charumam.com | Chapter 01 | Chapter Notes
Rational Numbers — Notes
Class 8 | Mathematics | NCERT / ICSE
📚 About These Notes
Class 8 | Mathematics | Chapter 01
Board: NCERT / ICSE
Subject: Rational Numbers
🎯 Learning Outcomes
Iss chapter ke baad aap rational numbers define, operate, aur number line pe represent kar paoge.
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🎯 Aaj Hum Seekhenge
- Rational numbers kya hote hain aur \(\frac{p}{q}\) form kya hoti hai
- Closure, Commutativity, Associativity, Distributivity properties
- Rational numbers ko number line pe represent karna
- Do rational numbers ke beech infinite rational numbers dhundhna
📑 Table of Contents
What are Rational Numbers?
🔥 Hook — Pehle Socho!
Aur agar 5 toffees 4 doston mein barabar baantni hain? \(5 \div 4 = \frac{5}{4}\). Yeh bhi integer nahi!
Jawab: Yeh dono Rational Numbers hain! Aao seekhte hain. 🚀
📘 Concept — Samjho Dhyan Se
📌 Definition
Ek number rational hota hai agar usse \(\dfrac{p}{q}\) form mein likha ja sake, jahan \(p\) aur \(q\) dono integers hain aur \(q \neq 0\).
Examples:
✅ \(\dfrac{3}{4}\) → \(p=3,\ q=4,\ q\neq 0\) ✔
✅ \(\dfrac{-7}{2}\) → \(p=-7,\ q=2,\ q\neq 0\) ✔
✅ \(0 = \dfrac{0}{1}\) → \(q=1 \neq 0\) ✔ (zero bhi rational hai!)
✅ \(5 = \dfrac{5}{1}\) ✔ (integers bhi rational hain!)
❌ \(\sqrt{2}\) → Rational nahi — \(\frac{p}{q}\) form mein nahi aata
❌ \(\pi\) → Rational nahi
📌 Standard Form (Simplest Form)
Ek rational number standard form mein hota hai jab:
1️⃣ \(q\) (denominator) positive ho
2️⃣ \(\gcd(p,\ q) = 1\) — yaani \(p\) aur \(q\) ka HCF sirf 1 ho
Example: \(\dfrac{-12}{16}\) → \(\text{HCF}(12,16) = 4\) → \(\dfrac{-12 \div 4}{16 \div 4} = \dfrac{-3}{4}\) ✅
✏️ Examples — Khud Try Karo
📝 Example 1 — Is \(-\dfrac{5}{7}\) a rational number?
Q: Kya \(-\dfrac{5}{7}\) ek rational number hai?
📝 Example 2 — Standard Form of \(\dfrac{-36}{24}\)
Q: \(\dfrac{-36}{24}\) ko standard form mein likho.
🌍 Real Life — Apne Aas Paas Dekho
🌡️ Temperature: \(-2.5°C = -\dfrac{5}{2}°C\) — negative rational number!
💰 Bank: \(-₹500 = \dfrac{-500}{1}\) — debt bhi rational hai!
📏 Measurement: \(2.5\text{ m} = \dfrac{5}{2}\text{ m}\) — har measurement rational hai!
🌟 Wah! Topic 1 ho gaya! Definition yaad raho: \(\dfrac{p}{q},\ q \neq 0\). Bas!
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Yeh notes bilkul free hain! Doston ke saath share karo 💜 | Visit NowProperties of Rational Numbers
🔥 Hook — Pehle Socho!
\(\dfrac{2}{3} \neq \dfrac{3}{2}\) — Nahi! 😲
Kyun kuch operations order nahi mante? Yahi samjhaati hain Properties! 💪
📘 Concept — Properties Table
| Property | Addition | Subtraction | Multiplication | Division |
|---|---|---|---|---|
| Closure | ✅ | ✅ | ✅ | ✅* |
| Commutativity | ✅ | ❌ | ✅ | ❌ |
| Associativity | ✅ | ❌ | ✅ | ❌ |
| Distributivity | \(a \times (b + c) = a \times b + a \times c\) ✅ | |||
*Division: divisor ≠ 0 hona chahiye
📌 Commutativity
\[\frac{a}{b} + \frac{c}{d} = \frac{c}{d} + \frac{a}{b} \quad \checkmark\] \[\frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b} \quad \checkmark\] \[\frac{a}{b} – \frac{c}{d} \neq \frac{c}{d} – \frac{a}{b} \quad \times\]
📌 Additive & Multiplicative Identity
Additive Identity = 0: \(\dfrac{a}{b} + 0 = \dfrac{a}{b}\)
Multiplicative Identity = 1: \(\dfrac{a}{b} \times 1 = \dfrac{a}{b}\)
📌 Inverse Elements
Additive Inverse of \(\dfrac{a}{b}\): \(-\dfrac{a}{b}\) kyunki \(\dfrac{a}{b} + \left(-\dfrac{a}{b}\right) = 0\)
Multiplicative Inverse of \(\dfrac{a}{b}\): \(\dfrac{b}{a}\) kyunki \(\dfrac{a}{b} \times \dfrac{b}{a} = 1\)
✏️ Examples — Khud Try Karo
📝 Example 3 — Verify Commutativity: \(\dfrac{2}{3} + \left(-\dfrac{5}{4}\right)\)
Q: Verify karo: \(\dfrac{2}{3} + \left(-\dfrac{5}{4}\right) = \left(-\dfrac{5}{4}\right) + \dfrac{2}{3}\)
📝 Example 4 — Additive & Multiplicative Inverse of \(-\dfrac{5}{9}\)
Q: \(-\dfrac{5}{9}\) ka additive inverse aur multiplicative inverse batao.
🌍 Real Life — Apne Aas Paas Dekho
➗ Division not commutative: \(10 \div 5 = 2\) lekin \(5 \div 10 = \dfrac{1}{2}\) — bilkul alag!
🔄 Inverse: Speed \(= 60\) km/hr → Time per km \(= \dfrac{1}{60}\) hr/km — real life multiplicative inverse! 🚀
💪 Properties done! Ek kaam karo — table ghar pe likhke rakh!
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Tumne 2 topics kar liye! Karo aage! 💜 | Visit NowRational Numbers on a Number Line
🔥 Hook — Pehle Socho!
📘 Concept — Samjho Dhyan Se
📌 Number Line pe Represent karna
\(\dfrac{2}{5}\) represent karo:
Step 1: 0 aur 1 ke beech 5 equal parts karo (denominator = 5)
Step 2: Left se 2nd part pe point lagao (numerator = 2)
Step 3: Woh point hai \(\dfrac{2}{5}\) ✅
\(-\dfrac{3}{4}\) represent karo:
Step 1: 0 aur \(-1\) ke beech 4 equal parts karo
Step 2: 0 se left mein 3rd part pe point → yeh hai \(-\dfrac{3}{4}\) ✅
✏️ Example — Khud Try Karo
📝 Example 5 — Represent \(-\dfrac{3}{7}\) on Number Line
Q: \(-\dfrac{3}{7}\) ko number line pe represent karo.
🌍 Real Life — Apne Aas Paas Dekho
🏦 Bank: \(-₹500\) yaani 0 se left pe — exactly number line jaisa!
🗺️ Altitude: Sea level \(= 0\). Mountain \(=\) positive. Underwater \(=\) negative. ⛰️
📏 Number line clear! Ek ruler lo aur \(\dfrac{1}{3},\ \dfrac{2}{3}\) khud mark karo!
charumam.com pe aur practice milegi! 💜Rational Numbers Between Two Rational Numbers
🔥 Hook — Pehle Socho!
📘 Concept — Do Methods
📌 Method 1 — Mean (Average) Method
\(\dfrac{a}{b}\) aur \(\dfrac{c}{d}\) ke beech ek rational number: \[\frac{\dfrac{a}{b} + \dfrac{c}{d}}{2}\] Example: \(\dfrac{1}{3}\) aur \(\dfrac{1}{2}\) ke beech: \[\frac{\dfrac{1}{3} + \dfrac{1}{2}}{2} = \frac{\dfrac{5}{6}}{2} = \frac{5}{12} \checkmark\]
📌 Method 2 — Common Denominator Method
Dono numbers ko same denominator pe laao, phir beech wale numerators ke numbers likho!
Example: \(\dfrac{1}{5}\) aur \(\dfrac{4}{5}\) ke beech → numerators 1 se 4 ke beech:
\[\frac{2}{5},\ \frac{3}{5}\]
Agar aur chahiye → \(\times 10\) karo → \(\dfrac{10}{50}\) aur \(\dfrac{40}{50}\) → \(\dfrac{11}{50},\ \dfrac{12}{50},\ \ldots\)
✏️ Example — Khud Try Karo
📝 Example 6 — Find 5 rational numbers between \(\dfrac{3}{5}\) and \(\dfrac{4}{5}\)
Q: \(\dfrac{3}{5}\) aur \(\dfrac{4}{5}\) ke beech 5 rational numbers batao.
🌍 Real Life — Apne Aas Paas Dekho
💊 Medicine: \(10\)mg aur \(11\)mg ke beech dosage — \(10.5\)mg, \(10.25\)mg… rational numbers life save karte hain!
💰 Finance: Share price ₹50 aur ₹51 ke beech — ₹50.25, ₹50.50… infinite prices! 📈
🎉 Poora Chapter Complete! Tum ne aaj bahut kuch seekha!
Rational numbers ki duniya \(\infty\) hai — bilkul tumhari possibilities ki tarah! 🌟
Practice ke liye MCQs aur Worksheets available hain. | charumam.com 💜💜 charumam.com — Har Sawaal Ka Jawab!
Poora chapter padh liya! Ab practice karo! Visit Now →Definition: \(\dfrac{p}{q}\) form, \(p,q \in \mathbb{Z},\ q \neq 0\)
Standard Form: \(q > 0\) aur \(\gcd(p,q) = 1\)
Closure: \(\frac{a}{b} + \frac{c}{d},\ \frac{a}{b} – \frac{c}{d},\ \frac{a}{b} \times \frac{c}{d}\) — result hamesha rational ✅
Commutativity: Addition & multiplication only ✅ | Subtraction & division ❌
Associativity: Addition & multiplication only ✅
Distributivity: \(a(b+c) = ab + ac\) ✅
Identity: Additive \(= 0\) | Multiplicative \(= 1\)
Inverse: Additive of \(\frac{a}{b} = -\frac{a}{b}\) | Multiplicative of \(\frac{a}{b} = \frac{b}{a}\)
Number Line: Denominator = parts | Numerator = count | Negative = left of 0
Between Two Numbers: Mean method: \(\dfrac{\frac{a}{b}+\frac{c}{d}}{2}\) | Or multiply denominator!
❓ FAQs — Frequent Doubts Cleared
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