Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.7

7. Find the following values of the greatest integer function.

a) [1/4]
b) [-3/4]
c) [22/7]
d) [-2]
e) [[1/2]+ [1/2]]
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 0

So, [1/4]=0

(b)[-3/4]

The greatest integer less than equal to -3/4 is -1

So, [-3/4]=-1

(c)[22/7]

The greatest integer less than equal to 22/7 is 3

So, [22/7]=3

(d) [-2]

The greatest integer less than equal to -2 is -2 itself

So,[-2]=-2

(e) [[1/2]+[1/2]]

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

So, [1/2]=0

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

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

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

So, [-1/2]=-1

Now,
[-3+[-1/2]]=[-3+(-1)]
=[-4]
=-4
So, [-3+[-1/2]]=-4

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