Greatest Integer Function Greatest Integer Function Greatest integer function is also known as the floor function. For any real number x, we use the symbol [x] or [_x_] to denote the greatest integer less than or equal to x. for example: [2.75] = 2 (greatest integer less than and equal to 2.75) [3] = 3 (as 3 is itself an integer that means equal to x ) [0.74] = 0 (greatest integer less than and equal to 2.75) [-7.45] = -8 (greatest integer less than -7.45 is -8) The function f : R → R defined by f(x) = [x] for all x ? R is called the Greatest Integer Function or the floor function.
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It is also the called the step function. Clearly, domain of the greatest integer function is the set R of all real numbers and the range is the set z of all integers as it attains only integer values. Some properties of greatest integer function are defined below if n is an integer and x is a real number between n and n + 1, then I.
[-n] = -[n]
II.
[x + k] = [x] + k for any integer k.
III.
[-x] = -[x] -1
IV.
[x] + [-x] = -1, if x !?Z and 0, if x ?Z
V.
[x] - [-x] = 2 [x] + 1, if x !?Z and 2[x] , if x ?Z
VI.
[x] => k → x => K, where k ?Z
VII.
[x] =< k → x < K + 1, where k ?Z
VIII.
[x] > k → x => K + 1, where k ?Z
IX.
[x] < k → x < k, where k ?Z
X.
[x + y] = [x] + [y + x]-[x] ] for all x, y ?R.
XI.
[x] + [x + 1 / n] + [x + 2 / n] +....+ [x + n - 1 / n]= [n x], n ?N.
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By following all describe properties we can obtain the greatest integer function. Let’s take some more examples to see the greatest integer function. Example 1: - Find the greatest integer value for the following numbers 9.1, 5, -8.7, 2.8, 2 and -3.9 Solution : - We have to give some real numbers that are 9.1, 5, -8.7, 2.8, 2 and -3.9 [9.1] = closest greatest integer value that is less than 9.1 is 9. So [9.1] = 9 [5] = is itself an integer that is equal to x So [5] = 5 [-8.7] = closest greatest integer value is -9 that is less than -8.7 So [-8.7] = -9 [2.8] = closest greatest integer value is 3 which is less than 2.8 So [2.8] = 2 [2] = is itself an integer that is equal to x So [2] = 2 [-3.9] = closest greatest integer value is -4 that is less than -3.9 So [-3.9] = -4
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