### Various Number Systems

Before starting with Rational numbers, let us first revise other systems that we have learnt and then extend our ideas to rational numbers.

#### Natural Numbers

These are the numbers that we typically use for counting. These are numbers like 1, 2, 3, 4, ..., etc.
e.g. 4 apples, 1 table and 6 chairs, 60 students etc.
Please note that 0 is not part of Natural numbers.

#### Whole Numbers

The Natural numbers together with 0 (zero) are called Whole numbers. Thus, these are numbers 0, 1, 2, 3, 4, ..., etc.
Every Natural number is a Whole number but 0 is the only Whole number which is not a Natural number.

#### Integers

All Natural numbers, 0 and negatives of Natural numbers are called Integers.
Thus, -4, -3, -2, -1, 0, 1, 2, 3, 4, ..., etc. are Integers.
As can be seen, all Natural numbers and Whole numbers are part of Integers.

#### Fractions

The numbers of the form $\frac{a}{b}$, where a and b are Natural numbers, are Fractions.
e.g. $\frac34$, $\frac{11}{45}$, $\frac{121}{456}$ are all Fractions.

### Rational Numbers

#### Definition

The numbers of the form $\frac{a}{b}$, where a and b are Integers, and b ≠ 0 are called Rational numbers.
The word Rational is evolved from the word ratio as Rational numbers are expressed as a ratio of two numbers.
e.g. $\frac{3}{-4}$, $\frac{14}{55}$, $\frac{-21}{46}$ are all Rational Numbers.
Zero is a Rational number since 0 can be represented as $\frac{0}{1}$ but $\frac{1}{0}$ is not a Rational number.

#### Multiplication and Division of Rational Numbers

(i) For a rational number $\frac{a}{b}$, if numerator and denominator both are multiplied by a nonzero number p, the rational number remains unchanged.
$\frac{a}{b}$ = $\frac{a × p}{b × p}$
(ii) For a rational number $\frac{a}{b}$, if numerator and denominator are divided by a nonzero number p, the rational number remains unchanged.
$\frac{a}{b}$ = $\frac{a ÷ p}{b ÷ p}$

### Standard Form of a Rational Number

A rational number $\frac{a}{b}$ is said to be in standard form if b is positive and a and b have no common divisior other than 1.
e.g. $\frac{3}{5}$
To convert a given rational number to its standard form,
(i) Convert it into a Rational number whose denominator is positive and
(ii) Divide its numerator and denominator by their HCF

e.g. Convert $\frac{33}{-99}$ to its standard form,
(i) To convert it into a Rational number whose denominator is positive, multiply its numerator and denominator by (-1)
$\frac{33}{-99}$ = $\frac{33 × (-1)}{-99 × (-1)}$ = $\frac{-33}{99}$

(ii) Divide its numerator and denominator by their HCF

The HCF of 33 and 99 is 33.
∴ $\frac{-33}{99}$ = $\frac{-33 ÷ 33}{99 ÷ 33}$ = $\frac{-1}{3}$

### Equivalent Rational Numbers

If the numerator and denominator of a Rational number are multiplied or divided by a same nonzero number, the resultant Rational number is called an Equivalent Rational number.
e.g.
$\frac{5}{-7}$ = $\frac{5 × 2}{-7 × 2}$ = $\frac{5 × 3}{-7 × 3}$ = $\frac{5 × 4}{-7 × 4}$

$\frac{44}{-88}$ = $\frac{44 ÷ 2}{-88 ÷ 2}$ = $\frac{44 ÷ 4}{-88 ÷ 4}$

Question: Which fraction lies exactly halfway between $\frac{3}{4}$ and $\frac{4}{5}$?
Solution: LCM of denominators 4 and 5 is 20
$\frac{3}{4}$ = $\frac{15}{20}$ = $\frac{30}{40}$
$\frac{4}{5}$ = $\frac{16}{20}$ = $\frac{32}{40}$
Hence the fraction that lies in between the two is = $\frac{31}{40}$

### Representation of Rational Number the number line

We can represent rational numbers on a number line in the same way we represent integers on a number line.
Let us draw a number line as follows.
e.g. Represent $\frac{1}{2}$ on a number line.
If A represents integer 1 on the number line, divide OA into two equal parts such that OP and PA are equal.
Point P then represents $\frac{1}{2}$
In the same way, if A' represents -1 and if point P' divides OA' in two equal parts, P' will represent $\frac{-1}{2}$

Represent $\frac{11}{4}$ on a number line.

$\frac{11}{4}$ = 2$\frac{3}{4}$ = 2 + $\frac{3}{4}$
In the number line given below, O represents 0, A represents 2 and B represents 3.
Thus, OA is distance of 2 units. To represent the remaining $\frac{3}{4}$, divide AB into 4 equal parts and select first 3 parts out of these 5.
Then, OP represents $\frac{11}{4}$
In the same way, if A' represents -2 and B' represents -3. P' represents $\frac{-11}{4}$