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.
Ans: 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
1. Determine the Ceiling Function of 3.5 and – 3.5. Also, explain your answer.
Ans: ⌈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 symbols, but with top and bottom parts missing.
Study Tips to Master the Topic
Attend classes on Floor and Ceiling Functions - Paying attention to the teacher in the class helps a student in understanding the chapter in the best possible way. A student must be willing to listen and learn carefully about Floor and Ceiling Functions in order to calculate the sums.
The Functions must be understood well- Students should be well versed with the Functions that are present in this chapter. The Ceiling Function is: F (x) = minimum {a ∈ Z ; a ≥ p }
It is essential for the student to read and study on a daily basis so as to remember the Functions of the Ceiling and Floor, not only that but the students should study regularly to stay in touch with the chapter to perform better in the exams.
Students should remember the symbols that are used in Floor and Ceiling Functions. To denote the Function of Floor ceil, the ⌈ ⌉ symbol is used. It can be used as ⌈x⌉, ceil (x) or f(x) = ⌈x⌉ The notations are used according and hence the students must not be confused and they should learn it accordingly.
Students should understand the properties - The properties of Ceiling Function can be confusing so a student should learn properties to get a clearer picture of the Function of the Ceiling and solve the sum accordingly.
Students should be well versed with the formulas - The students must note the formulas and then practice the sums accordingly. It is essential to remember the formulas because it is the key to solving any problem.
Make notes during the classes - A student must make notes simultaneously during the classes so that the notes can help later during revision sessions. There are various Functions and properties of Floor and Ceiling Functions so it is very important to note them along with the formulas in order to calculate the sums correctly and understand the chapter overall.
Check knowledge with the help of quizzes- After solving questions from the books, students may test their knowledge through quizzes which will help them know the time that is taken for solving one question. Quizzes are a good way for memorizing something because it is spontaneous and quick so it helps the mind remember things easily. It is good to challenge the skills on a lighter note.
RD Shama to the rescue - Solving questions from RD Sharma is always beneficial and it must be taken seriously. The textbook by RD Sharma gives a general idea regarding the possible questions that will be present in the exam question paper so it is essential for the students to be well versed with it. It gives out an idea regarding the toughest questions and the solutions to the question in present in RD Sharma regarding the chapter on Functions of Floor and Ceiling can be downloaded from the official website of Vedantu in PDF Formats with ease.
Solving previous year question papers is also a great way to get a general idea of the Functions and properties of Floor and Ceiling. Previous year question papers are great for guiding the students into understanding and expecting the kind of questions in the exam question paper.
Discussing the chapter with friends and in a study group can help the students grasp and remember the topic in a better way. Study groups are interactive and filled with students with diversified knowledge on the subject. It is the right place for a student if they want to gain knowledge in depth without a lot of pressure and teachers. It is also essential for the students who are preparing for the exams and revising.
FAQs on Function Floor Ceiling
1. What are the floor and ceiling functions in mathematics?
The floor function, denoted as ⌊x⌋, gives the greatest integer that is less than or equal to the real number x. It essentially rounds a number down to the nearest integer. The ceiling function, denoted as ⌈x⌉, gives the smallest integer that is greater than or equal to x. It rounds a number up to the nearest integer. For example, for the number 5.7, the floor is 5 and the ceiling is 6.
2. What is the main difference between the floor function and the ceiling function?
The main difference lies in the direction of rounding. Here is a clear comparison:
- Direction: The floor function always rounds down to the nearest integer, while the ceiling function always rounds up.
- Value Relationship: For any real number x, the floor function satisfies ⌊x⌋ ≤ x, whereas the ceiling function satisfies ⌈x⌉ ≥ x.
- Effect on Negative Numbers: For a negative number like -4.3, the floor is ⌊-4.3⌋ = -5 (rounding down to the next lower integer), while the ceiling is ⌈-4.3⌉ = -4 (rounding up to the next higher integer).
3. How are the floor and ceiling functions represented using symbols and formulas?
The standard symbols and mathematical definitions are:
- Floor Function: It is represented by the symbol ⌊x⌋ or sometimes floor(x). It is also known as the Greatest Integer Function. The formula is f(x) = max{n ∈ Z | n ≤ x}, where Z is the set of all integers.
- Ceiling Function: It is represented by the symbol ⌈x⌉ or ceil(x). It is also known as the Least Integer Function. The formula is f(x) = min{n ∈ Z | n ≥ x}.
4. Can you provide some examples of calculating floor and ceiling values?
Certainly. Here are examples for different types of numbers:
- For a positive decimal (e.g., 9.2):
Floor: ⌊9.2⌋ = 9
Ceiling: ⌈9.2⌉ = 10 - For a negative decimal (e.g., -3.8):
Floor: ⌊-3.8⌋ = -4
Ceiling: ⌈-3.8⌉ = -3 - For an integer (e.g., 7):
Floor: ⌊7⌋ = 7
Ceiling: ⌈7⌉ = 7
5. Why are the graphs of floor and ceiling functions called 'step functions'?
The graphs of both the floor and ceiling functions are called step functions because they look like a series of steps on a staircase. The graph is not a continuous line but a collection of horizontal line segments. For every interval between two consecutive integers, the function's value remains constant, creating a flat 'step'. At each integer value, the function 'jumps' to the next value, which creates the vertical rise of the step, making the graph discontinuous.
6. What are some real-world applications or uses of floor and ceiling functions?
Floor and ceiling functions are very important in fields that require rounding to whole numbers. Key applications include:
- Computer Science: Used in algorithms for dividing problems, allocating memory, and converting data types (e.g., from floating-point numbers to integers). Functions like `FLOOR()` and `CEILING()` are common in programming languages like Python, C++, and SQL.
- Financial Calculations: For calculating prices where costs must be rounded up to the nearest dollar or cent, or for determining the number of items that can be purchased with a given amount of money.
- Age Calculation: To calculate a person's age in whole years from their birth date, the floor function is implicitly used.
7. What is a common misconception when applying the ceiling function to negative numbers?
A common mistake is to think 'rounding up' for negative numbers means moving away from zero (e.g., thinking ⌈-5.2⌉ is -6). However, the correct definition of the ceiling function is the smallest integer greater than or equal to the number. On a number line, 'greater than' means moving to the right. Therefore, for -5.2, the first integer to its right is -5. The correct calculation is ⌈-5.2⌉ = -5, not -6.
