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666 bro

Member

From: Flatland

Registered: 2019-04-26

Posts: 706

Greatest integer function

I've a little bit confusion in understanding about greatest integer function my questions are:

(1) : What does greatest integer function means?

(2)  : How it's different from floor and ceiling functions?

( 3) :How could we plot greatest integer function in graph?

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"An equation for me has no meaning, unless it expresses a thought of God"- Srinivasa ramanujan

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,384

Re: Greatest integer function

Greatest integer function

The greatest integer function, also called step function, is a piecewise function whose graph looks like the steps of a staircase. The greatest integer function is denoted by f(x) = [x] and is defined as the greatest integer less or equal to x.

Example #1

[2.5] is the greatest integer less or equal to 2.5.

Notice that the number we are looking for must meet two conditions.

The number has to be an integer.
The number has to be less than or equal to 2.5. 

Which number can you think of that is less than 2.5 and is an integer? That number is 2!

[2.5] = 2

Example #2

[-2.5] is the greatest integer less or equal to -2.5.

Again, the number we are looking for must meet the following two conditions.

The number has to be an integer.
The number has to be less than or equal to -2.5. 
A common mistake by students is to say that [-2.5] = -2.
For [2.5] = 2, it appears that we just removed the .5. By the same token, students may think that all we need to do here is to remove the 0.5 and [-2.5] = -2. Wrong!

[-2.5] is not equal to -2. Remember that we are looking for a number smaller than -2.5. The number that is smaller than -2.5 is -3, so [-2.5]= -3

Example #3

[4] is the greatest integer less or equal to 4.

We are looking for a number less or equal to 4. Now, we need to pay more attention to the part or equal to 4. Since 4 is equal to 4, [4] = 4

Example #4

[0.5] is the greatest integer less or equal to 0.5.

Again, focus on the word less and try to ignore as much as possible the word greatest. Which integer is less than 0.5? Since it is 0, [0.5] = 0.

The greatest integer function is also called the floor function.

[x] = floor(x).

In mathematics and computer science, the floor function is the function that takes as input a real number

, and gives as output the greatest integer less than or equal to

, denoted

or

. Similarly, the ceiling function maps

to the least integer greater than or equal to

, denoted

or

For example,

and

, while

.

The integral part or integer part of x, often denoted

is

  if x is nonnegative, and

otherwise. In words, this is the integer that has the largest absolute value less than or equal to the absolute value of x.

2) Floor and Ceiling Functions

See the link in mathsisfun.com: Floor and Ceiling Functions.

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