Verify associativity of addition of rational numbers i.e., (x + y) + z = x + (y + z), when:
Solution
\[x = \frac{- 7}{11}, y = \frac{2}{- 5}, z = \frac{- 3}{22}\]
\[(x + y) + z = (\frac{- 7}{11} + \frac{2}{- 5}) + \frac{- 3}{22} = (\frac{- 35}{55} + \frac{- 22}{55}) + \frac{- 3}{22} = \frac{- 57}{55} + \frac{- 3}{22} = \frac{- 114}{110} + \frac{- 15}{110} = \frac{- 114 - 15}{110} = \frac{- 129}{110}\]
\[ x + (y + z) = \frac{- 7}{11} + (\frac{2}{- 5} + \frac{- 3}{22}) = \frac{- 7}{11} + (\frac{- 44}{110} + \frac{- 15}{110}) = \frac{- 7}{11} + \frac{- 59}{110} = \frac{- 70}{110} + \frac{- 59}{110} = \frac{- 70 - 59}{110} = \frac{- 129}{110}\]
\[ \therefore (\frac{- 7}{11} + \frac{2}{- 5}) + \frac{- 3}{22} = \frac{- 7}{11} + (\frac{2}{- 5} + \frac{- 3}{22})\]
\[\text{Hence verified} . \]
