What is Rational Numbers Class 7: Definition & Examples Explained

Definition of Rational Numbers for Class 7 Students

In Class 7 NCERT Mathematics, rational numbers are defined as numbers that can be written in the form:

$$\frac{p}{q}$$

where:

• $p$ and $q$ are integers
• $q \neq 0$

This means any number that can be expressed as a fraction with an integer numerator and a non-zero integer denominator is a rational number.

Examples:

• $\frac{3}{4}$ (positive rational number)
• $\frac{-7}{2}$ (negative rational number)
• $0$ (can be written as $\frac{0}{1}$)

Rational numbers include integers, fractions, and decimals that can be converted into fractions.

Types of Rational Numbers: Positive, Negative, and Zero

Rational numbers can be classified into three types:

• Positive Rational Numbers: Numbers greater than zero, e.g., $\frac{5}{3}$, $2$, $0.75$
• Negative Rational Numbers: Numbers less than zero, e.g., $-\frac{4}{7}$, $-3$, $-0.5$
• Zero: The number zero itself is a rational number because it can be written as $\frac{0}{1}$.

This classification helps in understanding their placement on the number line and their behavior during operations like addition and subtraction.

Properties of Rational Numbers Explained

Rational numbers have several important properties:

• Closure Property: The sum, difference, and product of two rational numbers is always a rational number.
• Commutative Property: $a + b = b + a$ and $ab = ba$ for rational numbers $a$ and $b$.
• Associative Property: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$.
• Distributive Property: $a(b + c) = ab + ac$.

These properties make rational numbers easy to work with in algebra and arithmetic.

Example:

If $a = \frac{2}{3}$ and $b = \frac{4}{5}$,

• Sum: $a + b = \frac{2}{3} + \frac{4}{5} = \frac{10}{15} + \frac{12}{15} = \frac{22}{15}$
• Product: $ab = \frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$

Rational Numbers vs Irrational Numbers: A Comparison

Understanding the difference between rational and irrational numbers is crucial:

This comparison helps Class 7 students grasp the scope of rational numbers in the number system.

How to Perform Operations on Rational Numbers

Operations with rational numbers follow specific rules:

• Addition: Find a common denominator, then add numerators.

$$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$

• Subtraction: Similar to addition, subtract numerators after finding a common denominator.

$$\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$$

• Multiplication: Multiply numerators and denominators directly.

$$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$$

• Division: Multiply by the reciprocal of the divisor.

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$$

Worked Example:

Add $\frac{2}{5}$ and $\frac{3}{10}$:

• Common denominator = 10
• Convert $\frac{2}{5} = \frac{4}{10}$
• Sum = $\frac{4}{10} + \frac{3}{10} = \frac{7}{10}$

Decimal Representation of Rational Numbers

Every rational number can be expressed as a decimal which is either:

• Terminating Decimal: Decimal expansion ends after a finite number of digits.

Example: $\frac{3}{4} = 0.75$

• Repeating Decimal: Decimal expansion has a repeating pattern.

Example: $\frac{1}{3} = 0.333...$

This property helps students identify rational numbers when given decimals.

Note: If a decimal neither terminates nor repeats, it is not rational.