#### The definition and explanation with examples of the Associative property of rational numbers.

Associative Property Of Rational Numbers

If we take the three rational numbers -3/4, 2/3 and 2 1/2, for example, then we have;

$\displaystyle \frac{-3}{4}+\left( \frac{2}{3}+2\frac{1}{2} \right)=\frac{-3}{4}+\left( \frac{4}{6}+2\frac{3}{6} \right)=\frac{-3}{4}+2\frac{7}{6}=\frac{-9}{12}+2\frac{14}{12}=2\frac{5}{12}$

### =

$\displaystyle \left( \frac{-3}{4}+\frac{2}{3} \right)+2\frac{1}{2}=\left( \frac{-9}{12}+\frac{8}{12} \right)+2\frac{1}{2}=\frac{-1}{12}+2\frac{1}{2}=\frac{-1}{12}+2\frac{6}{12}=2\frac{5}{12}$

RESULTS ARE EQUAL TO EACH OTHER

In general if a/b, c/d, e/f ∈ Q then;

$\displaystyle \frac{a}{b}+\left( \frac{c}{d}+\frac{e}{f} \right)=\left( \frac{a}{b}+\frac{c}{d} \right)+\frac{e}{f}$

The set of rational numbers is associative under addition.

subtraction;

$\displaystyle \frac{-3}{4}-\left( \frac{2}{3}-2\frac{1}{2} \right)=\frac{-3}{4}-\left( \frac{4}{6}-2\frac{3}{6} \right)=\frac{-3}{4}-\left( \frac{4}{6}+-1\frac{9}{6} \right)=\frac{-3}{4}-\left( -1\frac{5}{6} \right)=\frac{-9}{12}+\left( +1\frac{10}{12} \right)=1\frac{1}{12}$

### ≠

$\displaystyle \left( \frac{-3}{4}-\frac{2}{3} \right)-2\frac{1}{2}=\left( \frac{-9}{12}+\frac{-8}{12} \right)-2\frac{1}{2}=\frac{-17}{12}-2\frac{1}{2}=-1\frac{-5}{12}-2\frac{6}{12}=-3\frac{11}{12}$

RESULTS ARE NOT EQUAL TO EACH OTHER

In general if a/b, c/d, e/f ∈ Q then;

$\displaystyle \frac{a}{b}-\left( \frac{c}{d}-\frac{e}{f} \right)\ne \left( \frac{a}{b}-\frac{c}{d} \right)-\frac{e}{f}$

The set of rational numbers is not associative under subtraction.

multiplication;

$\displaystyle \frac{-3}{4}x\left( \frac{2}{3}x2\frac{1}{2} \right)=\frac{-3}{4}x\left( \frac{2}{3}x\frac{5}{2} \right)=\frac{-3}{4}x\frac{5}{3}=\frac{-5}{4}=-1\frac{1}{4}$

### =

$\displaystyle \left( \frac{-3}{4}x\frac{2}{3} \right)x2\frac{1}{2}=\left( \frac{-1}{2}x2\frac{1}{2} \right)=\frac{-1}{2}x\frac{5}{2}=\frac{-5}{4}=-1\frac{1}{4}$

RESULTS ARE EQUAL TO EACH OTHER

In general if a/b, c/d, e/f ∈ Q then;

$\displaystyle \frac{a}{b}x\left( \frac{c}{d}x\frac{e}{f} \right)=\left( \frac{a}{b}x\frac{c}{d} \right)x\frac{e}{f}$

The set of rational numbers is associative under multiplication.

division;

$\displaystyle \frac{-3}{4}\div \left( \frac{2}{3}\div 2\frac{1}{2} \right)=\frac{-3}{4}\div \left( \frac{2}{3}x\frac{2}{5} \right)=\frac{-3}{4}\div \frac{4}{15}=\frac{-3}{4}x\frac{15}{4}=-\frac{45}{16}=-2\frac{13}{16}$

### ≠

$\displaystyle \left( \frac{-3}{4}\div \frac{2}{3} \right)\div 2\frac{1}{2}=\left( \frac{-3}{4}x\frac{3}{2} \right)\div 2\frac{1}{2}=\frac{-9}{8}\div 2\frac{2}{5}=\frac{-9}{8}x\frac{2}{5}=-\frac{9}{20}$

RESULTS ARE NOT EQUAL TO EACH OTHER

In general if a/b, c/d, e/f ∈ Q then;

$\displaystyle \frac{a}{b}\div \left( \frac{c}{d}\div \frac{e}{f} \right)\ne \left( \frac{a}{b}\div \frac{c}{d} \right)\div \frac{e}{f}$

The set of rational numbers is not associative under division.

the set of rational numbers is associative under addition and muitiplication, but it is non-associative under subtraction and division.