Roots of unity — cube roots, nth roots, geometric interpretation

3.5.12 · Maths › Complex Numbers

1. Setting up from first principles

KYU exactly $n$ hote hain? Kyunki $z^{n} - 1 = 0$ ek degree- $n$ polynomial hai, aur ek degree- $n$ polynomial ke exactly $n$ roots hote hain $C$ mein (Fundamental Theorem of Algebra), aur yahan sab distinct hain.

KAISE dhundhte hain inhe? $z$ ko polar form mein likho. Yahi key move hai: polar form mein powers easy hain.

$z = r (cos θ + i sin θ) = r e^{i θ}$

De Moivre se, $z^{n} = r^{n} e^{in θ}$ . Hume chahiye ki yeh $1 = 1 \cdot e^{i \cdot 0}$ ke barabar ho.

Modulus aur argument alag-alag match karo:

Modulus: $r^{n} = 1 \Rightarrow r = 1$ (kyunki $r \geq 0$ real hai). Yeh step kyu? Saare roots unit circle pe hote hain.
Argument: $n θ = 0 + 2 π k$ , kisi bhi integer $k$ ke liye (kyunki angles $2 π$ ke baad repeat hote hain). Yeh step kyu? $e^{in θ} = 1$ ka matlab exactly $n θ = 0$ nahi hai — koi bhi full-turn multiple kaam karta hai.

$θ = \frac{2 π k}{n}, k = 0, 1, 2, \dots, n - 1$

Sirf $k = 0 \dots n - 1$ kyu? Uske baad angles repeat ho jaate hain ( $2 π$ add hota hai), wahi points milte hain.

2. Primitive root $ω$ aur clean formula

Maano $ω = e^{i 2 π / n}$ (circle ke around "pehla kadam"). Tab

$z_{k} = ω^{k}, k = 0, 1, \dots, n - 1.$

YEH powerful kyu hai? Har root ek ek root ki power hai. $n$ roots hain ${1, ω, ω^{2}, \dots, ω^{n - 1}}$ — baar baar $2 π / n$ rotate karke generate hote hain. Root $ω^{k}$ ko primitive kaha jaata hai jab uski powers saare $n$ roots generate karen (hota hai jab $g cd (k, n) = 1$ ).

Figure — Roots of unity — cube roots, nth roots, geometric interpretation

3. Cube roots of unity (famous case $n = 3$ )

$n = 3$ lo. Angles hain $θ = 0, \frac{2 π}{3}, \frac{4 π}{3}$ .

$z_{0} = 1, ω = e^{i 2 π /3} = - \frac{1}{2} + \frac{3}{2} i, ω^{2} = e^{i 4 π /3} = - \frac{1}{2} - \frac{3}{2} i$

$1 + ω + ω^{2} = 0$ kyu? Do tarike: (a) $ω$ solve karta hai $z^{2} + z + 1 = 0$ , toh seedha $ω^{2} + ω + 1 = 0$ ; ya (b) neeche diya geometric-series argument use karo. (c) Geometrically: teen unit vectors jo $12 0^{\circ}$ par hain cancel karte hain — peace sign ki tarah balance hote hain, sum hota hai center $0$ .

4. Saare $n$ th roots ka sum = 0 (general)

Derivation (geometric series): Maano $S = 1 + ω + ω^{2} + \dots + ω^{n - 1}$ . Yeh ek geometric series hai ratio $ω \neq = 1$ ke saath:

$S = \frac{ω ^{n} - 1}{ω - 1} .$

Yeh step kyu? Standard geometric-sum formula. Lekin $ω^{n} = 1$ , toh numerator $0$ hai, jabki denominator $ω - 1 \neq = 0$ . Isliye $S = 0$ . $■$

Intuition: $n$ position vectors circle ke around symmetric hain, toh cancel ho jaate hain — center of mass origin hai.

Product: $\prod_{k = 0}^{n - 1} ω^{k} = (- 1)^{n + 1}$ ( $z^{n} - 1$ ke roots ka product, Vieta se = $- (- 1)^{n} = (- 1)^{n + 1}$ ).

5. Geometric interpretation (Dual Coding)

$n$	Shape	Roots
2	line segment	$1, - 1$
3	equilateral triangle	$1, ω, ω^{2}$
4	square	$1, i, - 1, - i$
6	hexagon	$\pm 1, \pm e^{\pm iπ /3}$

6. Worked examples

7. Common mistakes (Steel-man + fix)

8. Active recall

Recall Quick self-test (answers chhupao)

Roots unit circle pe kyu hote hain? → modulus condition $r^{n} = 1 \Rightarrow r = 1$ .
$\sum ω^{k}$ kya hai? → $0$ .
Cube roots kaun sa polygon banate hain? → equilateral triangle.
$ω$ se multiply karna geometrically kya karta hai? → $2 π / n$ rotate karna.

Recall Feynman: 12-saal ke bachche ko samjhao

Ek ghadi socho jisme $n$ equally placed dots hain, ek "3 baje" pe. Ghadi ghoomao toh har dot agle dot ki jagah jump karta hai. Agar kaafi baar ghoomaao, toh har dot wapas ghar aa jaata hai. Woh dots "roots of unity" hain. Har ek, agar tum usse "khud se multiply karo" $n$ baar (ghoomaate raho), exactly wapas "1" pe land karta hai (3-baje waala dot). Aur agar tum har dot pe thoda weight rakh do, ghadi center pe perfectly balance karta hai — isliye sab add hokar zero dete hain!

Flashcards

n

th root of unity ko kaun si equation define karti hai?

z^{n} = 1

n

th roots of unity ka general formula?

z_{k} = e^{i 2 π k / n}

k = 0, 1, \dots, n - 1

Saare roots of unity ka modulus 1 kyu hota hai?

z^{n} = 1 \Rightarrow r^{n} = 1 \Rightarrow r = 1

(real, nonnegative).

Saare

n

th roots of unity ka sum kya hai (

n \geq 2

0

Sum 0 hai yeh ek line mein prove karo.

Geometric series

\frac{ω ^{n} - 1}{ω - 1} = \frac{0}{ω - 1} = 0

Cube roots of unity kya hain?

1, - \frac{1}{2} + \frac{3}{2} i, - \frac{1}{2} - \frac{3}{2} i

Key cube-root identity kya hai?

1 + ω + ω^{2} = 0

aur

ω^{3} = 1

ω = e^{i 2 π / n}

se multiply karne ka geometric matlab?

2 π / n

se rotation (length unchanged).

n

th roots of unity kaun sa polygon banate hain?

Ek regular

n

-gon unit circle mein inscribed, vertex

1

pe.

4th roots of unity kya hain?

1, i, - 1, - i

ω^{k}

primitive root kab hota hai?

Jab

g cd (k, n) = 1

z^{n} = a

ke roots, roots of unity ke terms mein?

Ek root times har

n

th root of unity.

Connections

De Moivre's Theorem — $n$ th powers/roots lene ka engine.
Polar Form of Complex Numbers — jahan se $r, θ$ aate hain.
Argand Diagram — geometric stage.
Geometric Series — sum-of-roots result prove karta hai.
Fundamental Theorem of Algebra — exactly $n$ roots guarantee karta hai.
Regular Polygons — roots jaisa shape banate hain.