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Rational Numbers


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.
$\frac{5}{-7}$ = $\frac{5 × 2}{-7 × 2}$ = $\frac{5 × 3}{-7 × 3}$ = $\frac{5 × 4}{-7 × 4}$

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




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.
Graphical representation of exponential functions

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 $\frac{1}{2}$
Graphical representation of exponential functions
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 $\frac{11}{4}$
Graphical representation of exponential functions
In the same way, if A' represents -2 and B' represents -3. P' represents $\frac{-11}{4}$