Real Numbers & Decimals (Deep Dive) - Class 9 Maths

👋 Welcome Back Students!

Pichle topics mein humne Rational aur Irrational numbers ko alag-alag padha. Aaj hum in dono ko milakar Real Numbers ka poora concept samjhenge.

Hum ye bhi seekhenge ki kaise hum sirf number ko dekh kar bata sakte hain ki wo rational hai ya irrational, aur kaise repeating decimals ko wapas fraction (p/q) banaya jata hai. Ye exam ke liye bahut important hai! 🌟

🎥 Video Explanation

English Explanation

Hindi Explanation

📊 Infographic Summary

🎒 The "Bag" Analogy: Real Numbers Kya Hain?

📖 Story Time: The Giant Bag

Imagine karo tumhare paas ek bahut bada bag hai. Sabse pehle tumne usme saare Rational numbers (1, 1/2, 0.5) daal diye. Phir tumne bache hue Irrational numbers (√2, π) bhi usi bag me bhar diye. Ab bag poora full hai! Is bhare hue bag ko hi hum Real Numbers kehte hain. Isme sab kuch hai!

Imagine karo aapke paas ek bada sa bag hai jiska naam hai "Real Numbers Bag".

Is bag ke andar aapne saare Rational Numbers (jaise 1/2, 5, -10) daal diye.
Phir aapne saare Irrational Numbers (jaise √2, π) bhi usi bag mein daal diye.

Ab ye bag poora bhar gaya hai. Iska matlab:

Real Numbers woh collection hai jisme Rational aur Irrational dono tarah ke numbers hote hain. Number line ka har ek point ek Real Number hota hai.

🔍 Decimal Expansions: The 3 Cases

📖 Story Time: The Three Runners

Teen tarah ke runners hote hain:

Runner 1 (Terminating): Thoda daudta hai aur finish line par ruk jaata hai (Remainder 0).
Runner 2 (Recurring): Gol-gol ghoomta rehta hai, wahi rasta baar-baar repeat karta hai (Repeating).
Runner 3 (Irrational): Seedha chalta jaata hai, kabhi nahi rukta aur kabhi wapas nahi aata (Non-terminating, Non-repeating).

Jab hum kisi bhi fraction p/q ko divide karte hain, toh remainder (shesh) ke saath kya hota hai? Sirf 2 cheezein ho sakti hain:

Remainder Zero ho jayega.
Remainder kabhi Zero nahi hoga, bas Repeat karega.

Chaliye inhein detail mein dekhte hain:

Case 1: Terminating Decimal (Rukne wala)

Jab division karte waqt kuch steps ke baad remainder 0 aa jata hai.

Example: 7/8

Agar hum 7 ko 8 se divide karein:

      0.875
    _______
  8 ) 7.000
     -6 4
     ----
        60
       -56
       ----
         40
        -40
        ----
          0  <-- Remainder is 0 (Stop!)

Result: 0.875. Ye khatam ho gaya, isliye ye Rational hai.

🚀 Pro Tip (Secret Rule):
Bina divide kiye kaise pata karein ki decimal terminate hoga?
Agar denominator (q) ke prime factors mein sirf 2 ya sirf 5 ya dono hon, toh wo pakka terminate karega!
Example: 1/20 (20 = 2×2×5) → Terminating.
Example: 1/6 (6 = 2×3) → Non-terminating (kyunki 3 aa gaya).

Case 2: Non-Terminating Recurring (Repeating)

Jab remainder kabhi zero nahi hota, aur wahi digits baar-baar aane lagte hain.

Example: 10/3

10 ÷ 3 = 3.3333...

Yahan '3' repeat ho raha hai. Hum isse bar laga kar likhte hain: 3.3̄

Example: 1/7

1 ÷ 7 = 0.142857142857...

Yahan poora block '142857' repeat ho raha hai. So, 0.142857̄

Ye bhi Rational numbers hote hain.

Case 3: Non-Terminating Non-Recurring (Non-Repeating)

Ye special numbers hain. Na to ye rukte hain, aur na hi inme koi pattern repeat hota hai.

Example: √2 & π

√2 = 1.41421356... (Random digits aate rahenge)

π = 3.14159265...

Hum aise numbers khud bhi bana sakte hain:

0.101001000100001... (Dekho, har baar zero badh raha hai, so pattern same nahi hai!)

Ye Irrational Numbers hote hain.

🛠️ How to Convert Decimals to Fractions (p/q)?

📖 Story Time: Catching the Ghost

Repeating decimals (jaise 0.333...) ek ghost ki tarah hote hain jo kabhi khatam nahi hote. Unhe pakadne ke liye hum 'Algebra ka Jaal' (x method) use karte hain. Hum number ko bada karte hain (multiply by 10) aur phir original number ko minus kar dete hain. Isse ghost gayab ho jaata hai aur humein simple fraction mil jaata hai!

Ye exam mein zaroor aata hai! Dhyan se samjhein.

Type A: Terminating Decimals (Easy Peasy)

Bas point hatao aur neeche 10, 100, 1000 laga do.

Q: Convert 0.15 to p/q

Step 1: Point ke baad 2 digits hain, toh neeche 100 lagayenge.

0.15 = 15/100

Step 2: Simplify karo (5 se kato).

15/100 = 3/20 (Answer)

Type B: Repeating Decimals (The "x" Method)

Jab number repeat ho raha ho (jaise 0.333...), toh hum algebra use karte hain.

Example 1: Convert 0.333... (0.3̄) to p/q

Step 1: Maan lo x = 0.3333... (Equation 1)
Step 2: Kyunki 1 digit repeat ho raha hai, dono side 10 se multiply karo.
10x = 3.3333... (Equation 2)
Step 3: Equation 2 mein se Equation 1 ko minus karo.
10x - x = 3.3333... - 0.3333...
9x = 3 (Decimal part cancel ho gaya!)
Step 4: x ki value nikalo.
x = 3/9 = 1/3 (Answer)

Example 2: Convert 0.2353535... (0.235̄) to p/q

Note: Yahan '2' repeat nahi ho raha, sirf '35' repeat ho raha hai.

Step 1: x = 0.2353535...
Step 2: Pehle non-repeating part (2) ko point ke left mein lao. 1 digit hai, toh 10 se multiply karo.
10x = 2.353535... (Eq 1)
Step 3: Ab repeating part (35) ko shift karne ke liye, 100 se aur multiply karo (kyunki 2 digits hain).
1000x = 235.353535... (Eq 2)
Step 4: Subtract Eq 1 from Eq 2.
1000x - 10x = 235.35... - 2.35...
990x = 233

Q1. Identify Rational or Irrational:

a) √23
b) √225
c) 0.3796
d) 7.478478...
e) 1.1010010001...

Show Answers

a) Irrational (Not a perfect square)
b) Rational (√225 = 15)
c) Rational (Terminating)
d) Rational (Repeating 478)
e) Irrational (Non-terminating non-repeating)

Q2. Convert to p/q form:

a) 0.6̄
b) 0.47̄ (Only 7 is repeating)

Show Answers

a) 2/3 (x=0.66.., 10x=6.66.., 9x=6, x=6/9)
b) 43/90 (x=0.477.., 10x=4.77.., 100x=47.77.., 90x=43)