Fractional numbers are a subset of Rational Numbers. The set of Rational Numbers consist of both positive fractions, negative fractions and zero. These numbers are numbers of the form $\frac{p}{q}$ where p, q - I and q ? 0. I is the set of integers.

I = {0, ±1, ±2, ±3,...}

The set of Rational numbers is represented by the letter Q. The terminating decimal numbers and the recurring decimal numbers can be written in the form of a fraction. Hence, they are also rational numbers. The rational numbers can be represented on a number line. Some examples of rational numbers are $\frac{2}{3}$, $\frac{-3}{5}$, 0 .

In general we write whole numbers, integers, fractions and decimal numbers in numerator over denominator form, that is $\frac{p}{q}$,q $\neq$ 0 where p and q are integers. Positive rational numbers are represented to the right of 0. Negative rational numbers are represented to the left of 0. The rational number zero is neither a negative rational number nor a positive rational number.

## Rational Numbers Definition

The numbers of the form $\frac{h}{k}$ , where h and k are integers and k ? 0 are called rational numbers. The sets of rational number is denoted by "Q".

Definition of classical rational numbers: Q={$\frac{h}{k}$ : h, k Z, k ? 0},
where Z= integer.

In general we says that every integer is a rational number.

For example $\frac{5}{8}$ ,$\frac{(-3)}{14}$ , $\frac{7}{(-15)}$ , $\frac{(-6)}{-11}$

### Positive rationals:

A rational number is said to be positive if its numerator and denominator are either both positive or negative.

Thus, $\frac{5}{7}$ and $\frac{(-2)}{-3}$ are both positive rationals.

### Negative rationals :

A rational number is said to be negative if its numerator and denominator are of opposite signs.

Thus, $\frac{(-4)}{9}$ and $\frac{5}{-12}$ are both negative rationals.

All problems on rational numbers are solved by the " PEDMAS " rule.

Observe the following:

 Numbers As a fraction Rational (Yes or No) 2 $\frac{1}{2}$ Yes $\sqrt{3}$ $\frac{3}{\sqrt{3}}$ No 0.25 $\frac{1}{4}$ Yes 0.40 $\frac{2}{5}$ Yes -0.35 $\frac{(-7)}{20}$ Yes

1. $\frac{10}{0}$ this is not a rational number because here the denominator =0 (as per the definition of a rational number)

2. Express $\frac{(-3)}{5}$ as a rational number with the denominator 20.

Since we need to make the denominator 20 we need to divide 20 by 5. Now by multiplying the numerator and the denominator by 4 we get $\frac{(-3)}{5}$ × $\frac{4}{4}$ = $\frac{(-12)}{20}$ [here we can observe that the denominator is 20 as per the question.

3. Express $\frac{48}{60}$ as a rational number with the denominator 5.

Since the denominator needs to be made 5, we divide 60 by 5 to get 12. Now by dividing the numerator and denominator by 12 we get, $\frac{(48\div12)}{(60\div12)}$ = $\frac{4}{5}$ [here we can observe that the denominator is 5 as per the question].

Notes: In the above two examples we multiplied as well as divided by same number in the numerator and the denominator the fraction looks different but the numerical value of the fraction remains the same.

### Irrational Numbers

 Density Property of Rational Numbers Operations with Rational Numbers Rational Exponents Rational Functions Rationalize a Denominator Even Number Adding and Subtracting Rational Expressions Dividing Rational Equations Domain and Range of Rational Functions End Behavior of Rational Functions How to Simplify Rational Expressions Real Number
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