Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.15

15. Show that if x and y are positive real numbers,then [xy] >= [x] [y]. What is the situation when both x and y are negative? When one of x and y is negative and the other positive?
Solution: We have to prove the inequality
[xy]>=[x][y]
Where x and y are positive real numbers.
We prove the above inequality by making assumptions.
Let us assume that
x=a+r
y=b+s
Where a,b are integers and r,s are real numbers such that 0<=r,s<1
Now, using the values of x and y , we get,
[xy]=[(a+r)(b+s)]
[xy]=[ab+as+br+rs]
[xy]=ab+[as+br+rs]
Whereas
[x][y]=[a][b]
[x][y]=ab
This is because a,b are integers and
[r]=0
[s]=0
Where 0<=r,s<1
Thus we have
[xy]<=[x][y]
If one of x and y is positive and the other negative, then depending on what the numbers are, the order sign can go in either direction.

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