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