samishr sankhya

kisi samishr sankhya ka argend aarekh par pradarshan

ganit mein samishr sankhyaaeain (complex number) vaastavik sankhyaaon ka vistaar hai. kisi vaastavik sankhya mein ek kaalpanik bhaag jod dene se samishr sankhya banti hai. samishr sankhya ke kaalpanik bhaag ke saath i juda hota hai jo nimnalikhit sambandh ko santusht karti hai:


kisi bhi samishr sankhya ko a + bi, ke roop mein vyakt kiya ja sakta hai jismein a aur b dono hi vaastavik sankhyaaen hain. a + bi mein a ko vaastavik bhaag tatha b ko kaalpanik bhaag kehte hain. udaaharan: 3 + 4i ek samishr sankhya hai.

anukram

samishr sankhya ka kaarteeya niroopan

samishr sankhya ko a + bi ke roop mein darshaane ko samishr sankhya ka kaarteeya svaroop (Cartesian Form) kehte hai.

polar svaroop (polar form)

samishr sankhya z = x + iy ko dhruveeya nirdeshaankon ke roop mein bhi niroopit kar sakte hain. dhruveeya nirdeshaank r = |z| ≥ 0, ko samishr sankhya ka nirpeksh maan (absolute value) ya maapaank (modulus) kehte hain. isi prakaar φ = arg(z) ko z ka konaank (argument) kehte hain.

kaartiya svaroop se dhruveeya svaroop mein parivartan

konaank φ = konaank 2π.n falan atan2 mukhya maan (−π, +π] ke beech deta hai. kintu yadi φ ka rinaatmak maan nahin chaahiye balki [0, 2π) ke beech mein chaahiye to us rinaatmak maan mein 2π jodkar praapt kiya ja sakta hai.

dhruveeya se kaarteeya svaroop mein parivartan

samishr sankhya ka polar svaroop (Notation of the polar form)

nimnalikhit roop dhruveeya svaroop kahalaata hai:

ise cis φ se bhi nirupit karte hain jo cos φ + i sin φ ka sankshipt roop hai.

yoolar ka sootr (Euler's formula) ka prayog karke ise nimnalikhit tareeke se bhi likh sakte hain:

is svaroop ko iksaponeinshiyl roop' (exponential form) kehte hain.

elektrauniki mein kisi fejar (phasor) ke liye samishr sankhya ke koneeya niroopan ka bahudha prayog hota hai. jismein A aayaam evam θ kala (phase) hai.

dhyaan rahe ki elektrauniki aur vidyut abhiyaantriki mein i ke bajaay j ka prayog kiya jaata hai kyonki i ke dvaara vidyut dhaara ka nirupan kiya jaata hai.

nirpeksh maan evam samishr-yugm

The absolute value (or modulus or magnitude) of a complex number is defined as . Algebraically, if , then

The absolute value has three important properties:

where if and only if
(triangle inequality)

for all complex numbers z and w. These imply that and . By defining the distance function , we turn the set of complex numbers into a metric space and we can therefore talk about limits and continuity.

The complex conjugate of the complex number is defined to be , written as or . As seen in the figure, is the "reflection" of z about the real axis, and so both and are real numbers. Many identities relate complex numbers and their conjugates:


kaarteeya svaroop mein samishr sankriyaaeain

yog (Addition)

.

antar

.

guna

.

bhaag (Division)

kuchh udaaharan

yog:

ghataana:

guna:

bhaag:

dhruveeya svaroop mein sankriyaaen

guna evam bhaag

trikonamitteeya svaroop mein

eksponeinshiyl roop (Exponentil form)


anya sankriyaaeain

ghaataank

praakrutik sankhya

ka vaaain ghaat is prakaar nikaala jaata hai

ya kaarteeya roop ke liye

kisi bhi samishr ghaataank

kisi samishr aadhaar par samishr ghaataank ke liye saamaanya sootr hai:

yahaaain samishr gaghuganak ka mukhya maan liya jaayega.

mool (roots)

yahaaain bahut saavadhaani ki jaroorat hoti hai; dekhiye -

nimnalikhit sootr samishr sankhya ka vaaain mool nikaalne ke liye prayukt hota hai:

jahaaain ka maan . is prakaar kisi sankhya ke vein moolon ki kul sankhya hoti hai.

laghuganak

samishr sankhya ke praakrutik laghuganak ka mukhya maan hoga:

samishr sankhyaaon se sambandhit kuchh sarvasamikaaeain

if and only if z is real
if and only if z is purely imaginary
yadi z ashoonya sankhya hai.

antim wala sootr kisi samishr sankhya ka vyutkram (invars) nikaalne ke liye bahut upayogi hai, yadi vah sankhya kaarteeya roop mein di gayi hai.

samishr sankhyaaon ke anuprayog

niyantran siddhaant (Control theory)

sanket vishleshan (Signal analysis)

Improper integrals

kvaantam yaantriki (Quantum mechanics)

saapekshikta (Relativity)

vyaavahaarik ganit (Applied mathematics)

taral gatiki (Fluid dynamics)


inhein bhi dekhein

  • di mauyavar ka prameya (De Moivre's formula)
  • yoolar ki sarvasamika (Euler's identity)


baahari kadiyaaain