In this chapter we will discuss some important properties of rational number with examples.

To understand the properties, you should have basic understanding of concept of rational number.

To learn about** rational numbers in detail,** click the red link.

## Rational numbers properties

Given below are properties of rational number with example;

(a) Rational number are **generally expressed in form of P / Q**

Where P & Q are integers.

Numbers like \mathtt{\frac{2}{3} ,\ \frac{-7}{6} \ \&\ \frac{\ 10}{9}} are some of the examples of rational numbers.

(b) **Decimal number are also rational numbers**

Decimal numbers can be easily converted in the form of P / Q. Hence they are also rational numbers.

**For example;**

The decimal 10.25 can be converted into fraction as follows.

\mathtt{10.25\ \Longrightarrow \ \frac{1025}{100}}

Hence, the above decimal is a rational number.

(c) If you **multiply number m on both numerator & denominator** of rational number,** you will get the rational number of same value**.

Let \mathtt{\frac{a}{b}} be the rational number.

Multiply m on both numerator and denominator, we get;

\mathtt{\Longrightarrow \frac{a}{b}}\\\ \\ \mathtt{\Longrightarrow \ \frac{a\times m}{b\times m}}\\\ \\ \mathtt{\Longrightarrow \ \frac{am}{bm}}

Here the resultant rational number \mathtt{\frac{am}{bm}} has the same value as the original number \mathtt{\frac{a}{b}}

\mathtt{\frac{a}{b} \Longrightarrow \ \frac{a\times m}{b\times m}} **For example;**

Consider the rational number 2 / 3

Multiply number 5 on both numerator and denominator.

\mathtt{\Longrightarrow \frac{2}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2\times 5}{3\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10}{15}}

Here both 10/15 and 2/3 have same value.

\mathtt{\frac{10}{15} \Longrightarrow \ 0.67}\\\ \\ \mathtt{\frac{2}{3} \ \Longrightarrow \ 0.67}

**Conclusion**

The multiplication of same number on numerator and denominator will produce rational number of same value.

(d) If you **divide numerator and denominator** of rational number by number m, you will **get rational number of same value.**

Let \mathtt{\frac{a}{b}} be the rational number.

Divide numerator and denominator by m.

\mathtt{\Longrightarrow \frac{a}{b}}\\\ \\ \mathtt{\Longrightarrow \ \frac{a\div m}{b\div m}}

This new rational number value is same as initial value a / b.

\mathtt{\frac{a}{b} =\ \frac{a\div m}{b\div m}} **For example**

Consider the rational number \mathtt{\frac{10}{16}}

Divide numerator and denominator by 2

\mathtt{\Longrightarrow \ \frac{10\div 2}{16\div 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5}{8}}

According to the property of rational number, both the number 10/16 and 5/8 have same values.

\mathtt{\frac{10}{16} \Longrightarrow \ 0.625}\\\ \\ \mathtt{\frac{5}{8} \ \Longrightarrow \ 0.625 }

(e) If **two rational numbers are equal**, then the **cross multiplication of rational number will also be equal**.

Let a/b & c/d be the two rational numbers.

\mathtt{\frac{a}{b} =\ \frac{c}{d}}

Doing the cross multiplication we get;

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

**For example;**

Let 2/6 and 8/24 be the equal rational numbers.

\mathtt{\frac{2}{6} =\ \frac{8}{24}}

On doing the cross multiplication, we get the equal numbers.

\mathtt{2\times 24\ =6\ \times 8}\\\ \\ \mathtt{48\ =\ 48}

(f) The **value of rational number a/b** **depend on the values of a & b**.

(i) if a > b, the value of rational number will be greater than 1

(ii) if a < b, the value if rational number is less than 1

(iii) if a = b ,the value of rational number is equal to 1**For example**;

Consider the rational number 5/4

Here numerator > denominator, so value of rational number is greater than 1.

\mathtt{\frac{5}{4} \Longrightarrow \ 1.25} **Example 02**

Consider rational number 4/5

Here numerator is less than denominator, so value of rational number is less than 1.

latex] \mathtt{\frac{4}{5} \Longrightarrow \ 0.8} [/latex]

(g) Suppose there are three rational numbers X, Y & Z such that, **X > Y and Y > Z**, then it automatically implies that** X > Z**.

**For example**;

Let the three rational numbers are \mathtt{\frac{2}{3} ,\ \frac{2}{4} ,\ \frac{2}{6}} .

Its given that;

\mathtt{\frac{2}{3} >\ \frac{2}{4}} and \mathtt{\frac{2}{4} >\ \frac{2}{6}} .

From the above information, we can automatically infer \mathtt{\frac{2}{3} >\ \frac{2}{6}}