Modulus - z - and argument arg(z) — Hinglish notes

3.5.4 · Maths › Complex Numbers

1. Modulus $∣ z ∣$

YEH FORMULA KYU? Point $z = x + i y$ origin se horizontal offset $x$ aur vertical offset $y$ par baitha hai. Pythagoras theorem lagao right triangle par jiske legs $x$ aur $y$ hain, toh hypotenuse (seedhi-line distance) $x^{2} + y^{2}$ hogi. Woh hypotenuse hi arrow ki length hai. Toh $∣ z ∣$ koi nai cheez nahi — yeh bas Pythagoras hai complex wala hat pehne hue.

2. Argument $ar g (z)$

DO equations KYU, sirf $tan$ nahi? Kyunki $tan θ = y / x$ information lose kar deta hai: $θ$ aur $θ + π$ ek hi tangent dete hain (opposite directions mein point karne wale arrows $tan$ ko same lagte hain). $x$ aur $y$ ke signs quadrant batate hain, toh aapko unhe rakhna hi hoga. Isliye blindly $ar g = arctan (y / x)$ likhna dangerous hai.

Figure — Modulus - z - and argument arg(z)

3. Har quadrant mein $Arg$ sahi karna

Maan lo $α = arctan \frac{y}{x} \in [0, \frac{π}{2}]$ reference angle hai. Toh:

Quadrant	sign $(x, y)$	$Arg (z)$
I	$(+, +)$	$α$
II	$(-, +)$	$π - α$
III	$(-, -)$	$- (π - α) = α - π$
IV	$(+, -)$	$- α$

SHIFTS KYU? Reference angle $α$ sirf nearest real axis tak ka acute angle measure karta hai. QII mein arrow left-aur-upar jhukta hai, toh positive real axis se uska asli angle $π - α$ hai, wagera. Har baar arrow draw karo — geometry memorize karne se behtar hai.

4. Worked examples

5. Key properties (har ek derive ki gayi)

6. Common mistakes (steel-manned)

Recall Feynman: 12-saal ke bachche ko samjhao

Socho tum football field ke beech mein khade ho. Ek complex number ek khazane ki jagah hai. Isko describe karne ke liye keh sakte ho "3 kadam east, 4 kadam north" (yahi $x + i y$ hai). Lekin tum yeh bhi keh sakte ho "5 kadam chalo ek direction mein jo 53° baayein ghuma ho." Woh 5 kadam modulus hai (kitni door), aur 53° ka ghoomna argument hai (kis taraf munh kiya). Ek hi khazana, do descriptions. Do complex numbers multiply karo? Bas turning angles add karo aur distances multiply karo — yehi magic trick hai.

7. Active-recall flashcards

#flashcards/maths

z = x + i y

ke liye

∣ z ∣

define karo.

∣ z ∣ = x^{2} + y^{2}

, origin se doori; hamesha

\geq 0

Prove karo

∣ z ∣^{2} = z \overset{z}{ˉ}

(x + i y) (x - i y) = x^{2} - i^{2} y^{2} = x^{2} + y^{2} = ∣ z ∣^{2}

ar g z

ke liye sirf

arctan (y / x)

kyu use nahi kar sakte?

Yeh sirf

(- \frac{π}{2}, \frac{π}{2})

return karta hai; quadrant fix karne ke liye

x, y

ke signs dekhne padte hain (aur

\pm π

adjust karna padta hai).

Principal argument

Arg (z)

ki range?

(- π, π]

Arg (- 1 + i 3) = ?

\frac{2 π}{3}

(QII, reference angle

\frac{π}{3}

z

ki polar form batao.

z = r (cos θ + i sin θ)

jahan

r = ∣ z ∣, θ = ar g z

∣ z_{1} z_{2} ∣ = ?

aur

ar g (z_{1} z_{2}) = ?

∣ z_{1} ∣∣ z_{2} ∣

; aur

ar g z_{1} + ar g z_{2} (mod 2 π)

ar g (0)

undefined kyu hai?

Zero-length arrow ki koi direction nahi hoti.

Arg (- 5) = ?

π

(

- π

nahi; range

(- π, π]

hai).

Modulus

r

, arg

θ

wale

z

se multiply karne ka geometric matlab?

r

se scale karo aur

θ

se rotate karo.

8. Forecast-then-Verify (check karne se pehle karo)

Predict karo, phir $z = - 3 + 4 i$ ke liye compute karo:

$∣ z ∣ = ?$ 2. Quadrant? 3. $Arg (z) = ?$
Answers: $∣ z ∣ = 9 + 16 = 5$ ; QII; $Arg = π - arctan \frac{4}{3} \approx π - 0.927 = 2.214$ rad.

Connections

Complex Numbers - Cartesian form z = x + iy
Polar and Trigonometric form of complex numbers
Euler's formula e^{iθ} = cosθ + i sinθ
De Moivre's Theorem
Conjugate of a complex number
Argand Diagram
Vectors and 2D coordinate geometry (Pythagoras, polar coordinates)