3.5.1
Intuition Ek hi breath mein core ideaHumne ek naya number i invent kiya jiska poora kaam hai i 2 = − 1 ≥ 0 i i
Intuition Hume naye number ki zaroorat kyun haiEquation x 2 = − 1 koi real solution nahi hai , kyunki kisi bhi real number ko square karne se kuch ≥ 0 invent karo jo ise solve kare. Nayi unsolvable equations solve karne ke liye naye numbers invent karna — bilkul isi tarah humne negatives liye (to solve x + 1 = 0 2 x = 1
Definition The imaginary unitImaginary unit i i = − 1 , i 2 = − 1. complex number woh hota hai jo a + bi a , b a real part kehte hain, b imaginary part kehte hain.
Common mistake Steel-man: "− 1 Yeh sahi kyun lagta hai: school mein x x ≥ 0 Fix: hum meaning ko extend karte hain. i purely i 2 = − 1 i i 2 − 1
Hum sirf i 2 = − 1
i 0 = 1 (any nonzero base to the 0) i 1 = i i 2 = − 1 (the definition)
Yeh step kyun? Yahan se hum bas i i 2 − 1
i 3 = i 2 ⋅ i = ( − 1 ) ⋅ i = − i
Yeh step kyun? i 2 − 1
i 4 = i 2 ⋅ i 2 = ( − 1 ) ( − 1 ) = 1
Yeh step kyun? Hum wapas 1 i
Kyunki i 4 = 1 i 5 = i 4 ⋅ i = 1 ⋅ i = i , i 6 = i 4 ⋅ i 2 = − 1 , …
Intuition Dual coding — ise rotation ki tarah dekhoi 9 0 ∘ 1 9 0 ∘ i − 1 − i 1 9 0 ∘ = 36 0 ∘ Isliye period 4 hai.
Worked example Example 1 —i 27 Step 1: Exponent ko 4 se divide karo: 27 = 4 × 6 + 3 r = 3 Kyun? Sirf remainder answer decide karta hai (i n = i n mod 4 Step 2: i 27 = i 3 = − i . Answer: i 27 = − i
Worked example Example 2 —i 100 Step 1: 100 = 4 × 25 0 Kyun? 100 Step 2: i 100 = i 0 = 1. Answer: 1
Worked example Example 3 —i 15 + i 16 + i 17 + i 18 Step 1: Har ek reduce karo: 15 → r = 3 ⇒ − i 16 → r = 0 ⇒ 1 17 → r = 1 ⇒ i 18 → r = 2 ⇒ − 1 Kyun? Remainder rule ko term by term apply karo.
Step 2: Add karo: ( − i ) + 1 + i + ( − 1 ) = 0. Answer: 0 Insight: i chaar consecutive powers ka sum 0 , kyunki woh { 1 , i , − 1 , − i } 1 + i − 1 − i = 0
Worked example Example 4 —− a − b − 4 ⋅ − 9 Galat raasta: ( − 4 ) ( − 9 ) = 36 = 6 Sahi raasta: − 4 = 2 i , − 9 = 3 i = 6 i 2 = − 6 Farq kyun hai? Neeche mistake box dekho.
Common mistake Steel-man: "a b = ab Yeh sahi kyun lagta hai: yeh ek sach rule hai — non-negative reals ke liye.
Fix: identity a b = ab tab fail hoti hai jab dono a , b < 0 . Hamesha pehle i − x = i x x > 0 pehle . Tab − 4 − 9 = ( 2 i ) ( 3 i ) = − 6
Common mistake Steel-man: "i 3 = i ⋅ i ⋅ i = i Yeh sahi kyun lagta hai: log i ⋅ i ⋅ i i ⋅ i = − 1 Fix: i ⋅ i = i 2 = − 1 i 3 = ( − 1 ) ⋅ i = − i nahi i
Mnemonic Cycle yaad rakho"1, i, minus-1, minus-i" — jaise 9 0 ∘ E → N → W → S (East=1 i − 1 − i "I N eed W hole S leep" order 1 , i , − 1 , − i
Recall Feynman: ek 12-saal ke bache ko explain karoImagine karo ek clock jisme sirf 4 numbers hain: 1 i − 1 − i i agali jagah kudta hai, ghoomte hua. 4 jumps ke baad tum wapas shuru wali jagah ho — bilkul jaise 12 ghante baad clock same time dikhata hai. To i baach jaate hain 4 ke poore loops hatane ke baad. Woh leftover (0, 1, 2, ya 3) exactly batata hai ki tum chaar mein se kis jagah land karte ho. Aur i 2 = − 1 decide kiya tha ek aisa number banao jo yeh kare — poora point tha "kya apne aap se multiply ho ke negative de sakta hai?" ka jawab dena.
Recall Active recall — answers cover karo
i i 2 = − 1 i = − 1 Powers ka period 4 kyun hai? ::: Kyunki i 4 = 1 i i n = i n mod 4
i 3 i − i i 2 = − 1 i 3 = ( − 1 ) i = − i
i i Har real number square karo to ≥ 0 i 2 = − 1 < 0
i 4 k , i 4 k + 1 , i 4 k + 2 , i 4 k + 3 1 , i , − 1 , − i
i n r = n mod 4 { 1 , i , − 1 , − i } i r
i 27 27 mod 4 = 3 ⇒ − i
i 100 100 mod 4 = 0 ⇒ 1
1/ i − i i i / i 2 = i / ( − 1 )
i 0 { 1 , i , − 1 , − i } 0
− 4 − 9 = − 6 6 Rule
a b = ab do negatives ke liye fail hoti hai; pehle convert karo:
( 2 i ) ( 3 i ) = 6 i 2 = − 6 .
i Complex plane mein 9 0 ∘
x squared = -1 has no real solution
i = sqrt(-1), i squared = -1
Multiply by i rotates 90 degrees
