Complex Numbers 🔢

√−1 impossible lagta tha — lekin mathematicians ne isko define karke poori mathematics badal di! Complex numbers = Real + Imaginary.

🤯 Ek Impossible Sawaal

x² + 1 = 0 ko solve karo. Koi bhi real number ka square negative nahi hota — toh koi solution hi nahi? Lekin mathematicians ne socha: "Kya hoga agar hum ek aisa number imagine karein jiska square −1 ho?" Unhone use 'i' (imaginary unit) naam diya. Aur yahi se Complex Numbers ka janam hua! Aaj complex numbers sirf imagination nahi — electricity, quantum physics, signal processing mein use hote hain.

1. Imaginary Unit 'i'

i ko define kiya gaya hai: i = √(−1), matlab i² = −1. Yeh real number nahi hai — yeh ek naya type ka number hai. i ki powers ek pattern follow karti hain:

i¹ = i | i² = −1 | i³ = −i | i⁴ = 1 | i⁵ = i ... (cycle of 4)

Tip: iⁿ nikalna hai → n ko 4 se divide karo, remainder dekhо: rem=1→i, rem=2→−1, rem=3→−i, rem=0→1

✅ i²³ ki value nikalo

23 ÷ 4 = 5 remainder 3
Toh i²³ = i³ = −i

2. Complex Number z = a + ib

Ek complex number z = a + ib hota hai jahan a aur b real numbers hain:

3 + 4i

a=3, b=4
Mixed complex

−2 + 0i

a=−2, b=0
Pure real = −2

0 + 5i

a=0, b=5
Pure imaginary = 5i

√3 − i

a=√3, b=−1
Mixed complex

3. Operations on Complex Numbers

z₁ = a + ib aur z₂ = c + id ke saath:

Addition: (a+ib) + (c+id) = (a+c) + i(b+d)
Subtraction: (a+ib) − (c+id) = (a−c) + i(b−d)
Multiply: (a+ib)(c+id) = (ac−bd) + i(ad+bc)

Multiplication mein remember karo: i·i = i² = −1 → real part mein jaata hai!

✅ z₁ = 3+2i, z₂ = 1−4i. z₁·z₂ nikalo.

(3+2i)(1−4i) = 3(1) + 3(−4i) + 2i(1) + 2i(−4i)
= 3 − 12i + 2i − 8i²
= 3 − 10i − 8(−1)    [kyunki i² = −1]
= 3 − 10i + 8 = 11 − 10i

4. Conjugate aur Modulus

Conjugate z̄: Imaginary part ka sign palat do. z = a+ib ka conjugate z̄ = a−ib. Trick: complex number mein bas i ki jagah −i rakh do.

Modulus |z|: z ka magnitude — Argand plane par origin se distance. Pythagoras theorem use hota hai:

z̄ = a − ib
|z| = √(a² + b²)

Important: z · z̄ = a² + b² = |z|² (always real aur positive!)

✅ z = 5 − 12i ke liye z̄ aur |z| nikalo

Conjugate z̄ = 5 + 12i
|z| = √(5² + (−12)²) = √(25+144) = √169 = 13

5, 12, 13 ek Pythagorean triplet hai — isliye |z| neat nikla!

5. Division of Complex Numbers

Complex numbers divide karne ke liye: denominator ka conjugate se multiply karo (rationalize):

z₁/z₂ = (z₁ · z̄₂) / (z₂ · z̄₂) = (z₁ · z̄₂) / |z₂|²

Denominator real ho jaata hai — phir aasani se simplify kar sakte hain.

✅ (3+2i)/(1+i) simplify karo

Conjugate of denominator: (1+i) ka conjugate = (1−i)
= (3+2i)(1−i) / (1+i)(1−i)
Numerator: 3−3i+2i−2i² = 3−i+2 = 5−i
Denominator: 1−i² = 1+1 = 2
= (5−i)/2 = 5/2 − i/2

6. Argand Plane (Complex Plane)

Complex numbers ko 2D plane par represent karte hain jahan x-axis = Real part aur y-axis = Imaginary part. Ise Argand Plane kehte hain.

Re Im z=3+4i b=4 (Im) a=3 (Re) |z|=5 z̄=3−4i 3 4 −4

Argand Plane: z=3+4i aur z̄=3−4i real axis ke reflection hain!

7. Practice Questions

Q1. i⁵⁵ ki value nikalo.

Solution

55 ÷ 4 = 13 remainder 3 → i⁵⁵ = i³ = −i

Q2. z₁ = 4+3i, z₂ = 2−i. z₁−z₂ aur z₁·z₂ nikalo.

Solution

z₁−z₂ = (4−2)+(3+1)i = 2+4i | z₁·z₂ = (8+3)+(4·(−1)·(−1)+6)i... = (8+3)+(-4+6)i = 11+2i

Q3. (2+3i)/(3−2i) simplify karo.

Solution

×(3+2i)/(3+2i): Num = 6+4i+9i+6i² = 6+13i−6 = 13i. Den = 9+4 = 13. Answer = i

Q4. |3−4i| aur conjugate nikalo.

Solution

|z| = √(9+16) = √25 = 5. Conjugate = 3+4i

Q5. x²+4 = 0 ke roots nikalo complex numbers mein.

Solution

x² = −4 → x = ±√(−4) = ±2i

← Trig Functions Next: Linear Inequalities →