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

1. The modulus $|z|$

WHY this formula? The point $z=x+iy$ sits at horizontal offset $x$ and vertical offset $y$ from the origin. By the Pythagoras theorem on the right triangle with legs $x$ and $y$ , the hypotenuse (the straight-line distance) is $\sqrt{x^2+y^2}$ . That hypotenuse is the length of the arrow. So $|z|$ is nothing new — it's just Pythagoras wearing a complex hat.

2. The argument $\arg(z)$

WHY two equations, not just $\tan$ ? Because $\tan\theta = y/x$ loses information: $\theta$ and $\theta+\pi$ give the same tangent (points in opposite directions look identical to $\tan$ ). The signs of $x$ and $y$ tell you the quadrant, so you must keep them. That is exactly why blindly writing $\arg = \arctan(y/x)$ is dangerous.

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

3. Getting $\operatorname{Arg}$ right in each quadrant

Let $\alpha = \arctan\!\big|\tfrac{y}{x}\big| \in [0,\tfrac\pi2]$ be the reference angle. Then:

Quadrant	sign $(x,y)$	$\operatorname{Arg}(z)$
I	$(+,+)$	$\alpha$
II	$(-,+)$	$\pi - \alpha$
III	$(-,-)$	$-(\pi - \alpha) = \alpha-\pi$
IV	$(+,-)$	$-\alpha$

WHY the shifts? The reference angle $\alpha$ only measures the acute angle to the nearest part of the real axis. In QII the arrow leans left-and-up, so its true angle from the positive real axis is $\pi-\alpha$ , etc. Draw the arrow every time — geometry beats memorising.

4. Worked examples

5. Key properties (each derived)

Formula Additive argument

Claim: $\arg(z_1z_2) = \arg z_1 + \arg z_2 \pmod{2\pi}$ . Derive: write $z_k = r_k(\cos\theta_k + i\sin\theta_k)$ . Multiply and use the angle-addition identities: $z_1z_2 = r_1r_2\big[(\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2) + i(\sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2)\big]$ $= r_1r_2\big[\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)\big].$ So moduli multiply, arguments add. This is the geometric heart of complex multiplication: scale and rotate.

6. Common mistakes (steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine standing at the middle of a football field. A complex number is a treasure spot. To describe it you can say "3 steps east, 4 steps north" (that's $x+iy$ ). But you could also say "walk 5 steps in a direction turned 53° to the left." The 5 steps is the modulus (how far), and the 53° turn is the argument (which way you face). Same treasure, two descriptions. Multiplying two complex numbers? Just add the turning angles and multiply the distances — that's the magic trick.

7. Active-recall flashcards

#flashcards/maths

Define

|z|

for

z=x+iy

|z|=\sqrt{x^2+y^2}

, the distance from origin; always

\ge 0

Prove

|z|^2 = z\bar z

(x+iy)(x-iy)=x^2-i^2y^2=x^2+y^2=|z|^2

Why can't you use

\arctan(y/x)

alone for

\arg z

It only returns

(-\tfrac\pi2,\tfrac\pi2)

; you must use the signs of

x,y

to fix the quadrant (add/subtract

\pi

Range of the principal argument

\operatorname{Arg}(z)

(-\pi,\ \pi]

\operatorname{Arg}(-1+i\sqrt3)=?

\tfrac{2\pi}{3}

(QII, ref angle

\tfrac\pi3

State the polar form of

z

z=r(\cos\theta+i\sin\theta)

with

r=|z|,\ \theta=\arg z

|z_1z_2|=?

and

\arg(z_1z_2)=?

|z_1||z_2|

; and

\arg z_1+\arg z_2 \ (\bmod 2\pi)

Why is

\arg(0)

undefined?

A zero-length arrow has no direction.

\operatorname{Arg}(-5)=?

\pi

(not

-\pi

; range is

(-\pi,\pi]

Geometric meaning of multiplying by

z

of modulus

r

, arg

\theta

Scale by

r

and rotate by

\theta

8. Forecast-then-Verify (do before checking)

Predict, then compute for $z=-3+4i$ :

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

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek complex number $z = x + iy$ ko simple samjho ek teer (arrow) ki tarah jo origin se kisi point tak jaata hai Argand plane par. Is teer ke bare mein do cheezein important hain: kitna lamba hai, aur kis direction mein point kar raha hai. Lambai ko bolte hain modulus $|z| = \sqrt{x^2+y^2}$ — ye bas Pythagoras theorem hai, hypotenuse nikaalna. Aur direction ko bolte hain argument $\arg(z)$ — ye woh angle hai jo teer positive real axis se banata hai, anticlockwise measure karke.

Sabse bada trap ye hai ki students seedha $\arctan(y/x)$ maar dete hain calculator par. Galat! Calculator ka $\arctan$ sirf QI aur QIV ke answers deta hai. Isliye pehle quadrant dekho — $x$ aur $y$ ke signs se. Agar QII mein ho to $\pi - \alpha$ , QIII mein $\alpha - \pi$ , QIV mein $-\alpha$ , jahan $\alpha$ reference (acute) angle hai. Hamesha ek chhota sa arrow bana lo, geometry se kabhi galti nahi hogi.

Ye topic itna zaroori kyun hai? Kyunki jab do complex numbers ko multiply karte ho, to moduli multiply hoti hain aur arguments add hote hain — matlab multiplication ka matlab hai "scale karo aur ghumao (rotate karo)". Yehi cheez aage De Moivre theorem, roots, aur Euler formula $e^{i\theta}$ ka base hai. Ek baar $|z|$ aur $\arg(z)$ ka intuition pakka ho gaya, to poora Complex Numbers chapter aasaan lagega. Yaad rakho: Modulus = length (kabhi negative nahi), Argument = turn (negative ho sakta hai), range $(-\pi,\pi]$ .

Modulus - z - and argument arg(z)

1. The modulus $|z|$

2. The argument $\arg(z)$

3. Getting $\operatorname{Arg}$ right in each quadrant

4. Worked examples

5. Key properties (each derived)

6. Common mistakes (steel-manned)

7. Active-recall flashcards

8. Forecast-then-Verify (do before checking)

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Complex Numbers

Connections