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, Note: - It is to be noted that this function returns us closest integer value of the input as the output and it is also noted that this function is piece wise defined. To understand greatest integer function more we can plot them on a graph paper.

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