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

The Commutative Property Of Rational Numbers

If we take the two rational numbers -3/7 and 2 1/2 , for example, then we have :

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

# =

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

RESULTS ARE EQUAL TO EACH OTHER.

In general; if a/b, c/d ∈ Q then;

$\displaystyle \frac{a}{b}+\frac{c}{d}=\frac{c}{d}+\frac{a}{b}$

The set of rational numbers is commutative under addition.

subtraction;

$\displaystyle \left( \frac{-3}{7} \right)-2\frac{1}{2}=\frac{-6}{14}-2\frac{7}{14}=\frac{-6}{14}+\left( -2\frac{7}{14} \right)=-2\frac{13}{14}$

# ≠

$\displaystyle 2\frac{1}{2}-\left( \frac{-3}{7} \right)=2\frac{7}{14}-\left( \frac{-6}{14} \right)=2\frac{7}{14}+\frac{6}{14}=2\frac{13}{14}$

RESULTS ARE NOT EQUAL TO EACH OTHER.

In general; if a/b, c/d ∈ Q then;

$\displaystyle \frac{a}{b}-\frac{c}{d}\ne \frac{c}{d}-\frac{a}{b}$

The set of rational numbers is not commutative under subtraction.

multiplication;

$\displaystyle \left( \frac{-3}{7} \right)x2\frac{1}{2}=\frac{-3}{7}x\frac{5}{2}=\frac{-3x5}{7x2}=\frac{-15}{14}=-1\frac{1}{14}$

# =

$\displaystyle 2\frac{1}{2}x\left( \frac{-3}{7} \right)=\frac{5}{2}x\frac{-3}{7}=\frac{5x\left( -3 \right)}{2x7}=\frac{-15}{14}=-1\frac{1}{14}$

RESULTS ARE EQUAL TO EACH OTHER.

In general; if a/b, c/d ∈ Q then;

$\displaystyle \frac{a}{b}x\frac{c}{d}=\frac{c}{d}x\frac{a}{b}$

The set of rational numbers is commutative under multiplication.

division;

$\displaystyle \left( \frac{-3}{7} \right)\div 2\frac{1}{2}=\frac{-3}{7}\div \frac{5}{2}=\frac{-3}{7}x\frac{2}{5}=\frac{-6}{35}$

# ≠

$\displaystyle 2\frac{1}{2}\div \left( \frac{-3}{7} \right)=\frac{5}{2}\div \left( \frac{-3}{7} \right)=\frac{5}{2}x\left( \frac{-7}{3} \right)=\frac{-35}{6}=-5\frac{5}{6}$

RESULTS ARE NOT EQUAL TO EACH OTHER.

In general; if a/b, c/d ∈ Q then;

$\displaystyle \frac{a}{b}\div \frac{c}{d}\ne \frac{c}{d}\div \frac{a}{b}$

The set of rational numbers is not commutative under division.

the set of rationa! numbers is commutative under addition and multiplication, but it is non-commutative under subtraction and division.