Algebra rules for rational numbers

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 ?

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