A Rational Number is any number expressible in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). In Class 8 NCERT Mathematics, rational numbers obey closure, commutativity, and associativity under addition and multiplication, with \(0\) as the additive identity, \(1\) as the multiplicative identity, and distributive property \(a(b + c) = ab + ac\).
High-Yield Chapter Key Takeaways
- Standard Form: \(\frac{p}{q}\) where \(p, q\) are integers and \(q \neq 0\).
- Addition and multiplication are commutative and associative; subtraction and division are NOT.
- Additive Identity is \(0\) (\(a + 0 = a\)); Multiplicative Identity is \(1\) (\(a \times 1 = a\)).
- Additive Inverse of \(\frac{a}{b}\) is \(-\frac{a}{b}\); Multiplicative Inverse (reciprocal) is \(\frac{b}{a}\).
- Zero (\(0\)) has NO reciprocal because \(\frac{1}{0}\) is undefined.
- There are infinitely many rational numbers between any two given rational numbers.
1. Definition and Standard Representation
A number that can be expressed in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\), is defined as a Rational Number (\(\mathbb{Q}\)). All natural numbers, whole numbers, integers, and terminating/repeating decimals are rational numbers.
| Number Category | Rational Form \(\frac{p}{q}\) | Explanation |
|---|---|---|
| Zero (\(0\)) | \(\frac{0}{1}, \frac{0}{5}, \frac{0}{-8}\) | Rational because denominator \(q \neq 0\). |
| Integers (e.g., \(-7, 5\)) | \(\frac{-7}{1}, \frac{5}{1}\) | Any integer \(a\) can be written as \(\frac{a}{1}\). |
| Terminating Decimals (\(0.75\)) | \(\frac{75}{100} = \frac{3}{4}\) | Expressible as ratio of two integers. |
| Repeating Decimals (\(0.333\dots\)) | \(\frac{1}{3}\) | Non-terminating repeating decimals are rational. |
2. Core Properties of Rational Numbers
Operations on rational numbers follow strict algebraic properties across addition, subtraction, multiplication, and division:
| Property | Addition (\(+\)) | Subtraction (\(-\)) | Multiplication (\(\times\)) | Division (\(\div\)) |
|---|---|---|---|---|
| Closure | ✅ Closed: \(a + b \in \mathbb{Q}\) | ✅ Closed: \(a - b \in \mathbb{Q}\) | ✅ Closed: \(a \times b \in \mathbb{Q}\) | ❌ Not Closed (if divisor is \(0\)) |
| Commutative | ✅ \(a + b = b + a\) | ❌ \(a - b \neq b - a\) | ✅ \(a \times b = b \times a\) | ❌ \(a \div b \neq b \div a\) |
| Associative | ✅ \((a + b) + c = a + (b + c)\) | ❌ \((a - b) - c \neq a - (b - c)\) | ✅ \((ab)c = a(bc)\) | ❌ \((a \div b) \div c \neq a \div (b \div c)\) |
| Distributive | \(a(b + c) = ab + ac\) and \(a(b - c) = ab - ac\) (Multiplication distributes over addition and subtraction). | |||
3. Identities and Inverses
Zero is the additive identity for rational numbers:
Adding zero to any rational number leaves the number unchanged.
One (\(1\)) is the multiplicative identity for rational numbers:
Multiplying any rational number by 1 yields the original number.
For any rational number \(\frac{a}{b}\), its additive inverse is \(-\frac{a}{b}\):
Example: Additive inverse of \(\frac{-5}{9}\) is \(\frac{5}{9}\).
The reciprocal of a non-zero rational number \(\frac{a}{b}\) is \(\frac{b}{a}\):
Warning: Zero (\(0\)) has no reciprocal because \(\frac{1}{0}\) is undefined.
4. Representation on a Number Line
Any rational number can be plotted on the real number line:
- To represent a proper fraction like \(\frac{4}{7}\), divide the unit distance between \(0\) and \(1\) into \(7\) equal segments. The \(4^{\text{th}}\) mark from \(0\) represents \(\frac{4}{7}\).
- To represent an improper fraction like \(\frac{8}{3}\), convert to a mixed fraction: \(\frac{8}{3} = 2\frac{2}{3}\). Locate the segment between \(2\) and \(3\), divide it into \(3\) equal intervals, and pick the \(2^{\text{nd}}\) subdivision.
- Negative rational numbers (e.g., \(-\frac{5}{6}\)) are plotted symmetrically to the left of zero.
5. Finding Rational Numbers Between Two Rational Numbers
Between any two rational numbers, there exist infinitely many rational numbers. Two reliable methods are used in NCERT Class 8:
Goal: Find \(5\) rational numbers between \(\frac{-2}{5}\) and \(\frac{1}{2}\).
Step 1: Equalize denominators using \(\text{LCM}(5, 2) = 10\):
94237\frac{-2}{5} = \frac{-2 \times 2}{5 \times 2} = \frac{-4}{10}, \quad \frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}94237Step 2: Select integers between \(-4\) and \(5\): \(-3, -2, -1, 0, 1, 2, 3, 4\).
Result: Five rational numbers are: \(\frac{-3}{10}, \frac{-2}{10}, \frac{-1}{10}, \frac{1}{10}, \frac{2}{10}\).
For two rational numbers \(a\) and \(b\), the average \(\frac{a + b}{2}\) always lies strictly between \(a\) and \(b\) (\(a < \frac{a+b}{2} < b\)). Repeat successively to find as many numbers as required.
6. Solved NCERT Exemplar Problems
Problem: Evaluate using appropriate algebraic properties:
94237\frac{2}{5} \times \left(-\frac{3}{7} \right) - \frac{1}{14} - \frac{3}{7} \times \frac{3}{5}94237Step 1: Rearrange terms using Commutative Property:
94237= \frac{2}{5} \times \left(-\frac{3}{7} \right) - \frac{3}{7} \times \frac{3}{5} - \frac{1}{14}94237Step 2: Factor out \(\left(-\frac{3}{7} \right)\) using Distributive Property:
94237= \left(-\frac{3}{7} \right) \times \left[\frac{2}{5} + \frac{3}{5} \right] - \frac{1}{14}94237Step 3: Simplify inside brackets:
94237= \left(-\frac{3}{7} \right) \times \left[\frac{5}{5} \right] - \frac{1}{14} = \left(-\frac{3}{7} \right) \times 1 - \frac{1}{14} = -\frac{3}{7} - \frac{1}{14}94237Step 4: Common denominator (\(14\)):
94237= \frac{-6 - 1}{14} = -\frac{7}{14} = -\frac{1}{2}942377. Common Exam Pitfalls & Trap Avoidance
- Trap 1: Assuming Division is Commutative or Associative: \(8 \div 4 = 2\), but \(4 \div 8 = \frac{1}{2}\). Subtraction and division are NEVER commutative or associative.
- Trap 2: Writing Reciprocal of Zero: The reciprocal of \(0\) is \(\frac{1}{0}\), which is undefined. \(0\) is the only rational number with no reciprocal.
- Trap 3: Forgetting Sign when Computing Multiplicative Inverse: The reciprocal of \(-\frac{3}{7}\) is \(-\frac{7}{3}\). The negative sign is preserved (only the numerator and denominator flip).
- Trap 4: Confusing Additive and Multiplicative Inverses: Additive inverse changes the sign (sum is \(0\)); multiplicative inverse flips the fraction (product is \(1\)).
