3.5.8
Ek complex number bas ek 2D arrow hai plane pe. Rectangular form z = a + bi do axes ke shadows se name karta hai (horizontal a b z = r ( cos θ + i sin θ ) arrow ko uski length r θ se name karta hai.
Add/subtract = arrows ko tip-to-tail line up karo → rectangular mein sabse aasaan.Multiply/divide = lengths scale karo aur angles rotate karo → polar mein sabse aasaan.
Poora subtopic yeh hai: woh form chuno jo operation se match kare.
Definition Rectangular aur polar form
Rectangular: z = a + bi a = Re ( z ) b = Im ( z ) i 2 = − 1 Modulus: r = ∣ z ∣ = a 2 + b 2 Argument: θ = arg ( z ) cos θ = r a , sin θ = r b Polar: z = r ( cos θ + i sin θ ) = r cis θ = r e i θ
KAISE convert karte hain dono taraf:
a = r cos θ , b = r sin θ , r = a 2 + b 2 , θ = atan2 ( b , a ) .
θ = arctan ( b / a ) Kyun sahi lagta hai: tan θ = b / a Trap yeh hai: plain arctan ( − 9 0 ∘ , 9 0 ∘ ) z = − 1 − i a , b < 0 arctan ( 1 ) = 4 5 ∘ 22 5 ∘ − 13 5 ∘ Fix: a b a < 0 ± 18 0 ∘ atan2 tumhare liye karta hai.
Intuition Arrows add karna = components add karnaDo arrows tip-to-tail add kiye: total horizontal reach horizontals ka sum hai, total vertical reach verticals ka sum hai. Toh tum parts add karte ho , bilkul vectors ki tarah.
( 3 + 2 i ) + ( 1 − 5 i ) = ( 3 + 1 ) + ( 2 − 5 ) i = 4 − 3 i Yeh step kyun? Real parts 3 , 1 2 , − 5
Common mistake Polar form mein add karnaKyun sahi lagta hai: polar "acchi form" hai baaki sab ke liye. Trap: tum moduli ya angles add nahi kar sakte: r 1 + r 2 θ 1 + θ 2 Fix: rectangular mein convert karo, add karo, zaroorat ho toh wapas convert karo.
( 1 + i ) ( 1 + i ) Rectangular: 1 + i + i + i 2 = 1 + 2 i − 1 = 2 i Polar: 1 + i r = 2 θ = 4 5 ∘ r = ( 2 ) 2 = 2 θ = 9 0 ∘ 2 ( cos 9 0 ∘ + i sin 9 0 ∘ ) = 2 i Yeh step kyun? Dono raaste agree karne chahiye — polar geometry dikhata hai (angle double karo, length square karo).
Intuition Denominator rationalise karnaHum "ek arrow se divide" nahi kar sakte, lekin upar aur neeche conjugate z ˉ = c − d i real number ban jaata hai, kyunki ( c + d i ) ( c − d i ) = c 2 + d 2 = ∣ z 2 ∣ 2
1 − i 3 + 2 i 1 + i ( 1 − i ) ( 1 + i ) ( 3 + 2 i ) ( 1 + i ) = 1 − i 2 3 + 3 i + 2 i + 2 i 2 = 2 ( 3 − 2 ) + 5 i = 2 1 + 5 i = 0.5 + 2.5 i . Yeh step kyun? Denominator 1 − i 2 = 2
Poore denominator ka conjugate lena bhool jaana
Kyun sahi lagta hai: "i Trap: students likhte hain c + d i a + bi ⋅ c − d i 1 numerator ko bhi multiply karna bhool jaate hain — isse value badal jaati hai. Fix: upar aur neeche dono ko c − d i 1
Recall Feynman: 12-saal ke bacche ko explain karoHar complex number ko graph paper pe ek arrow samjho. Do arrows add karne ke liye tum bas unhe tip-to-tail stack kar do — kitna right aur kitna upar sirf pile up hota hai, toh sideways numbers add karo aur upar wale numbers add karo. Multiply karne ke liye yeh alag game hai: arrow ko stretch karo aur spin karo. Answer ki length do lengths ka product hai, aur direction do angles ka sum hai. Divide karne se yeh undo hota hai: lengths divide karo aur angles subtract karo. Yahi poori kahani hai — add karna hai "line them up," multiply karna hai "stretch and spin."
Mnemonic Kaun si form kaun se kaam ke liye?"Add in the box, multiply on the clock."
Box = rectangular (add/subtract components). Clock = polar (multiply/divide → hands ghoomao , angles add/subtract hote hain).
Rectangular form of a complex number z = a + bi a = Re b = Im i 2 = − 1
Polar form of a complex number z = r ( cos θ + i sin θ ) = r e i θ r = ∣ z ∣ θ = arg z
Formula for modulus r a , b Why can't you use plain arctan ( b / a ) Yeh quadrant ignore karta hai; a , b ± 18 0 ∘ a < 0
Which form is best for addition and why? Rectangular — tum bas real aur imaginary parts independently add karte ho (vectors ki tarah).
Rectangular multiplication result of ( a + bi ) ( c + d i ) ( a c − b d ) + ( a d + b c ) i
Why does a c − b d Kyunki b d i 2 = − b d
Polar rule for multiplication Moduli multiply karo, arguments add karo: r 1 r 2 cis ( α + β )
Which trig identities make polar multiplication work? cos ( α + β ) sin ( α + β )
How do you divide two complex numbers in rectangular form? Numerator aur denominator dono ko denominator ke conjugate se multiply karo.
Result of ( c + d i ) ( c − d i ) c 2 + d 2 = ∣ z ∣ 2
Polar rule for division Moduli divide karo, arguments subtract karo: r 2 r 1 cis ( α − β )
Geometric meaning of multiplying two complex numbers Length scale (stretch) hoti hai aur angle rotate (spin) hota hai.
named by length and angle
multiply moduli add angles
Complex number as 2D arrow