Question

Verify associativity of addition of rational numbers i.e., (x + y) + z = x + (y + z), when:

\[x = \frac{1}{2}, y = \frac{2}{3}, z = - \frac{1}{5}\]

Answer 1

\[\text{We have to verify that} (x + y) + z = x + (y + z) . \]
\[x = \frac{1}{2}, y = \frac{2}{3}, z = \frac{- 1}{5}\]
\[(x + y) + z = (\frac{1}{2} + \frac{2}{3}) + \frac{- 1}{5} = (\frac{3}{6} + \frac{4}{6}) + \frac{- 1}{5} = \frac{7}{6} + \frac{- 1}{5} = \frac{35}{30} + \frac{- 6}{30} = \frac{35 - 6}{30} = \frac{29}{30}\]
\[ x + (y + z) = \frac{1}{2} + (\frac{2}{3} + \frac{- 1}{5}) = \frac{1}{2} + (\frac{10}{15} + \frac{- 3}{15}) = \frac{1}{2} + \frac{7}{15} = \frac{15}{30} + \frac{14}{30} = \frac{15 + 14}{30} = \frac{29}{30}\]
\[ \therefore ( \frac{1}{2} + \frac{2}{3}) + \frac{- 1}{5} = \frac{1}{2} + (\frac{2}{3} + \frac{- 1}{5})\]
\[ \text{Hence verified} . \]