dvayaadhaari sankhya paddhati

dvayaadhaari sankhya paddhati (do number ka system ya binary numeral system; dvayaadhaari = dvi + aadhaari = '2' aadhaar wala) keval do ankon (0 tatha 1) ko kaam mein lene waali sthaaneeya maan sankhya paddhati hai. ismein sankhya ka maan nikaalne ka aadhaar (raidiks) 2 liya jaata hai. choonki do sthiti (haai / lo) vaale ilektraanik gate in sankhyaaon ko badi saralata se niroopit kar dete hain, is kaaran computer ke haardaveyar evam saaftaveyar mein is paddhati ka bahutaayat se prayog hota hai.

dvayaadhaari sankhyaaon ko dashamlav nanbaron mein badalne ke ganiteeya tareeke hote hain. iske tahat kai ganiteeya upakaran hain jinse dvayaadhaari sahit anya vidhiyon mein jama, ghata, guna, bhaag va anya ganiteeya aakalan hote hain. dvayaadhaari nanbaron se dashamlav mein ankon ko badalna jahaan jatil hai, vaheen dvayaadhaari ko anya vidhiyon mein antaran karna apekshaakrut saral hota hai.

anukram

itihaas

Bhaarat ke vidvaan pingal (lagbhag 5veen se - 2ri shati isaapoorv) ne chhandon ke varnan mein dvayaadhaari sankhya paddhati ka atyant buddhimtaapoorvak prayog kiya hai. is prakaar pingal dvayaadhaari sankhya paddhati ka varnan karne vaale pratham vyakti hain.

dashamlav paddhati maanaveeya upayog ke liye saral hai, isaliye aaranbhik roop yahi prachalit hui aur baad mein bhi jab ganana ke kai tareeke saamane aae to dashamlav paddhati ko pramukh sthaan mila tha. haalaanki dvayaadhaari bhi kaafi had tak ek praakrutik paddhati hai. kai aadhyaatmik paranparaaon mein, jaise paaithaagoras school aur praacheen bhaarateeya sant parampara mein bhi iska prayog hota tha. dvayaadhaari paddhati ka aarambh isa poorv chhathi shataabdi se maana jaata hai. san 1854 mein ganitjnya George bool ne dvayaadhaari paddhati par aadhaarit ek patra prakaashit kiya tha. isi ke saath booliyn elajebra (beejaganit) ki aadhaarashila padi thi. san 1937 mein klaud shainan ne dvayaadhaari beejaganit ke aadhaar par theory of sarkit ki neenv rakhi thi. 1940 mein baainari kanpyooting ki shuruaat bail laibs complex number computer ke saath hui thi.

ankeeya ginti (Digital counting)

ginti	baait	hartj
$2^{1}=2\cdot 1=2$ $2^{1}=2\cdot 1=2$
$2^{2}=2\cdot 2=4$ $2^{2}=2\cdot 2=4$
$2^{3}=2\cdot 2\cdot 2=8$ $2^{3}=2\cdot 2\cdot 2=8$
$2^{4}=2\cdot 2\cdot 2\cdot 2=16$ $2^{4}=2\cdot 2\cdot 2\cdot 2=16$
$2^{5}=2\cdot 2\cdot 2\cdot 2\cdot 2=32$ $2^{5}=2\cdot 2\cdot 2\cdot 2\cdot 2=32$
$2^{6}=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=64$ $2^{6}=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=64$
$2^{7}=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=128$ $2^{7}=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=128$
$2^{8}=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=256$ $2^{8}=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=256$
$2^{9}=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=512$ $2^{9}=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=512$
$2^{10}=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=1,024$ $2^{10}=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=1,024$	ek kilo-baait	ek kilo-hartj
$2^{20}=1024^{2}=1024\cdot 1024=10,48,576$ $2^{20}=1024^{2}=1024\cdot 1024=10,48,576$	ek mega-baait	ek mega-hartj
$2^{30}=1024^{3}=1024\cdot 1024\cdot 1024=10,73,741$ $2^{30}=1024^{3}=1024\cdot 1024\cdot 1024=10,73,741$	ek geega-baait	ek geega-hartj
$2^{40}=1024^{4}=1024\cdot 1024\cdot 1024\cdot 1024=10,99,51,16,27,776$ $2^{40}=1024^{4}=1024\cdot 1024\cdot 1024\cdot 1024=10,99,51,16,27,776$	ek tera-baait	ek tera-hartj
$2^{50}=1024^{5}=1024\cdot 1024\cdot 1024\cdot 1024\cdot 1024=1,12,58,99,90,68,42,620$ $2^{50}=1024^{5}=1024\cdot 1024\cdot 1024\cdot 1024\cdot 1024=1,12,58,99,90,68,42,620$	ek peta-baait	ek peta-hartj

dvayaadhaari niroopan

kisi dvayaadhaari sankhya ke maan ki ganana nimnalikhit prakaar se karte hain-

[1101]_{2}=1\cdot 2^{3}+1\cdot 2^{2}+0\cdot 2^{1}+1\cdot 2^{0}=[13]_{10}

dvayaadhaari paddhati mein niroopit sankhya ke aage ya peechhe 'kuchh' jodkar yeh spasht kiya jaata hai ki sankhya dvi-aadhaari hai (na ki daashamik, ashtaadhaari ya shodashaadhaari). neeche likhe hue sabhi 'sanketon ka samooh' sankhya 'chh: sau sarasath (667) ko niroopit kar rahe hain. kintu pehla wala niroopan sabse adhik prachalit hai.

1 0 1 0 0 1 1 0 1 1
| − | − − | | − | |
x o x o o x x o x x
y n y n n y y n y y

bhram se bachaane ke 0 aur 1 ka prayog karke likhe gaye dvi-aadhaari sankhyaaon ke saath kuchh aur bhi laga diya jaata hai taaki usaka aadhaar (2) spasht rahe. is prakaar, nimnalikhit sabhi niroopan ek hi sankhya ko niroopit karte hain-

100101 dvayaadhaari (aadhaar ka spasht ullekh kar diya hai)

100101b (yahaaain pratyaya jod diya hai jo dvayaadhaari sankhya ko soochit kar raha hai.)

100101B (yahaaain bhi pratyaya jod diya hai jo dvayaadhaari sankhya ko soochit kar raha hai.)

bin 100101 ((yahaaain sankhya ke pehle upasarg bin jod diya hai jo dvayaadhaari sankhya ko soochit kar raha hai.)

100101₂ (yahaaain aadhaar-2 ko soochit karne wala 'sabaskript' jod diya gaya hai.)

%100101 (dvayaadhaari sankhya bataane wala ek upasarg (preefiks) laga diya gaya hai.)

0b100101 (dvayaadhaari sankhya bataane wala ek upasarg (preefiks) laga diya gaya hai. ; prograaman bhaashaaon mein praay: prayukt)

6b100101 (dvayaadhaari sankhya bataane wala ek upasarg (preefiks) laga diya gaya hai. ; prograaman bhaashaaon mein praay: prayukt)

dvayaadhaari sankhyaaon ko jab shabdon mein uchchaarit karna padta hai to unhein ankash: (digit-by-digit) padhte hain jisse daashamik sankhyaaon se bhinnata samajh mein aa sake. udaaharan ke liye, baainari sankhya 100 ka uchchaaran 'ek shoonya shoonya' (one zero zero) kareinge na ki 'ek sau'. isse is sankhya ka dviaadhaari prakruti ka pata bhi chal jaata hai aur 'shuddhata' bhi rahati hai. '100', ek sau naheen hai, yeh keval chaar hai. isaliye ise 'ek sau' pukaarana galat hai.

dvayaadhaari ginti (Counting in binary)

neeche dvayaadhaari sankhya paddhati mein shoonya se solah tak ki ginti (likhne ka tareeka) diya gaya hai.

dvayaadhaari paitarn	0	1	10	11	100	101	110	111	1000	1001	1010	1011	1100	1101	1110	1111	10000
daashamik sankhya	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16

baahari kadiyaaain

Binary Numbers in Ancient India
Counting Computer Memory computer memori ka ginti