(i) The addition of rational number is commutative

i.e, if \(\frac{a}{b}\) and \(\frac{c}{d}\) are any two rational numbers, then

\(\frac{a}{b}+\frac{c}{d}=\frac{c}{d}+\frac{a}{b}\)

Verification: In order to verify this property,

Let us consider two expressions:

\(\frac{-11}{5}+\frac{4}{7}\)

And,

\(\frac{4}{7}+\frac{-11}{5}\)

We have:

\(\frac{-11}{5}+\frac{4}{7}=\frac{-77}{35}+\frac{20}{35}\)

= \(\frac{-77+20}{35}\)

= \(\frac{-57}{35}\)

And,

\(\frac{4}{7}+\frac{-11}{5}=\frac{20}{35}+\frac{-77}{35}\)

= \(\frac{20-77}{35}\)

= \(\frac{-57}{35}\)

Therefore,

\(\frac{-11}{5}+\frac{4}{7}=\frac{4}{7}+\frac{-11}{5}\)

(ii) The addition of rational number is commutative i.e, if \(\frac{a}{b}\) and \(\frac{c}{d}\) are any two rational numbers, then

\(\frac{a}{b}+\frac{c}{d}=\frac{c}{d}+\frac{a}{b}\)

Verification: In order to verify this property,

Let us consider two expressions:

\(\frac{4}{9}+\frac{-7}{12}\)

And,

\(\frac{-7}{12}+\frac{4}{9}\)

We have:

\(\frac{4}{9}+\frac{-7}{12}=\frac{16}{36}+\frac{-21}{36}\)

= \(\frac{16-21}{36}\)

= \(\frac{-5}{36}\)

And,

\(\frac{-7}{12}+​​\frac{4}{9}=\frac{-21}{36}+\frac{16}{36}\)

\(=\frac{-21+16}{36}\)

= \(\frac{-5}{36}\)

Therefore,

\(\frac{4}{9}+\frac{-7}{12}=\frac{-7}{12}+\frac{4}{9}\)

(iii) The addition of rational number is commutative i.e, if \(\frac{a}{b}\) and \(\frac{c}{d}\)are any two rational numbers, then

\(\frac{a}{b}+​​\frac{c}{d}=\frac{c}{d}+\frac{a}{b}\)

Verification: In order to verify this property,

Let us consider two expressions:

\(\frac{3}{5}+\frac{-2}{-15}\)

And,

\(\frac{-2}{-15}+\frac{3}{5}\)

We have:

\(\frac{3}{5}+​​\frac{-2}{-15}=\frac{-3}{5}+\frac{2}{15}\)

= \(\frac{-9+2}{15}\)

= \(\frac{-7}{15}\)

And,

\(\frac{-2}{-15}+​​\frac{3}{5}=\frac{-2}{-15}+\frac{3}{5}\)

= \(\frac{2-9}{15}\)

= \(\frac{-7}{15}\)

Therefore,

\(\frac{3}{5}+​​\frac{-2}{-15}=\frac{-2}{-15}+\frac{3}{5}\)

(iv) The addition of rational number is commutative i.e, if \(\frac{a}{b}\) and \(\frac{c}{d}\) are any two rational numbers, then

\(\frac{a}{b}+​​\frac{c}{d}=\frac{c}{d}+\frac{a}{b}\)

Verification: In order to verify this property,

Let us consider two expressions:

\(\frac{2}{-7}+\frac{12}{-35}\)

And,

\(\frac{12}{-35}+\frac{2}{-7}\)

We have:

\(\frac{2}{-7}+​​\frac{12}{-35}=\frac{-10}{35}+\frac{-12}{35}\)

= \(\frac{-10-12}{35}\)

= \(\frac{-22}{35}\)

And,

\(\frac{12}{-35}+​​\frac{2}{7}=\frac{-12}{35}+\frac{-10}{35}\)

= \(\frac{-12-10}{35}\)

= \(\frac{-22}{35}\)

Therefore,

\(\frac{2}{-7}+​​\frac{12}{-35}=\frac{12}{-35}+\frac{2}{-7}\)

(v) The addition of rational number is commutative i.e, if \(\frac{a}{b}\) and \(\frac{c}{d}\) are any two rational numbers, then

\(\frac{a}{b}+​​\frac{c}{d}=\frac{c}{d}+\frac{a}{b}\)

Verification: In order to verify this property, Let us consider two expressions:

\(4+\frac{-3}{5}\)

And,

\(\frac{-3}{5}+4\)

We have:

\(4+\frac{-3}{5}\)=\(\frac{20}{5}-\frac{3}{5}\)

= \(\frac{20-3}{5}\)

= \(\frac{17}{5}\)

And,

\(\frac{-3}{5}+4\)= \(\frac{-3}{5}+\frac{20}{5}\)

= \(\frac{-3+20}{5}\)

= \(\frac{17}{5}\)

Therefore,

\(4+\frac{-3}{5}\)= \(\frac{-3}{5}+4\)

(vi) The addition of rational number is commutative i.e, if \(\frac{a}{b}\) and \(\frac{c}{d}\) are any two rational numbers, then

\(\frac{a}{b}+​​\frac{c}{d}=\frac{c}{d}+\frac{a}{b}\)

Verification: In order to verify this property,

Let us consider two expressions:

\(-4+\frac{4}{-7}\)

And,

\(\frac{4}{-7}-4\)

We have:

\(-4+\frac{4}{-7}\)=\(\frac{-28}{7}-\frac{4}{7}\)

= \(\frac{-28-4}{7}\)

= \(\frac{-32}{7}\)

And,

\(\frac{4}{-7}-4\)=\(\frac{-4}{7}-\frac{28}{7}\)

= \(\frac{-4-28}{7}\)

= \(\frac{-32}{7}\)

Therefore,

\(-4+\frac{4}{-7}\)=\(\frac{4}{-7}-4\)