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Ceiling function is referred to as the least integer function. The ceiling function can be defined in simple words as the smallest function of an integer belonging to a real number 'x' which is not smaller than 'x.' Ceiling function is denoted by [x].
Read Also: Greatest Common Divisor
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Key Terms: Integer, function, round off, real number, successive, ceiling.
Ceiling Function Definition
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The ceiling function brings back the smallest successive integer that is larger or equal to the given number x. It is defined numerically as follows:
f (x) = minimum {a ∈ Z; a ≥ x}
- It is also known as the smallest integer function since it returns the smallest integer closest to the specified value. It is denoted by two brackets: [ ]
- It can also be denoted numerically as [x], ceil (x) or f(x) = [x]
Properties of Ceiling Function
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Following are the properties of the ceiling function:
- ⌈x⌉ + ⌈y⌉ – 1 ≤ ⌈x + y⌉ ≤ ⌈x⌉ + ⌈y⌉
- ⌈x + a⌉ = ⌈x⌉ + a
- ⌈x⌉ = a; if x ≤ a < x + 1
- ⌈x⌉ = a; if x – 1 < a ≤ x
- a < ⌈x⌉ if a < x
- a ≤ ⌈x⌉ if x < a
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Formulae of Ceiling Function
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The formula to find the ceiling function is:
f (x) = minimum {a ∈ Z; a ≥ x }
f (x) = [x] smallest successive integer of the value specified.
Ceiling Function Graph
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The graph consists of discontinuous segments with the left end denoted with an open dot and the right side with a shaded dot.
Ceiling Function Graph
Check Also: Argument of Complex Numbers
Difference between Floor and Ceiling Function
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The differences are:
| Floor Function | Ceiling Function |
|---|---|
| Returns the largest value which is equal to or less than the specified value. | Returns the smallest value which is equal to or more than the specified value. |
| The graph of the function has a distinct dot on the left and an open one to the right. | The graph of the function has a distinct dot on the right and an open one on the left. |
| Example f (4.2) = 4 | Example f (4.2) = 5. |
Similarities between Floor and Ceiling Function
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- The integers of both floor and ceiling function are the same. For instance the floor and ceiling of 5 will be 5.
- They are denoted by square brackets.
Read More: Operations of Integers
Things to Remember
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- Ceiling function is used in computer programs and mathematics.
- It is a rounding function.
- It returns the smallest integer value of a real number.
- It is denoted as [x], ceil (x) or f (x) = [x]
- Graphically denoted as a discontinuous staircase.
- The integer of a ceiling function is the same as the specified number.
Sample Questions
Ques: Give the ceiling function of 2.5 and -2.5. Explain your answer. (3 marks)
Ans: f [2.5] = 3 and f [-2.5] = -2.
The ceiling function returns the smallest integer value equal to or greater than of the specified value.
Let’s take up [2.5] the greater integers are 3,4,5,6 and so on but the smallest integer greater than it is 3. Therefore, its ceiling function is going to be 3.
In the case of [-2.5] the integers greater than it are -2, -1 and so on and the smallest integer greater than it is -2, therefore its ceiling function is 2.
Ques: If two numbers x and y are whole numbers but both are less than one, what is the maximum ceiling function of (x+y)? (2 marks)
Ans: If both x and y are whole numbers and x<1 and y<1, then (x+y) should be less than 2. Therefore, the maximum ceiling function of (x+y) = 2.
Ques: Show the ceiling function of [2.31] with the help of a graph. (3 marks)
Ans: The ceiling function of [2.31] is [3]. It’s shown in the graph below.
![the ceiling function of [2.31]](https://images.collegedunia.com/public/image/f54e734e0b9fc4cb8056d09517a1cbcd.jpeg)
Ques: Evaluate the ceiling functions of -0.9 and 1.1. Explain your answer. (2 marks)
Ans: The ceiling function of [-0.9] is 0 and that of [1.1] is 2. Ceiling function returns the nearest integer closer to the value which is equal to or greater than the specified value [x]. The greater integer to [-0.9] would be 0,1,2,3 and counting and that of [1.1] to be 2,3,4,5 and so on. However, the nearest one for [-0.9] would be 0 and for [1.1] is 2. Hence, their ceiling function is 0 and 2 respectively.
Ques: Given x = 5.3 and a whole number a=6. Is the ceiling function of x equal to a? Explain. (2 marks)
Ans: We know the ceiling function of [x] is equal to 6. It can be explained using the property:
[x] = a; if x ≤ a < x + 1.
The value of x = 5.3 is less than ‘a’ which is 6 and it is even lesser than that of x+1 which is 6.3. Thus, we can prove the ceiling function [x] is equal to a.
Ques: What’s the sum of Floor (1.3) and Ceil (-2.1)? (2 marks)
Ans: The floor of [1.3] is 1 since the largest integer closest to it is 1, and the ceiling function of [-2.1] is -2 since the smallest integer closest to it is -2.
The sum of the two will be [1+-2] is -1.
Ques: What does a ceiling function map a real number to? Explain with examples. (4 marks)
Ans: The ceiling function brings up the nearest integer greater than or equal to the specified number. In simpler words, ceil function f(x) is the smallest integer that is not less than x, but greater than or equal to it.
For instance, let’s take two numbers x = -2.9 and y = 1.2. Let’s use the ceil function on them.
- F(x) = -2.
- F(y) = 2.
For the first case, the number x is a negative integer. Therefore its ceiling function brings the nearest integer closest to it. The nearest integer to x which is -2.9 is -2, -1, 0 and the nearest one among them is -2.
For the second case, the number y is a positive integer. Therefore, its ceiling function brings the nearest integer closest to it which includes 2, 3, and 4 and so on and the nearest one among them is 2.
Hence, the ceiling function of x is -2 and that of y is 2.
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