nirpeksh maan

vaastavik sankhyaaon ke liye nirpeksh maan falan ka graaf

ganit mein kisi vaastavik sankhya ka nirpeksh maan ya 'nirpeksh moolya' (absolute value) ya 'maapaank' (modulus) |a| us sankhya ke chinh ke bina uske aankik maan ke baraabar hota hai. udaaharan ke liye 3 ka nirpeksh maan 3 hai, tatha -3 ka bhi nirpeksh maan bhi 3 hi hai. kisi sankhya ke nirpeksh maan ko us sankhya ki shoonya se doori ke baraabar samjha ja sakta hai.

udaaharan
  • |5| = abs(5) = 5
  • |-2| = abs(-2) = 2

vaastavik sankhyaaon ke liye nirpeksh maan ki upar di gayi paribhaasha ko kai anya ganiteeya kshetron mein saamaanyeekaran (Generalization) kiya gaya hai. udaaharan ke liye samishr sankhyaaon ke liye bhi nirpeksh maan paribhaashit kiya jaata hai. iske alaava quaternions, ordered rings, fields aur sadish avakaash (vector spaces) ke liye bhi nirpeksh maan paribhaashit kiya jaata hai.

anukram

vaastavik sankhyaaon ka nirpeksh maan

vaastavik sankhya a ka nirpeksh maan | a a| (raashi ke dono or urdhv rekha dvaara) niroopit kiya jaata hai. tatha iski paribhaasha nimnalikhit prakaar se ki jaati hai-

is paribhaasha se spasht hai ki kisi bhi raashi ka nirpeksh maan ya to dhanaatmak hoga ya shoonya hoga ; yeh rinaatmak kabhi bhi naheen ho sakta.

chooainki bina chinh ke vargamool ka sanket us raashi ke dhanaatmak vargamool ko ingit karta hai; iska arth hua ki-

yahi kabhi-kabhi nirpeksh maan ki paribhaasha ke taur par istemaal kiya jaata hai.[1]

nirpeksh maan ke nimnalikhit chaar moolabhoot gun hote hain-

Non-negativity
Positive-definiteness
Multiplicativeness
Subadditivity

nirpeksh maan ke anya mahatvapoornan gun nimnalikhit hain-

Symmetry
Identity of indiscernibles (equivalent to positive-definiteness)
Triangle inequality (equivalent to subadditivity)
Preservation of division (equivalent to multiplicativeness)
(equivalent to subadditivity)

yadi b > 0, to asamataaon (inequalities) se sambandhit do anya upayogi gun ye hain-

uparokya sambandhon ka prayog un asamataaon ka hal nikaalne ke liye kiya ja sakta hai jinmein nirpeksh maan ka prayog hua ho. udaaharan ke liye,

samishr sankhyaaon ka nirpeksh maan

kisi samishr sankhya z ka nirpeksh maan moolabindu (origin) se z ki doori r ke baraabar hoti hai. chitr se yeh bhi spasht hai ki z aur uske samishr yugm (complex conjugate) z dono ka nirpeksh maan ekasamaan hota hai.

kisi samishr sankhya,

jahaaain x aur y vaastavik sankhyaaeain hain, z ka maapaank |z| dvaara pradarshit kiya jaata hai. isko nimnalikhit prakaar se paaribhaashit karte hain-

samishr sankhya ke maapaank mein upar varnit ve sab gun hain jo vaastavik sankhya ke maapaank mein hain.

yadi,

ho aur

z ka samishr yugm ho to,

tatha


z ke maapaank ka varg nimnalikhit hai-

sandarbh

  1. Stewart, James B. (2001). Calculus: concepts and contexts. Australia: Brooks/Cole. aai॰aऍsa॰abee॰aऍna॰ 0-534-37718-1. , p. A5

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