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Mathematics

For each pair of rational number, verify commutative property of addition of rational numbers.

-2 and 3−5\dfrac{3}{-5}

Rational Numbers

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Answer

To prove:

−2+3−5=3−5+−2-2 + \dfrac{3}{-5} = \dfrac{3}{-5} + -2

Taking LHS: −2+3−5=−21+−35-2 + \dfrac{3}{-5} \\[1em] = \dfrac{-2}{1} + \dfrac{-3}{5}

LCM of 1 and 5 is 5

=−2×51×5+−3×15×1=−105+−35=−10+(−3)5=−135= \dfrac{-2 \times 5}{1 \times 5} + \dfrac{-3 \times 1}{5 \times 1} \\[1em] = \dfrac{-10}{5} + \dfrac{-3}{5} \\[1em] = \dfrac{-10 + (-3)}{5} \\[1em] = \dfrac{-13}{5}

Taking RHS: 3−5+−2=−35+−21\dfrac{3}{-5} + -2 \\[1em] = \dfrac{-3}{5} + \dfrac{-2}{1}

LCM of 5 and 1 is 5

=−3×15×1+−2×51×5=−35+−105=(−3)+(−10)5=−135= \dfrac{-3 \times 1}{5 \times 1} + \dfrac{-2 \times 5}{1 \times 5} \\[1em] = \dfrac{-3}{5} + \dfrac{-10}{5} \\[1em] = \dfrac{(-3) + (-10)}{5} \\[1em] = \dfrac{-13}{5}

∴ LHS = RHS

Hence, −2+3−5=3−5+−2-2 + \dfrac{3}{-5} = \dfrac{3}{-5} + -2

So, the commutative property for the addition of the rational number is verified.

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