nth roots of complex numbers — finding all n roots

3.5.11 · Maths › Complex Numbers

HUM kya solve kar rahe hain?

Special case $w = 1$ se milte hain $n$ -th roots of unity ( $z^{n} = 1$ ke solutions).

Formula KAISE derive karte hain (scratch se)

Step 1 — $w$ ko polar form mein likho. Koi bhi nonzero $w = r (cos θ + i sin θ)$ , jahan $r = ∣ w ∣ > 0$ aur $θ = ar g w$ . Ye step kyun? Complex numbers ka multiplication polar form mein sabse easy hai: moduli multiply hote hain, arguments add hote hain. Roots, multiplication ko undo karne ke baare mein hain, isliye polar natural language hai.

Step 2 — Unknown root ko bhi polar form mein likho. Maano $z = ρ (cos ϕ + i sin ϕ)$ , jahan unknown modulus $ρ > 0$ aur unknown angle $ϕ$ hai.

Step 3 — $z^{n}$ par De Moivre apply karo. $z^{n} = ρ^{n} (cos n ϕ + i sin n ϕ) .$ Ye step kyun? De Moivre's theorem $(cos ϕ + i sin ϕ)^{n} = cos n ϕ + i sin n ϕ$ batata hai ki power lene se angle $n$ se multiply hota hai aur modulus power ho jaata hai.

Step 4 — $w$ ke saath match karo. Hume chahiye $z^{n} = w$ , to $ρ^{n} (cos n ϕ + i sin n ϕ) = r (cos θ + i sin θ) .$ Polar form mein do complex numbers equal hote hain tab aur sirf tab jab unke moduli equal hon aur unke arguments $2 π$ ke multiple se differ karein: $ρ^{n} = r, n ϕ = θ + 2 π k, k \in Z .$ $+ 2 π k$ kyun? YE crucial move hai. Angles sirf full turns tak defined hain; $cos$ aur $sin$ ka period $2 π$ hai. Agar hum $2 π k$ bhool jaayein to sirf ek root milega. Ise rakhne se baaki saare $n$ roots milte hain.

Step 5 — Solve karo. $ρ = r^{1/ n} (real positive n -th root), ϕ = \frac{θ + 2 π k}{n} .$

Step 6 — Kitne distinct roots hain yeh pata karo. Jab $k = 0, 1, 2, \dots, n - 1$ to hume $n$ alag angles milte hain. $k = n$ ke liye angle hai $\frac{θ}{n} + 2 π$ — $k = 0$ jaisi hi direction. To values period $n$ ke saath repeat hoti hain: sirf $n$ distinct roots hain.

Figure — nth roots of complex numbers — finding all n roots

Roots of unity (special case $w = 1$ )

Yahan $r = 1$ , $θ = 0$ , to $z_{k} = cos \frac{2 π k}{n} + i sin \frac{2 π k}{n} = e^{2 π ik / n}, k = 0, \dots, n - 1.$ $ω = e^{2 π i / n}$ likhne par, roots hain $1, ω, ω^{2}, \dots, ω^{n - 1}$ — ek geometric progression.

Worked Examples

Common Mistakes

Recall Feynman: ek 12-saal ke bacche ko explain karo

Ek ghadi ki sooee imagine karo. " $n$ ki power karna" matlab hai sooee ko $n$ guna tez ghoomana. Ise undo karne ke liye (root lene ke liye) aapko $n$ guna dheere ghoomana hoga. Par yahan trick hai: sooee 12 baje ki taraf point kar rahi hai — yeh 12 baje waali position ek baar poora chakkar lagate, do baar, teen baar... ke baad bhi same hi hai. Jab aap un saari "same" positions ko $n$ se slow down karte ho, to wo $n$ alag-alag jagah faili jaati hain, ghadi ke chakkar ke aas-paas evenly — jaise $n$ ped ek gol taalaab ke aas-paas equally lagaaye jaayein. Saare ped centre se same doori par hain (woh doori length ka slow-down hai), bas alag-alag angles par.

Active Recall Flashcards

#flashcards/maths

Ek nonzero complex number ke kitne distinct

n

-th roots hote hain?

Exactly

n

z^{n} = w

jahan

w = r cis θ

k

-th root ka formula kya hai?

z_{k} = r^{1/ n} cis (\frac{θ + 2 π k}{n})

k = 0, \dots, n - 1

Saare

n

roots ka common modulus kya hota hai?

r^{1/ n}

(

∣ w ∣

ka real positive

n

-th root).

Consecutive roots ke beech angular separation kitni hoti hai?

\frac{2 π}{n}

radians.

Argand plane mein

n

roots se kaunsi geometric shape banti hai?

Radius

r^{1/ n}

ke circle par ek regular

n

-gon ke vertices.

n

se divide karne se pehle

2 π k

kyun add karna zaroori hai?

Kyunki

ar g

sirf mod

2 π

tak defined hai; full turns add karne se baaki distinct roots milte hain.

Saare

n

-th roots of unity ka sum (

n \geq 2

0

(symmetry / GP sum).

Roots of unity ko GP ke roop mein kaise likhte hain?

1, ω, ω^{2}, \dots, ω^{n - 1}

jahan

ω = e^{2 π i / n}

Kisi bhi complex number ke square roots mein kya relation hota hai?

Ye ek doosre ke negatives hote hain,

z_{1} = - z_{0}

z^{3} = 8

ke liye complex roots kyun hain, jabki

8

real hai?

C

par ek cubic ke 3 roots hote hain; real value ≠ sirf ek root.

Connections

De Moivre's Theorem — angle multiplication/division ka engine.
Polar Form of Complex Numbers — prerequisite representation.
Roots of Unity — special case $w = 1$ ; cyclotomic structure.
Argand Diagram — regular polygons ke roop mein geometric visualisation.
Euler's Formula — $e^{i θ}$ GP structure ko obvious banata hai.
Fundamental Theorem of Algebra — kyun $z^{n} - w$ ke exactly $n$ roots hote hain.
Geometric Progression — roots of unity ka sum.