In this chapter we will learn some algebra rules for rational numbers with examples.

These rules are important to perform different mathematical operations between the rational numbers

## Rational number rules

Let numbers a, b and c are the given rational numbers.

The numbers can be expressed as; a, b & c 𝜖 Q

It means that the numbers a, b & c belongs to rational number Q.

Given below are properties of rational number with examples;

**(a) Addition of rational number will also produce a rational number**.

a + b 𝜖 Q

**For example**;

Consider the rational number 4/3 and 7/3. Adding these numbers we get;

\mathtt{\frac{4}{3} +\frac{7}{3} =\frac{11}{3}}

Here 11/3 is also a rational number.

**(b) Addition of rational number is commutative**

It means that changing the order of rational number in addition will not change the end result.

**a + b = b + a**

For example, **consider the rational number 3/5 and 4/5.**

Add 3/5 + 4/5;

\mathtt{\frac{3}{5} +\frac{4}{5} =\frac{7}{5}}

Now reverse the order of number and add 4/5 + 3/5.

\mathtt{\frac{4}{5} +\frac{3}{5} =\frac{7}{5}}

Hence, changing the order of number will produce the same result.

**(c) Addition of rational numbers follows associative property.**

It means that forming different groups while addition of rational number will produce the same result.

(a + b) + c = a + (b + c)

**For example**, consider the rational numbers 2/3, 5/3, 8/3

Add the numbers \mathtt{\left(\frac{2}{3} +\frac{5}{3}\right) +\frac{8}{3}}

\mathtt{\Longrightarrow \ \left(\frac{2}{3} +\frac{5}{3}\right) +\frac{8}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7}{3} +\frac{8}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{15}{3}}

Now add \mathtt{\ \frac{2}{3} +\left(\frac{5}{3} +\frac{8}{3}\right)}

\mathtt{\Longrightarrow \ \frac{2}{3} +\left(\frac{5}{3} +\frac{8}{3}\right)}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{3} +\frac{13}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{15}{3}}

Hence, changing the group during addition will produce the same result.

**(d) Adding rational number with its negative number will results in 0.**

a + (-a) = 0

**For example;**

Let 3/5 is the given rational number.

\mathtt{\Longrightarrow \ \frac{3}{5} +\left( -\frac{3}{5}\right)}\\\ \\ \mathtt{\Longrightarrow \ \frac{3}{5} -\frac{3}{5}}\\\ \\ \mathtt{\Longrightarrow 0}

**(e) Multiplication of rational number will also produce rational number.**

\mathtt{a\times b\ \epsilon \ Q}

**For example,**

Multiplying the rational numbers 5/6 and 11/3, we get;

\mathtt{\Longrightarrow \ \frac{5}{6} \times \frac{11}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5\times 11}{6\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{55}{18}}

Here 55/18 is also a rational number.

**(f) Multiplication of rational number is commutative**

It means that changing the order of multiplication of rational number will produce the same result.

\mathtt{a\ \times \ b\ =\ b\ \times \ a}

**For example**,

Multiply the rational number 3/7 and 8/5

\mathtt{\Longrightarrow \ \frac{3}{7} \times \frac{8}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3\times 8}{7\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{24}{35}}

Now change the order of number.

\mathtt{\Longrightarrow \ \frac{8}{5} \times \frac{3}{7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{8\times 3}{5\times 7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{24}{35}}

Hence, we get the same result.

**(g) Multiplication of rational number is associative in nature.**

It means that forming different groups during multiplication will produce the same result.

The property can be expressed as;

\mathtt{( a\times b) \times c=\ a\times ( b\times c)}

Let 2/3, 4/3 and 7/3 are the given rational numbers.

Multiplying \mathtt{\left(\frac{2}{3} \times \frac{4}{3}\right) \times \frac{7}{3}}

\mathtt{\Longrightarrow \ \left(\frac{2}{3} \times \frac{4}{3}\right) \times \frac{7}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{8}{9} \times \frac{7}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{56}{27}}

Now form different group and multiply.

\mathtt{\Longrightarrow \ \frac{2}{3} \times \left(\frac{4}{3} \times \frac{7}{3}\right)}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{3} \times \frac{28}{9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{56}{27}}

Hence, we get the same result.

**(h) Distributive law of rational number**

According to the distributive law;

a ( b + c ) = ab + ac

**( i ) The multiplication of rational number with inverse results in number 1**.

\mathtt{a\times \frac{1}{a} =\ 1}

**Next chapter** : **How to compare rational numbers ?**