Question

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

\[x = \frac{- 2}{5}, y = \frac{4}{3}, z = \frac{- 7}{10}\]

Answer 1

\[x = \frac{- 2}{5}, y = \frac{4}{3}, z = \frac{- 7}{10}\]
\[(x + y) + z = (\frac{- 2}{5} + \frac{4}{3}) + \frac{- 7}{10} = (\frac{- 6}{15} + \frac{20}{15}) + \frac{- 7}{10} = \frac{14}{15} + \frac{- 7}{10} = \frac{28}{30} + \frac{- 21}{30} = \frac{28 - 21}{30} = \frac{7}{30}\]
\[ x + (y + z) = \frac{- 2}{5} + (\frac{4}{3} + \frac{- 7}{10}) = \frac{- 2}{5} + (\frac{40}{30} + \frac{- 21}{30}) = \frac{- 2}{5} + \frac{19}{30} = \frac{- 12}{30} + \frac{19}{30} = \frac{- 12 + 19}{30} = \frac{7}{30}\]
\[ \therefore \frac{- 2}{5} + \frac{4}{3}) + \frac{- 7}{10} = \frac{- 2}{5} + (\frac{4}{3} + \frac{- 7}{10})\]
\[ \text{Hence verified} . \]