Operations on Real Numbers

👋 Hello Students!

Ab tak humne numbers ko pehchanna seekha. Ab hum unke saath khelna seekhenge! 🎮

Is chapter mein hum Real Numbers par Addition (+), Subtraction (-), Multiplication (×), aur Division (÷) karna seekhenge. Ye bilkul algebra jaisa hai, bas yahan x aur y ki jagah √2 aur √3 hote hain.

🎥 Video Explanation

English Explanation

Hindi Explanation

📊 Infographic Summary

Infographic Summary

⚖️ The Golden Rules (Rational vs Irrational)

📖 Story Time: Oil and Water

Rational numbers (paani) aur Irrational numbers (tel) kabhi poori tarah mix nahi hote. Agar tum unhe add ya subtract karoge, toh result hamesha 'oily' (Irrational) hi rahega. Lekin agar do Irrational numbers (tel aur tel) milte hain, toh kabhi-kabhi magic hota hai aur woh Rational ban jaate hain!

Jab ek Rational number ek Irrational number se milta hai, toh kya hota hai? Yaad rakhein:

  • Rational + Irrational = Irrational (e.g., 2 + √3 is Irrational)
  • Rational - Irrational = Irrational (e.g., 2 - √3 is Irrational)
  • Rational × Irrational = Irrational (e.g., 2√3 is Irrational, agar rational number 0 nahi hai)
  • Rational ÷ Irrational = Irrational (e.g., 2/√3 is Irrational)

Exception: Agar hum do Irrational numbers ko add, subtract, multiply ya divide karein, toh result Rational bhi ho sakta hai aur Irrational bhi.

Examples:
  • √2 + √2 = 2√2 (Irrational)
  • √2 - √2 = 0 (Rational!) 😲
  • √3 × √3 = √9 = 3 (Rational!) 😲
  • √2 × √3 = √6 (Irrational)

➕➖ Addition & Subtraction of Square Roots

📖 Story Time: Apples and Oranges

Square roots ko fruits ki tarah samjho. √2 ek Apple hai aur √3 ek Orange hai. Tum 2 Apples aur 3 Apples ko add kar sakte ho (5 Apples), lekin 2 Apples aur 3 Oranges ko mix nahi kar sakte. Isliye, sirf same root wale numbers hi add ya subtract hote hain.

Square roots ko add ya subtract karte waqt, unhe "Variables" (jaise x, y) ki tarah treat karein.

Rule: Sirf Like Terms (same root wale numbers) hi add ya subtract ho sakte hain.

Example 1: Add 2√2 + 5√3 and √2 - 3√3

Solution:

Step 1: Saath mein likhein: (2√2 + 5√3) + (√2 - 3√3)

Step 2: Like terms ko group karein (√2 wale ek saath, √3 wale ek saath):

= (2√2 + √2) + (5√3 - 3√3)

Step 3: Solve karein (Imagine √2 is x, √3 is y):

= (2+1)√2 + (5-3)√3

= 3√2 + 2√3 (Answer)

Note: Hum 3√2 aur 2√3 ko aage add nahi kar sakte kyunki wo alag-alag hain.

✖️➗ Multiplication & Division

Yahan rule simple hai: "Bahar wala bahar se, andar wala andar se".

Identities:

  • √a × √b = √(a × b)
  • √a / √b = √(a / b)
Example 2: Multiply 6√5 by 2√5

= 6 × 2 × √5 × √5

= 12 × 5 (Kyunki √5 × √5 = 5)

= 60

Example 3: Divide 8√15 by 2√3

= (8/2) × (√15 / √3)

= 4 × √(15/3)

= 4√5

🧩 Important Identities (Formulas)

📖 Story Time: The Formula Friends

Ye formulas tumhare purane dost hain (Algebraic Identities), bas naye kapde pehen kar aaye hain. (a+b)² wahi hai, bas 'a' ki jagah √a aa gaya hai. Daro mat, bas value put karo!

Ye formulas calculations ko fast karne ke liye hain. Ye bilkul algebraic identities (a+b)² jaise hain.

Identity Example
(√a + √b)(√a - √b) = a - b (√5 + √2)(√5 - √2) = 5 - 2 = 3
(a + √b)(a - √b) = a² - b (3 + √7)(3 - √7) = 3² - 7 = 9 - 7 = 2
(√a + √b)² = a + 2√ab + b (√3 + √2)² = 3 + 2√6 + 2 = 5 + 2√6

🧹 Rationalizing the Denominator

📖 Story Time: Cleaning the Floor

Denominator ghar ka farsh (floor) hai. Root ek pathar (stone) jaisa hai. Hum farsh par pathar nahi chahte. Rationalization ka matlab hai us pathar ko utha kar chhat (numerator) par rakh dena, taaki farsh saaf aur smooth ho jaye.

Maths mein hume denominator (neeche wale number) mein root pasand nahi hai. Usse hatane ke process ko Rationalization kehte hain.

Type 1: Single Term (Easy)

Q: Rationalize 1/√2

Trick: Jo root neeche hai, usse upar aur neeche multiply kar do.

1/√2 × (√2/√2)

= √2 / (√2 × √2)

= √2 / 2 (Answer)

Type 2: Binomial Term (Conjugate Method)

Agar denominator mein a + √b hai, toh hum uske Conjugate a - √b se multiply karte hain (Sign change kar do).

Q: Rationalize 1 / (2 + √3)

Step 1: Conjugate dhundo. 2 + √3 ka conjugate hai 2 - √3.

Step 2: Multiply numerator and denominator by conjugate.

= [1 / (2 + √3)] × [(2 - √3) / (2 - √3)]

Step 3: Solve.

Top: 1 × (2 - √3) = 2 - √3

Bottom: (2 + √3)(2 - √3) (Identity: a² - b²)

= 2² - (√3)² = 4 - 3 = 1

Result: (2 - √3) / 1 = 2 - √3 (Answer)

✏️ Practice Questions

Solve these:

  1. Add: (3√2 + 7√3) and (√2 - 5√3)
  2. Multiply: 5√11 by 3√11
  3. Simplify: (5 + √5)(5 - √5)
  4. Rationalize: 1 / (√7 - √6)
Show Answers

1. 4√2 + 2√3 (3√2+√2 = 4√2, 7√3-5√3 = 2√3)

2. 165 (5×3=15, √11×√11=11, 15×11=165)

3. 20 (5² - (√5)² = 25 - 5 = 20)

4. √7 + √6 (Multiply by √7+√6. Denominator becomes 7-6=1)