Real Numbers की दुनिया: आसान Tricks और मजेदार Facts! 📖

Real Numbers and Their Decimal Expansions

Introduction

Real numbers include both rational and irrational numbers. Understanding their decimal expansions is crucial to identifying and distinguishing between these types of numbers. Let’s break it down step by step with detailed explanations, additional examples, and step-by-step solutions for practice questions.

1. Decimal Expansion of Rational Numbers

Rational numbers are numbers that can be written in the form of p/q (where p and q are integers and q ≠ 0).

Types of Decimal Expansions in Rational Numbers

Rational numbers have two types of decimal expansions:

(a) Terminating Decimal Expansion

• A terminating decimal is one that comes to an end.
• Example:
  • 1/2 = 0.5 (It stops after one decimal place)
  • 3/4 = 0.75 (It stops after two decimal places)
• A rational number has a terminating decimal expansion if and only if its denominator (after simplification) has only the prime factors 2 or 5 (or both).

Additional Examples:

• 7/8 = 0.875 (Terminating Decimal)
• 1/40 = 0.025 (Terminating Decimal)
• 5/16 = 0.3125 (Terminating Decimal)

Practice Questions with Solutions:

1. Identify if the following numbers have a terminating or non-terminating decimal expansion:
   • 3/5 → Prime factor of 5 = 5 → Terminating
   • 11/25 → Prime factors of 25 = 5 × 5 → Terminating
   • 13/32 → Prime factors of 32 = 2 × 2 × 2 × 2 × 2 → Terminating

(b) Non-Terminating but Repeating Decimal Expansion

• These decimals go on forever but follow a repeating pattern.
• Example:
  • 1/3 = 0.3333… (3 repeats)
  • 7/11 = 0.6363… (63 repeats)
• A rational number has a repeating decimal expansion if its denominator has prime factors other than 2 or 5.

Additional Examples:

• 5/7 = 0.714285… (Pattern: 714285)
• 1/6 = 0.1666… (Pattern: 6)
• 4/9 = 0.444… (Pattern: 4)

Practice Questions with Solutions:

1. Identify the repeating pattern for these numbers:
   • 2/7 = 0.285714… (Pattern: 285714)
   • 5/12 = 0.4166… (Pattern: 6)
   • 11/30 = 0.3666… (Pattern: 6)

2. Decimal Expansion of Irrational Numbers

Irrational numbers are numbers that cannot be written in the form of p/q. Their decimal expansion is:

• Non-Terminating and Non-Repeating
• Example:
  • √2 = 1.41421356… (No pattern)
  • π (Pi) = 3.141592653… (No pattern)

Important Observation

• Decimal expansions of irrational numbers neither terminate nor repeat any pattern.
• Examples of some common irrational numbers:
  • √3 = 1.732050807…
  • √5 = 2.23606797…
  • e (Euler’s Number) = 2.718281828…

Practice Questions with Solutions:

1. Identify whether the following numbers are rational or irrational:
   • √7 → Irrational (Non-repeating, Non-terminating)
   • π → Irrational (Non-repeating, Non-terminating)
   • 0.101001000100001… → Irrational (Non-repeating, Non-terminating)

3. Link Between Decimal Expansion and Type of Number

• Terminating Decimal Expansion → Rational Number
• Non-Terminating but Repeating Decimal Expansion → Rational Number
• Non-Terminating and Non-Repeating Decimal Expansion → Irrational Number

Practice Questions with Solutions:

1. Identify the type of number for each of these decimal expansions:
   • 0.125 → Terminating → Rational
   • 0.666… → Repeating → Rational
   • 1.414213… → Non-repeating, Non-terminating → Irrational

4. Conversion from Fraction to Decimal

To determine the type of decimal expansion for a given rational number:

1. Convert the fraction to its simplest form.
2. Check the prime factors of the denominator.
   • If the denominator has only 2 or 5 as its prime factors → Terminating Decimal
   • If the denominator has prime factors other than 2 or 5 → Repeating Decimal

Practice Questions with Solutions:

1. Determine the type of decimal expansion for these fractions:
   • 7/20 → Prime factor of 20 = 2 × 2 × 5 → Terminating
   • 1/11 → Prime factor of 11 = 11 → Repeating
   • 3/25 → Prime factor of 25 = 5 × 5 → Terminating

5. Real-Life Examples for Better Understanding

• Shopping Bill: Prices like Rs. 12.50 (Terminating)
• Distance Measurements: Values like 3.14159 km (Pi, Non-repeating, Non-terminating)
• Splitting a Cake: Dividing 7 slices among 3 people gives 2.333… (Repeating)

6. Quick Recap Table Summary

Real Numbers की मजेदार दुनिया! 🚀 | हर Concept को आसान बनाएं!

🎯 शुरुआत एक सवाल से!

क्या आप भी Math के इस सवाल को देखकर 😵‍💫 कन्फ्यूज़ हो जाते हैं? –

“0.3333… ये नंबर आखिर कब रुकेगा? क्या ये किसी fraction से जुड़ा है?”

अगर हां, तो चिंता मत करो! हम Real Numbers और उनकी Decimal Expansions को इतनी मजेदार और आसान भाषा में समझेंगे कि यह आपके लिए गेम बन जाए! 😎

🔥 Real Numbers क्या होते हैं?

Real Numbers यानी वे सारे नंबर जो आपको कहीं भी देखने को मिलते हैं –

• Shopping Bill 🛒 – Rs. 199.99
• पाई (π) का मान – 3.141592653…
• Square Root वाले नंबर – √2 = 1.414…

👉 Simple Language में: Real Numbers में Rational और Irrational दोनों आते हैं। अब इन्हें विस्तार से समझते हैं! 🚀

🎯 Rational Numbers की Decimal Expansions

Rational Numbers वे होते हैं, जिन्हें p/q के रूप में लिखा जा सकता है (जहाँ p, q integers होते हैं और q ≠ 0)। इनके दो cases होते हैं:

✅ (a) Terminating Decimal Expansion (जहाँ decimal रुक जाता है!)

• Example:
  • 1/2 = 0.5
  • 3/4 = 0.75
  • 7/8 = 0.875

🎯 Trick – अगर fraction के denominator में सिर्फ 2 या 5 (या दोनों) हों, तो decimal expansion terminate करेगा! 🔥

✍️ Practice Question:

1. क्या 13/32 का decimal expansion terminate करेगा? 🤔
   • Answer: हाँ! क्यूंकि 32 = 2⁵ होता है! ✅

🔄 (b) Non-Terminating, Repeating Decimal Expansion (जहाँ decimal कभी नहीं रुकता, लेकिन repeat ज़रूर करता है!)

• Example:
  • 1/3 = 0.333… (3 बार-बार आता है)
  • 7/11 = 0.636363…
  • 5/7 = 0.714285714285…

🎯 Trick – अगर fraction के denominator में 2 या 5 के अलावा कोई और prime factor हो, तो decimal repeat करेगा! 🔁

✍️ Practice Question:

1. 2/7 का decimal expansion कैसा होगा?
   • Answer: यह 0.285714285714… repeat करेगा! 🔄

🌀 Irrational Numbers की Decimal Expansions

Irrational Numbers ऐसे होते हैं, जिन्हें fraction p/q में नहीं लिखा जा सकता। इनके decimal expansions:

• Non-Terminating होते हैं (मतलब decimal कभी खत्म नहीं होता!)
• Non-Repeating होते हैं (कोई pattern नहीं बनता!)

👉 Example:

• π (Pi) = 3.141592653… (infinite, no pattern!)
• √2 = 1.41421356…
• e (Euler’s Number) = 2.718281828…

✍️ Practice Question:

1. क्या √5 का decimal expansion terminate करेगा?
   • Answer: नहीं! यह irrational है, इसलिए non-terminating & non-repeating होगा! ✅

🤔 कैसे पता करें कि कोई Decimal Expansion किस Type का है?

🎯 Trick – बस denominator को देखो!

• सिर्फ 2 और 5 → Terminating ✅
• कोई और Prime Factor? → Repeating 🔄
• Fraction में नहीं लिखा जा सकता? → Irrational 🌀

✍️ Practice Question:

1. 1/11 का decimal expansion कैसा होगा?
   • Answer: यह repeating होगा! 0.090909… 🔄

🏆 Final Challenge – क्या आप इसे Crack कर सकते हैं? 🎯

1. 7/20 का decimal expansion कैसा होगा?
2. 1/17 repeating होगा या terminating?
3. π + 2 rational है या irrational?

👉 अपने जवाब नीचे कमेंट करें! 📢 😃

🚀 अब आप Real Numbers के EXPERT बन गए! अगर मजा आया तो शेयर ज़रूर करें! 😍

Check other topics below

Introduction to Numbers and the Number Line

Zero से Hero तक! 🚀 Rational Numbers का धमाल!

🚀 Numbers का Magic! Real vs Irrational – जानिए सच 🔢