The floor and ceiling functions in mathematics and computer science, respectively, transfer a real number to the largest preceding or least following integer.

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While “Floor” delivers the biggest integer less than or equal to x, “Ceil” returns the lowest integer equal to or greater to x (that is, gets round up to the closest integer)

**Ceil vs Floor Functions**

**The main differences between a ceil and a floor function are that of their rounding up function and graphical representation. Ceil function takes the lowest integer number, meanwhile, the floor function invokes the largest integer number.**

Ceil function is defined as the lowest integer expression. It reduces the integer to its lowest value and rounds them up.

In a graphical representation of ceil functions, there is a solid dot towards the right and an open dot towards the left. The formula for a ceil function is f (x) = min. {a ∈ Z; a ≥ x }.

The floor function is also recognized as the leading integer function. It rounds down the integer and returns it to its largest value.

The floor function is represented graphically as a solid dot towards the left and an open dot towards the right. The floor function formula is as follows: f(x) = x = Highest Nearest Integer with the supplied value.

**Comparison Table Between Ceil And Floor Functions**

Parameters of Comparison | Ceil Functions | Floor Functions |

Functions | Returns to the smallest value | Returns to the largest value |

Rounding | It rounds up the integer | It rounds down the integer |

Graphical Representation | Upon the right, there is a solid dot, then towards the left, there is an open dot. | Towards the right is an open dot, and on the left is a substantial dot. |

Formula | f (x) = minimum { a ∈ Z ; a ≥ x } | f(x) = ⌊x⌋ = highest closest integer of said value |

Other Name | It is also known as the smallest integer function | It is also known as the greatest integer function |

**What is a Ceil Function?**

A ceiling function is one that yields the fewest record for consecutive digits. In other terminology, the ceiling function of a genuine figure x is the simplest integer that is either equal to or greater than x.

The ceiling function is as described in the following: f (x) = min {a ∈ Z ; a ≥ x }

The least integer function is another name for the ceiling function. This function is denoted by the notation.

It is possible to write it as x, ceil (x), or f(x) = x. The floor function symbol is a type of square bracket as well.

The foregoing are among the most important characteristics of the ceiling functions:

[x⌉ + ⌈y⌉ – 1 ≤ ⌈x + y⌉ ≤ ⌈x⌉ + ⌈y⌉; ⌈x + a⌉ = ⌈x⌉ + a; a < ⌈x⌉ iff a < x; a ≤ ⌈x⌉ iff x < a; ⌈x⌉ = a; and iff x ≤ a < x + 1

The ceiling function graph is a categorical graph made up of jagged parallel lines, each end of which is designated by a black dot (closed interval) and the other by use of an open circle.

Because it resembles a stairwell, the ceiling function is a subtype of the scaling factor.

**What is a Floor Function?**

The inverse of the ceiling function is the floor function. It delivers the closest integer or several of the specified number’s relevance.

The biggest integer less than or equal to xx is denoted by the floor function (also known as the greatest integer function) of a real number xx.

Assume x is a real number. The [x] or floor [x] function of x is defined as the biggest integer that is less than or equal to x.

The formula for determining the floor value for any given value is as follows, indicated by: f(x) = ⌊x⌋ = Highest Nearest Integer of specified value

The floor function in C language may be used in the ANSI/ISO 9899-1990 version. The floor function is useful for arithmetic functions such as the Moebius -function and the Mangoldt -function, among other things.

As financial analysts, we may utilize the floor math function to determine price after providing discounts or converting currencies, among other things.

It allows us to round up figures to the next multiple or integer as needed while creating financial models.

**Main Differences Between Ceil And Floor Functions**

- The Ceil function is also known as the lowest integer operator. Meanwhile, the biggest integer value is another name for the floor function.
- The ceil function rounds up the integer whereas the floor function rounds down the integer.
- The Ceil algorithm invokes the smallest integer value and invokes the greatest integer number.
- The ceiling function is visually represented by a substantial dot towards right and an open dot towards left, whereas the floor function is graphically displayed by an open dot towards left.
- While speaking of formulae for Ceil function, it is f (x) = lowest a Z ; an x for the floor function and f(x) = x = highest closest integer of stated value

**Conclusion**

Both, the ceil (short for ceiling) and the floor functions are mathematical functions.

It is frequently used in mathematical calculations as well as in computer science in applications like spreadsheets, database systems, and computer languages such as C, C+, and Python.

The Ceil function returns the lowest integer value higher than or equivalent to the supplied value. Meanwhile, the floor function returns the biggest value that is less than or equal to the supplied integer.

The number supplied is always a double-precision number.

While difficult to grasp at first, the greatest and least integer functions are valuable tools. The bulk of these functions’ usages are as you describe, and they may be modified in the same way as any other function.

The functions are frequently seen in Postal Rate Slabs, Income Tax Slabs, Goods and Services Tax Slabs, Examination Grades, and so on.

Though the functions of the floor and ceiling functions change, the integer of both the floor and ceiling functions stays the same. In other words, they both have a floor and a ceiling of four.

**References**

1. https://arxiv.org/abs/2003.06885

2.https://www.sciencedirect.com/science/article/pii/B9780444821065500247