Function Floor Ceiling

download app banner

Floor and Ceiling Functions

In the field of Mathematics and Computer Programming, floor function and ceiling function are the two important functions used quite often. As an example of the floor and ceiling functions, the floor and ceiling of a decimal 4.41 will be 4 and 5 respectively.  

So using these two functions, we are able to obtain the nearest integer in a number line of an assigned decimal. Here, we will discuss the function floor ceiling definition, notation, graphs, symbols, properties, and examples.


Introduction to Ceiling Function

The ceiling math Function is classified under Trigonometry functions and Excel Math. Floor ceil enables returning a number that is rounded up to the closest enough integer or multiple of significance. The ceiling function was first introduced in MS Excel 2013.

It is a function where the smallest successive integer is returned successfully. This is to say; the ceiling function of a real number ‘p’ is the least integer that is greater than or equals to (≥) the given number ‘p’. Mathematically, the ceiling function is thus described as:

F (x) = minimum {a ∈ Z ; a ≥ p }


Symbol and Notation of Ceiling Function

The ceiling function is also referred to as the smallest integer function. The notation used to denote the function of floor ceil is ⌈ ⌉. It can be used as ⌈x⌉, ceil (x) or f(x) = ⌈x⌉

The symbol of the function floor ceiling is also a kind of square bracket. i.e.⌊ ⌋.


Properties of Ceiling Function

Let us take into account that p and q are two real numbers and ceil (x) = ⌈x⌉. Some of the essential properties of the function floor ceiling are as follows:

  • ⌈p⌉ + ⌈q⌉ – 1 ≤ ⌈p + q⌉ ≤ ⌈p⌉ + ⌈q⌉

  • ⌈p + a⌉ = ⌈p⌉ + a

  • ⌈p⌉ = a; iff x ≤ p < p + 1

  • ⌈p⌉ = a; iff p – 1 < a ≤ p

  • a < ⌈p⌉ iff a < p

  • a ≤ ⌈p⌉ iff p < a

Ceiling Function Formula

The formula used to find the ceiling value for any given value is:

f (x) = minimum { a ∈ Z ; a ≥ x }

It is denoted by:

F(x) = ⌈x⌉ = Smallest Closest Successive Integer of specified value

Let’s undertake a ceiling function example to understand the concept better.

Example: Determine the ceiling value of 4.8.

Solution:

Given, x = 4.8

As we can notice, the integers greater than 4.8, are 5, 6,7,8,9...and so on.

The closest enough integer here is 5.

Therefore, f(4.8) = ⌈4.8⌉ = 5


Introduction To Floor Function

Floor function is the reverse function of the ceiling function. It provides us with the largest nearest integer or multiple of significance of the specified value.


Floor Function Formula

The formula used to find the floor value for any given value is as below and is denoted by:

f(x) = ⌊x⌋ = Highest Nearest Integer of specified value

Let’s take an example

Example: Find the floor value of 4.8.

Answer: Given, x = 4.8

If we observe, the number of integers less than 4.8 is 4, 3,2,1,0, and -1,-2 and so on.

So, the largest integer will be 4.

f(4.8) = ⌊4.8⌋ = 4

Ceiling Function Graph

The graph of ceiling function is a discrete graph that contains discontinuous line segments with one end having a dark dot (closed interval) and another end having an open circle (open interval). The ceiling function is a type of a step function because it looks like a staircase.

The ceiling function graph is shown below:


[Image will be uploaded soon]


Solved Examples

Example 1:

Determine the ceiling function of 3.5 and – 3.5. Also, explain your answer.

Solution:

⌈3.5⌉ = 4 and ⌈- 3.5⌉ = – 3

Explanation:

The ceiling function of a real number is the least integer number greater than or equal to  (≥ ) an assigned number.

In the case of 4.5, the integers more than 3.5 are 4, 5, 6, 7, 8, 9 …..

The smallest of all is 4.

Thus, ⌈3.5⌉ = 4.

In the case of -3.5, the integers that are greater than – 3.5 are – 4, – 3, – 2, – 1…

Therefore, the smallest of them is – 4

Hence, ⌈- 3.5⌉ = – 3

Fun Facts

  • Though floor function and ceiling function differ in function, the integer of both floor and ceiling remains the same. In other words, the floor and ceiling of 3 are 3 for both of them.

  • Both the floor and ceiling functions are denoted by square brackets symbol, but with top and bottom parts missing.

FAQs (Frequently Asked Questions)

 1. What is the Difference Between the Ceiling Function and the Floor Function?

Answer: The floor and ceiling function do not only have a different definition but they also differ in other ways too that are as below:


Basis

Ceiling Function

Floor Function

Definition

Returns the smallest value for the specified number

Returns the highest value for a given number

Graphical Representation

In the ceiling function graph, it consists of a solid dot on the right and an open dot on the left

In the floor function graph, the solid dot is on the left whereas the open dot is on the right

Value

If 3.6 is a specified value, then the ceiling value is 4.

If 3.6 is a specified value, then the floor value is 3

2. What is the Use of the Floor Ceil Function?

Answer: Being a financial analyst, we can use the ceiling math function in establishing the pricing after offering discounts or converting currency etc. When preparing financial models, it enables us to round up the numbers to the nearest multiple or integer as per the requirement.

3. What is Meant By Integer Floor Function?

Answer: The integer part or also known as the integral of x, often represented [x] is ⌈x⌉, if x is nonnegative, and ⌈x⌉ otherwise. In other words, this is the integer that contains the highest absolute value less than or equal to the absolute value of x.