: Any fractional part causes rounding up.
Example: ⌈2.3⌉ = 3, ⌈7.1⌉ = 8.
For negative numbers: It rounds toward zero (upward in value).
Example: ⌈-2.3⌉ = -2, ⌈-7.6⌉ = -7.
For integers: The value remains unchanged.
Example: ⌈5⌉ = 5.

This function is also known as rounding toward positive infinity and is widely used in computer science, finance, and discrete mathematics for ensuring values are not underestimated.

In Programming:

C#: Math.Ceiling(double) or Math.Ceiling(decimal) returns the smallest integer ≥ the input.
JavaScript: Math.ceil(x) performs the same operation.
Excel: CEILING.MATH(number, [significance], [mode]) rounds up to the nearest multiple of significance, with optional mode control for negative numbers.

Key Identity:

The ceiling function is related to the floor function by the identity:
⌈x⌉ = -⌊-x⌋.

This relationship helps in deriving ceiling values using floor operations.

Microsoft Learn

learn.microsoft.com › en-us › dotnet › api › system.math.ceiling

Math.Ceiling Method (System) | Microsoft Learn

public: static System::Decimal Ceiling(System::Decimal d); ... A decimal number. ... The smallest integral value that is greater than or equal to d. Note that this method returns a Decimal instead of an integral type.

MDN Web Docs

developer.mozilla.org › en-US › docs › Web › JavaScript › Reference › Global_Objects › Math › ceil

Math.ceil() - JavaScript | MDN

The Math.ceil() static method always rounds up and returns the smallest integer greater than or equal to a given number.

Discussions

What is the purpose of the Floor and Ceiling Functions?

You'd be amazed how much we use the floor function in number theory. After all, our main area of concern is the natural numbers, so whenever we step outside them and use other kinds of math (which is all the time), we need a way to translate our answer back to the integers. One simple example: counting digits of numbers. The first n-digit number is always 10n-1, so if we have an unknown number x and want to know how many digits it has, what we're "really" asking is which two powers of 10 it lies between. Thus 10n-1 <= x <= 10n n-1 <= log x <= n Thus n-1 is floor(log x), or n is ceil(log x), whichever you like. We've answered our strictly integral question by stepping outside the integers, then using the floor function to step back in. These are base-10 logs, by the way. Counting digits is the one context in pure math where we use them. More on reddit.com

r/askmath

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0

February 2, 2014

I just discovered this pattern in the floor and ceiling functions

de Morgan's laws but epic More on reddit.com

r/mathematics

19

401

June 28, 2023

Low-floor, High-ceiling Problems for High Schoolers

Here are some good & brief puzzles that can usually be solved in about the time it takes to drink a cup of coffee: http://www.datagenetics.com/blog/june22014/index.html Actually, many of the rest of the posts at datagenetics might provide some material for good challenges (the one about distributing passwords will open their eyes to the possibilities of polynomials!): http://www.datagenetics.com/blog.html Martin Gardner's Aha! Gotcha! (about paradoxes) and Aha! Insight! (about finding simpler ways to solve complex problems) is another good source of the type of puzzles you're seeking (especially the latter). There's also: MindYourDecision.com's Sunday Puzzles - Try springing the evil geometry puzzle on them! James Tanton's Curriculum Inspirations - Don't miss the video archive ! More on reddit.com

r/math

20

27

January 11, 2017

How to return the ceiling of a number without using ceil() function?

double ceil(double a) { auto b = double((long long)a); if (a < 0 || b == a) { return b; } else { return b+1; } } EDIT: fixed negative number behavior and casted to larger to integral type per u/bad_investor13 More on reddit.com

r/cpp_questions

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September 12, 2023

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