3.5.3
Ek complex number z = x + i y do independent pieces of information hote hain: real part x y plane pe point ki tarah draw kiya ja sakta hai. Toh hum x y Argand plane complex numbers ki algebra ko points aur arrows ki geometry mein badal deti hai. Yehi poora trick hai: addition ban jaata hai vector addition, modulus ban jaata hai distance, aur argument ban jaata hai angle.
Definition Argand plane (complex plane)Woh plane jisme complex number z = x + i y point P ( x , y ) position vector O P
Horizontal axis ko real axis kehte hain (points x + 0 i
Vertical axis ko imaginary axis kehte hain (points 0 + i y
Naam rakha gaya Jean-Robert Argand ke naam pe (aur bhi Wessel, Gauss ne iska kaam kiya).
WHY ek plane, line nahi? Real numbers ek line pe fit ho jaate hain kyunki ek number = ek coordinate. Ek complex number do real numbers carry karta hai, isliye use do axes chahiye → ek plane.
Intuition Modulus geometrically kiya hai?Point P ( x , y ) O ∣ z ∣
z argument arg z θ O P positive real axis ke saath anticlockwise direction mein banata hai.
tan θ = x y Principal argument arg z ( − π , π ]
Common mistake Steel-man: "θ = tan − 1 ( y / x ) Why ye sahi lagta hai: triangle mein actually tan θ = y / x tan − 1 Why ye galat hai: tan − 1 ( − π /2 , π /2 ) z tan θ = y / x Fix: pehle reference angle α = tan − 1 x y
Quadrant
( x , y ) arg z
I
+ , + α
II
− , + π − α
III
− , − − ( π − α )
IV
+ , − − α
Intuition Addition = tip-to-tail vectorsz 1 + z 2 x 1 + x 2 y 1 + y 2 z 1 + z 2 O P 1 O P 2 parallelogram ka diagonal hota hai.
Intuition Do numbers ke beech distance∣ z 1 − z 2 ∣ P 1 P 2 straight-line distance hai. Isliye ∣ z − z 0 ∣ = r circle represent karta hai jiska radius r z 0
Intuition Conjugate = reflectionz ˉ = x − i y z − z 18 0 ∘
Worked example Example 1 —z = 3 + 4 i Step 1: Point hai ( 3 , 4 ) Ye step kyun? Dono parts positive hain, toh point origin ke upar aur daayein hai.
Step 2: ∣ z ∣ = 3 2 + 4 2 = 25 = 5 Kyun? Legs 3 , 4 Step 3: α = tan − 1 ( 4/3 ) ≈ 53.1 3 ∘ arg z = 53.1 3 ∘ Step 4: Polar form: z = 5 ( cos 53.1 3 ∘ + i sin 53.1 3 ∘ )
Worked example Example 2 — quadrant trap:z = − 1 − i Step 1: Point ( − 1 , − 1 ) Kyun? Dono parts negative hain.
Step 2: ∣ z ∣ = ( − 1 ) 2 + ( − 1 ) 2 = 2 Step 3: α = tan − 1 ( 1/1 ) = π /4 tan − 1 ( 1 ) = π /4 galat sign! ).
Kyun correction zaroori hai: point Q III mein hai, toh arg z = − ( π − π /4 ) = − 4 3 π Step 4: z = 2 ( cos ( − 4 3 π ) + i sin ( − 4 3 π ) )
Worked example Example 3 — ek locus (algebra se geometry)∣ z − ( 2 + i ) ∣ = 3 Step 1: ∣ z − z 0 ∣ = r z 0 = 2 + i r = 3 Kyun: ∣ z − z 0 ∣ z z 0 Step 2: "( 2 , 1 ) 3 circle , centre ( 2 , 1 ) 3
Recall Feynman: ek 12-saal ke bacche ko explain karoSocho ek treasure map hai. Ek normal number line sirf left ya right chalne deti hai. Lekin ek complex number tumhe DO cheezein batata hai: right/left chalo AUR upar/neeche chalo. Toh ye ek flat map pe ek jagah point karta hai, sirf ek line pe nahi. "Modulus" hai kitni door woh jagah starting corner se hai (seedha ruler se maapa hua), aur "argument" hai kaunsi direction mein tumhe haath uthaana hoga us jagah ki taraf. Do complex numbers add karna waisa hai jaise do walks ek ke baad ek lena — tum wahan pahuncho jahan arrows tip-to-tail stack hote hain.
"Real Runs, Imaginary Ists Up" — Real axis horizontal run karta hai, Imaginary axis upar uthta hai. Aur MAD : M odulus = A D istance (origin se).
Recall Quick self-test (answers chhupa lo!)
Argand plane ke do axes kaunse hain?
Pythagoras se ∣ z ∣
tan − 1 ( y / x ) arg z ∣ z − z 0 ∣ = r
Argand plane ka horizontal axis kiya represent karta hai? z = x + i y x
Vertical axis kiya represent karta hai? Imaginary part y
z = x + i y Point
P ( x , y ) / position vector
O P origin se.
z Right triangle with legs
x , y ; hypotenuse
= x 2 + y 2 = ∣ z ∣ by Pythagoras.
z Anticlockwise angle
θ jo
O P positive real axis ke saath banata hai;
tan θ = y / x .
Principal argument ki range? ( − π , π ]
θ = tan − 1 ( y / x ) tan − 1 ( − π /2 , π /2 )
Complex number ki polar form? z = r ( cos θ + i sin θ ) r = ∣ z ∣ θ = arg z
∣ z 1 − z 2 ∣ Points P 1 P 2
∣ z − z 0 ∣ = r z 0 r
Conjugation z ˉ Point ka real axis ke paas reflection.
z = − 1 − i arg z − 3 π /4
angle from positive real axis
tan theta equals y over x
z equals r cos theta plus i sin theta
