Floor And Ceiling Function

The floor function and ceiling function help convert a real number or decimal into a simpler whole number or integer. These functions are useful when we need to round numbers up or down to the nearest integer.

Floor function: The floor function gives the greatest integer that is less than or equal to the given real number. It rounds the number down to its nearest whole number. The floor function is represented as floor(x) or ⌊x⌋. 
For example, ⌊5.9⌋ = 5, the largest whole number less than 5.9 is 5.
⌊-1.6⌋ = -2, here, -2 < -1.6, so it becomes the floor value.
⌊3⌋ = 3, we can see that the number is already an integer, so there is no change in floor value.

Ceiling function: The ceiling function rounds up a real number to the nearest integer. This means that the integer is the smallest number that is greater than the ceiling value. It is written as ceil(x) or ⌈x⌉, but can also be written as ]]x[[.
For example, ⌈2.1⌉ = 3, the smallest integer that is greater than 2.1 is 3, so it is the ceiling value.
⌈-4.3⌉ = -4, here, -4 is the smallest integer greater than -4.3.
⌈6⌉ = 6, similar to the floor function; if the number in the ceiling function is already an integer, then the ceiling value remains the same.
 

The floor and ceiling functions are examples of step functions. This means that their graphs look like a sequence of horizontal steps. Each ‘step’ is a range of input values that gives a single integer as output. 
For any given interval of values:
Each segment of the floor function graph starts with a dot () on the left and an open dot () on the right, meaning that the edge is not included.
For example, for ⌊x⌋ between 2 and 3, the input is all values between 2 and 3, including 2 but not 3. 
The output is ⌊x⌋ = 2
The graph shows horizontal line y = 2 from x = 2 () to x = 3 ()
While each step in the ceiling function graph begins with an open dot () and ends with (), indicating the range of input values mapping to the greatest integer in that interval. 
For example, for ⌈x⌉ between 2 and 3, the input includes all values between 2 to 3, including 3 but not 2.
 The output is ⌈x⌉ = 3
So, the graph has a horizontal line at y = 3 from x = 2(°) to x = 3().

Properties Of Floor Function And Ceiling Function

Some key properties of floor and ceiling functions are as follows:

Properties of floor function:

• ⌊x⌋ gives the greatest integer less than or equal to x.
• The value of x lies between its floor and the next integer, ⌊x⌋ ≤ x < ⌊x⌋ + 1.
• When one number is smaller than the other, its floor will also be smaller or equal. The floor function does not decrease unexpectedly. If x  y, ⌊x⌋  ⌊y⌋.
• Adding a whole number to x shifts the floor up by that number. ⌊x + n⌋ = ⌊x⌋ + n, where n is an integer.
• For any real number x, the floor of -x is equal to its negative ceiling. This means the floor and the ceiling mirror each other across zero. ⌊−x⌋ = −⌈x⌉.
• If x is a whole number, the floor of x and −x cancel each other out: ⌊x⌋ + ⌊−x⌋ = 0. If x has decimals, the negative side drops further, giving −1 ⌊x⌋ + ⌊−x⌋ = -1.
• The floor of a sum is at least the sum of the floors, but can go up by 1 due to the leftover decimal parts. ⌊x⌋ + ⌊y⌋ ≤ ⌊x + y⌋ ≤ ⌊x⌋ + ⌊y⌋ + 1.
• If the ceiling is already an integer, then its floor doesn't change its value. ⌊⌈x⌉⌋ = ⌈x⌉.
• Any number x is made up of its integer part and decimal part. The decimal part always lies between 0 and 1. x = ⌊x⌋ + {x}.

Properties of the ceiling function:

• ⌈x⌉ moves x up to the nearest integer that is greater than or equal to x.
• x lies between one less than its ceiling and the ceiling itself. ⌈x⌉ − 1 < x ≤ ⌈x⌉.
• Like the floor, the ceiling function does not decrease if x increases. If x ≤ y, then ⌈x⌉ ≤ ⌈y⌉.
• Adding a whole number shifts the ceiling up by the same amount. ⌈x + n⌉ = ⌈x⌉ + n, where n is an integer.
•  The ceiling of −x is the negative of the floor of x, showing symmetry.⌈−x⌉ = −⌊x⌋.
• ⌈x⌉ + ⌈−x⌉ = 1 if x is not an integer and ⌈x⌉ + ⌈−x⌉ = +1 if x is an integer.
• The ceiling of a sum is not smaller than the sum of ceilings minus 1, and not more than the sum of ceilings. ⌈x⌉ + ⌈y⌉ − 1 ≤ ⌈x + y⌉ ≤ ⌈x⌉ + ⌈y⌉
• Applying the ceiling more than once makes no difference. ⌈⌈x⌉⌉ = ⌈x⌉.
• The ceiling of the floor stays the same. ⌈⌊x⌋⌉ = ⌊x⌋.
• If x is an integer, the relationship between floor and ceiling is ⌈x⌉ = ⌊x⌋; if x is not an integer, ⌈x⌉ = ⌊x⌋ + 1. Ceiling and floor differ by 1 only if x is a decimal.
   

The formula for finding the ceiling value of a given value is;
⌈x⌉ = min{aZ| ax}
Where, 
⌈x⌉ = The ceiling of x
Min = minimum value from a given set
a is an element of the set of integers denoted by z
| is read as such that, and it separates the condition from the variable
a x is the condition; in this formula, we only consider greater than or equal to.
Let’s apply this formula to an example. 
Question: Find the ceiling of x = 3.2
Using the formula, we look at the smallest integer a such that a  3.2. The integers greater than or equal to 3.2 are 4, 5, 6,... etc.
Here, 4 is the min of the set.
So, ⌈3.2⌉ = 4.

The formula for finding the floor value for a given value is ⌊x⌋ = max {a∈Z ∣ a ≤ x}.
Where,
⌊x⌋ = the floor 
Max refers to finding the maximum value of a given set
a is an element of the set of integers denoted by z
| separates the condition from the variable and is read as such that.
ax is the condition for the formula, and
{az | ax} is the set of all integers less than or equal to x.
Let’s take x = 3.8. To find its floor, we will use the formula. The set of integers less than or equal to 3.8 is {..., 1, 2, 3}
So, ⌊3.8⌋ = max {1, 2, 3} = 3
 

From billing systems to packaging logistics, floor and ceiling functions help make calculations simpler and help in critical decision-making.

• Fare rounding in finance and transport.
  Ceiling functions are used to round up bills or fares to the nearest whole unit. For instance, a cab fare is rounded from ＄14.2 to ＄15 using ⌈14.3⌉.
• Item packaging in manufacturing and logistics
  In manufacturing and logistics, ceiling functions help determine the number of boxes required for packaging, as packaging cannot be done in fractions. For instance, if 47 items need to be packed and each box can hold only 10 items, we cannot directly split these numbers, so we use ⌈47 ÷ 10⌉ = ⌈4.7⌉ = 5 boxes to find the number of boxes required.
• Time tracking in the service industry
  Floor functions are used to calculate working hours in the service industry for making billing reports and calculating paychecks based on hours.
• Page division in computer science for UI design
  Content can be divided into parts using ceiling functions. For instance, a brochure has 47 items divided as 10 per page, ⌈47/10⌉ = 5 pages. So, the brochure is 5 pages long.
• Level assignment in gaming
  In video games, levels are assigned based on the score range, and the floor functions are used to assign levels. For example, if a game advances levels after 1000 points, then a player with 3200 points will be at level ⌊3200/1000⌋ = 3.

Listed below are some useful tips and tricks for students to help increase their efficiency while working with floor and ceiling functions.

Tip 1: Memorize the definitions clearly to avoid confusion.
Tip 2: For tricky values or negatives, refer to the number line to be sure of values.
Tip 3: Floor rounds down, ceiling rounds up always.