1.1 - Introduction to Rational Numbers

Rational numbers are a type of number that can be expressed in the form of a fraction or ratio of two integers. In this section, we will explore the definition, properties, and examples of rational numbers as per the CBSE syllabus for Grade 8 Mathematics.

A rational number is any number that can be written as the ratio of two integers, where the denominator is not zero. It is typically represented as:

$\dfrac{p}{q}$, where $p$ and $q$ are integers, and $q \neq 0$.

The number $p$ is called the numerator, and $q$ is called the denominator. Here, $p$ and $q$ are both integers, and $q$ must not be zero because division by zero is undefined.

Examples of Rational Numbers:

• $\dfrac{3}{4}$, where 3 is the numerator and 4 is the denominator.
• $\dfrac{-7}{2}$, where -7 is the numerator and 2 is the denominator.
• $\dfrac{5}{1}$, which is equal to 5 (since any number divided by 1 is the number itself).
• 0 can be written as $\dfrac{0}{1}$, which is a rational number.
• Negative numbers like $-\dfrac{2}{3}$ are also rational numbers.

Properties of Rational Numbers:

• Closure Property: The sum, difference, and product of any two rational numbers are also rational numbers.
• Commutative Property: The sum and product of rational numbers follow the commutative property, meaning $a + b = b + a$ and $a \times b = b \times a$ for any rational numbers $a$ and $b$.
• Associative Property: The sum and product of rational numbers follow the associative property, meaning $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$.
• Existence of Identity: The number 0 is the additive identity, and 1 is the multiplicative identity for rational numbers.
• Existence of Inverse: Every rational number has an additive inverse and a multiplicative inverse. For any rational number $\dfrac{p}{q}$, the additive inverse is $\dfrac{-p}{q}$, and the multiplicative inverse (if $p \neq 0$) is $\dfrac{q}{p}$.

Types of Rational Numbers:

Rational numbers can be classified based on their nature:

• Positive Rational Numbers: These are rational numbers where both the numerator and denominator are positive integers. For example, $\dfrac{5}{3}$.
• Negative Rational Numbers: These are rational numbers where either the numerator or the denominator is negative, but not both. For example, $\dfrac{-7}{4}$.
• Zero: Zero is a rational number because it can be expressed as $\dfrac{0}{1}$, and it is neither positive nor negative.
• Recurring and Terminating Decimals: Rational numbers can also be expressed as decimals. Some rational numbers have a terminating decimal representation, such as $\dfrac{1}{4} = 0.25$, while others have a recurring decimal representation, such as $\dfrac{1}{3} = 0.333\ldots$.

Representation of Rational Numbers on the Number Line:

Rational numbers can be represented on the number line. Every rational number corresponds to a unique point on the number line. The positive rational numbers lie to the right of zero, while the negative rational numbers lie to the left of zero.

For example, $\dfrac{2}{3}$ would lie between 0 and 1 on the number line, and $\dfrac{-5}{2}$ would lie between -2 and -3.

Conversion between Fractions, Decimals, and Percentages:

Rational numbers can also be converted between fractions, decimals, and percentages:

• To convert a fraction to a decimal, divide the numerator by the denominator.
• To convert a decimal to a fraction, express the decimal as a fraction and simplify.
• To convert a fraction to a percentage, multiply the fraction by 100 and add the percentage sign.