Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.8

8. Find the following values of the greatest integer function.
(a) [-1/4]
(b) [-22/7]
(c) [5/4]
(d) [[1/2]]
(e) [[3/2] + [-3/2]]
(f) [3 – [1/2]]

Solution:  The greatest integer, denoted by [x], is the largest integer less than or equal to x satisfying the inequation
[x] \leq x<[x]+1
Where x is a real number

(a)[-1/4]

The greatest integer less than equal to -1/4 is -1

So, [-1/4]=1

(b) [-22/7]

The greatest integer less than equal to -22/7 is -4

So, [-22/7]=-4

(c) [5/4]

The greatest integer less than equal to 5/4 is 1

So, [5/4]=1

(d) [[1/2]]

The greatest integer less than equal to 1/2 is 0

Now, [[1/2]]=[0]
=0

So, [[1/2]]=0

(e) [[3/2]+[-3/2]]

The greatest integer less than equal to 3/2 is 1

So, [3/2]=1

Also, the greatest integer less than equal to -3/2 is -2

Now,
[[3/2]+[-3/2]]=[1+(-2)]
=[-1]
=-1
So, [[3/2]+[-3/2]]=-1

(f) [3-[1/2]]

The greatest integer less than equal to 1/2 is 0

So, [1/2]=0

Now,
[3-[1/2]]=[3-0]
=[3]
=3
So, [3-[1/2]]=3

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