Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.10

10. Find the fractional part of each of these numbers:
(a) -8/5
(b) 22/7
(c) -1
(d) -1/3
Solution: 
The fractional part of a real number x, denoted by {x}, is the difference between x and the largest integer less than or equal to x, that is,[x]
In other words, {x}=x-[x]

(a)-8/5

The fractional part of -8/5 is
{-8/5}=-8/5-[-8/5]
=-8/5-(-2)
=(-8+10)/5
=2/5
Therefore,
{-8/5}=2/5

(b) 22/7

The fractional part of 22/7 is
{22/7}=22/7-[22/7]
=22/7-3
=(22-21)/7
=1/7
Therefore,
{22/7}=1/7

(c)-1

The fractional part of -1 is
{-1}=-1-[-1]
=-1-(-1)
=-1+1
=0
Therefore,
{-1}=0

(d)-1/3

The fractional part of -1/3 is
{-1/3}=-1/3-[-1/3]
=-1/3-(-1)
=(-1+3)/3
=2/3
Therefore,
{-1/3}=2/3

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