e.g. 4 apples, 1 table and 6 chairs, 60 students etc.

Please note that 0 is not part of Natural numbers.

Every Natural number is a Whole number but 0 is the only Whole number which is not a Natural number.

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.

e.g. $\frac34$, $\frac{11}{45}$, $\frac{121}{456}$ are all Fractions.

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.

$\frac{a}{b}$ = $\frac{a × p}{b × p}$

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

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.

∴ $\frac{-33}{99}$ = $\frac{-33 ÷ 33}{99 ÷ 33}$ = $\frac{-1}{3}$

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 = $\frac{31}{40}$

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

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 = $\frac{31}{40}$

Let us draw a number line as follows.

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 $\frac{1}{2}$

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 $\frac{11}{4}$