number system and digital device (chapter 3)

1

ICT Private Program; Ashek Mahmud khan; 01725-402592;

Z_¨ I †hvMv‡hvM cÖhyw³

Z„Zxq Aa¨vq:

msL¨v c×wZ I wWwRUvj wWfvBm

weÁvb e¨emvq wkÿv gvbweK

GKv`k-Øv`k ‡kÖYx

†kÖYx

ICT Private Program

01725-402592

Ashek Mahmud Khan

Dept. of CSE (JUST)

[email protected]

ÒZ_¨ I ‡hvMv‡hvM cÖhyw³Ó

h‡kvi weÁvb I cÖhyw³ wek¦we`¨vj‡qi Computer Science & Engineering wefv‡Mi QvÎ‡`i, GKv`k

I Øv`k †kÖYxi Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó Gi Private Program…….

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2


iƒcvšÍi:

1. `kwgK ‡_‡K †h †Kvb msL¨vq iƒcvšÍi|

2. ‡h‡Kvb msL¨v c×wZ †_‡K `kwgK msL¨v c×wZ‡Z i~cvšÍi|

3. evBbvix †_‡K AKUvj/†n·v-‡Wwm‡gj|

4. AKUvj/†n·v-‡Wwm‡gj †_‡K evBbvix |

5. AKUvj/†n·v-‡Wwm‡gj †_‡K †n·v-‡Wwm‡gj/AKUvj|

`kwgK ‡_‡K †h †Kvb msL¨vq iƒcvšÍi:

`kwgK †_‡K evBbvix:

‡h‡Kvb msL¨i ywU Ask _vK‡Z cv‡i| h_v:

(1) c~Y©vsk

(2) fMœvsk

Avgiv cÖ_‡g †`L‡ev c~Y©vs‡ki †ÿ‡Î-

‡h `kwgK c~Y© msL¨v‡K cwieZ©b Ki‡Z n‡e Zv‡K 2 Øviv fvM Ki‡Z n‡e Ges fvM‡klwU‡K msiÿb Ki‡Z n‡e|

cÖ&vß fvMdj‡K Avevi `yB Øviv fvM Ki‡Z n‡e Ges fvM‡klwU msiÿb Ki‡Z n‡e|

A_v©r cÖvß fvMdj†K 2 Øviv fvM cÖwµqv ZZÿb ch©šÍ Pj‡Z _vK‡e hZÿb bv fvMdj k~b¨ nq|

cÖ&vß fvMdj¸‡jv‡K ( †k‡l cÖvß fvM‡k‡li w`K †_‡K ïiæ‡Z cÖvß fvM‡k‡li w`‡K) mvwR‡q wjL‡jB i~cvšÍwiZ msL¨vi c~Yv©sk

cvIqv hv‡e|

cÖkœ: (527)10 †K evBbvix ev wØwgK msL¨vq i~cvšÍi Ki|

myZivs (527)10 = (1000001111)2

fMœvs‡ki †ÿ‡Î -

‡h `kwgK fMœvsk‡K cwieZ©b Ki‡Z n‡e Zv‡K 2 Øviv ¸b Ki‡Z n‡e Ges cÖvß ¸bd‡ji c~Y©vskwU‡K msiÿb Ki‡Z n‡e|

cÖvß ¸bd‡ji fMœvsk‡K (c~Yv©skwU‡K bq) Avevi 2 Øviv ¸b Ki‡Z n‡e Ges cÖvß ¸bd‡ji c~Y©vskwU‡K msiÿb Ki‡Z n‡e|

A_v©r cÖvß fMœvsk‡K 2 Øviv ¸b Kivi cÖwµqv ZZÿb ch©šÍ Pj‡Z _vK‡e hZÿb bv ¸bdj k~b¨ nq|

Gevi cÖvß ¸Yd‡ji c~Yv©sk ¸wj‡K ïiæ †_‡K †kl ch©šÍ mvRv‡q wjL‡jB i~cvšÍwiZ msL¨vi fMœvsk cvIqv hv‡e|

h‡kvi weÁvb I cÖhyw³ wek¦we`¨vj‡qi

Computer Science & Engineering

wefv‡Mi QvÎ‡`i, GKv`k I Øv`k †kÖYxi

Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó

Gi Private Program…….

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cÖkœ: (0.75)10 †K evBbvix ev wØwgK msL¨vq iƒcvšÍi Ki|

cÖkœ: (175.15)10 msL¨vwU‡K AKUvj msL¨vq iƒcvšÍi Ki|

mgvavb: c~Y© As‡ki †ÿ‡Î-

. . (175)10=(257)8

fMœvs‡ki †ÿ‡Î-

(.15)10=(.11463.....)8

myZivs wb‡Y©q djvdj: (175.15)10 =(257.11463....)8 Ans

fMœvsk2 ¸bdj cÖvß fMœvsk c~Y© msL¨v

0.752

.502

.002

1.50

1.00

.00

.50

.00

.00

1 MSB

1

0 LSB


0.158

.208

.608

.808

.408

.208

.........

.........

1.20

1.60

4.80

6.40

3.20

1.60

.........

.........

.20

.60

.80

.40

.20

.60

.........

.........

1 MSB

1

4

6

3

1

.......

....... LSB

myZivs (.75)10 = (.110)2

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cÖkœ: (2479.50)10 msL¨vwU‡K ‡n·v-‡Wwmg¨vj msL¨vq iƒcvšÍi Ki|

mgvavb: c~Y© As‡ki †ÿ‡Î-

(2479)10=(9AF)16

fMœvs‡ki †ÿ‡Î-

(.50)10= (.80)16

myZivs wb‡Y©q djvdj: (2479.50)10 =(9AF.80)16 Ans

‡h‡Kvb msL¨v c×wZ †_‡K `kwgK msL¨v c×wZ‡Z i~cvšÍi:

mvavib wbqg:

‡h‡Kvb msL¨v (evBbvix/ A±j/†n·v‡Wwm‡gj) †_‡K `kwgK msL¨vq iƒcvšÍ‡ii †ÿ‡Î wb¤§wjwLZ avc AbymiY Ki‡Z cvwi| c~Yv©sk I fMœvs‡ki

Rb¨ GB wbqg cÖ‡hvR¨|

avc-1: cÖ`Ë msL¨vi wfwË mbv³ K‡i msL¨vwUi AšÍ©MZ cÖwZwU As‡Ki ¯’vbxqgvb †ei Ki‡Z n‡e|

avc-2: msL¨vwUi AšÍ©MZ cÖwZwU As‡Ki wbR¯ gvb‡K Zvi ’vbxq gvb w`‡q ¸b Ki‡Z n‡e|

avc-3: Gfv‡e cÖvß ¸Ydj ¸wj‡K †hvM Ki‡jB wb‡Y©q msL¨vwU cvIqv hvq|

evBbvix †_‡K `kwgK iƒcvšÍi:

c~Y©v‡ki †ÿ‡Î:

(1010)2 = 123+02

2+12

1+02

0

= 8 + 0 + 2 + 0

= (10)10

fMœv‡ki †ÿ‡Î:

(.101)2 =12-1 + 02

-2 + 12

-3

= 1/2 + 0 + 1/8


0.5016

.0016

8.00

0.00

.00

.00

8

0

GLv‡b wfwË 2

A_v©r n Gi gvb 2






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5


= 0.5 + 0.125

= (0.625)10

cÖkœ: (11011.101)2msL¨vwU‡K `kwg‡K iƒcvšÍi Ki|

mgvavb: (11011.101)2 = 124 + 12

3 + 02

2 + 12

1 + 12

0 + 12

-1 + 02

-2+12

-3

= 16 + 8 + 2 + 1 + 1/2+1/8

= 27+0.5+0.125

= (27.625)10

myZivs (11011.101)2 = =(27.625)10

AKUj †_‡K `kwgK iƒcvšÍi:

cÖkœ: (513.75)8 msL¨vwU‡K `kwg‡K iƒcvšÍi Ki|

mgvavb: (513.75)8 = 582 + 18

1 + 38

0 + 78

-1 + 58

-2

= 564 + 18 + 31 + 7 (1/8) + 5 (1/64)

= 320 + 8 + 3 +(7/8) +(5/64)

= 331 + (61/64)

= 331 + 0.953125

= (331.953125)10

myZivs (513.75)8 = (331.953125)10

†n·v‡Wwm‡gj †_‡K `kwg‡K iƒcvšÍi:

cÖkœ: (B5D.48)16 msL¨vwU‡K `kwg‡K iƒcvšÍi Ki|

mgvavb: (B5D.48)16 = B 162 +516

1 + D 16

0 + 416

-1 + 816

-2

= 11256 + 80 + 13 + 0.25 + 0.03125

= 2909.28125

= (2909.28125)10

myZivs (B5D.48)16 = (2909.28125)10






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6


A±vj †_‡K evBbvix:

1. (365.150)8 msL¨vwU‡K mgZzj¨ evBbvix msL¨vq cÖKvk Ki?

mgvavb: cÖ_‡g Avgiv A±vj msL¨vi cÖwZwU As¼‡K 3 (wZb) we‡Ui evBbvix msL¨vq iƒcvšÍi K‡i cvB-

3 6 5 . 1 5 0

011 110 101 . 001 101 000

(365.150)8 = (011110101.001101000)2

A±vj †_‡K †n·v‡Wwm‡gj:

2. (602.52)8 msL¨vwU‡K ‡n·v-‡Wwmgvj msL¨vq cÖKvk Ki?

mgvavb: cÖ_‡g A±vj msL¨vwU‡K ‡Wwmgvj msL¨vq iƒcvšÍi K‡i cvB-

= 682+08

1+28

0 +58

-1+28

-2

= 664+0+21+51/8 + 21/64

= 384+2+0.625+0.03125

= 386.65625

(602.52)8= (386.65625)10

GLb cÖvß †Wwmgvj msL¨vwU‡K ‡n·v-‡Wwmgvj msL¨vq iƒcvšÍi K‡i cvB-

evBbvix †_‡K A±vj:

1. (1111110001.10011)2 msL¨vwU‡K AKUvj msL¨vq cªKvk Ki?

mgvavb: cÖ_‡g Avgiv Radix Point †_‡K evg w`‡Ki msL¨v ¸‡jv‡K µgvš‡q evg w`‡K 3wU K‡i Ges Wvb w`‡Ki msL¨v ¸‡jv‡K

µgvš‡q Wvb w`‡K 3wU K‡i MYbv K‡i cvB

7


001 111 110 001 . 100 110

1 7 6 1 4 6

(1111110001.10011)2 = (1761.46)8

evBbvix †_‡K †n·v-‡Wwm‡gj:

2. (10111101101.0010111111)2 msL¨vwU‡K †n·v-‡Wwm‡gj msL¨vq cªKvk Ki?

mgvavb: cÖ_‡g Avgiv Radix Point †_‡K evg w`‡Ki msL¨v ¸‡jv‡K µgvš‡q evg w`‡K 4wU K‡i Ges Wvb

w`‡Ki msL¨v ¸‡jv‡K µgvš‡q Wvb w`‡K 4wU K‡i MYbv K‡i cvB-

0101 1110 1101 . 0010 1111 1100

5 E D 2 F C

(10111101101.0010111111)2 = (5ED.2FC)16

evBbvix †hvM-we‡qvM

evBbvix msL¨v †hvMdj nv‡Z _vKv msL¨v

0+0 0 0

0+1 1 0

1+0 1 0

evBbvix msL¨v †hvMdj nv‡Z _vKv msL¨v

1+1 0 1

1+1+1 1 1






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8


evBbvix ¸b , fvM (Multiplication and Division in Binary System)

wPýhy³ msL¨v(Signed Number)

1 Gi cwic~iK (1’s Complement form)

2 Gi cwicyiK (2’s Complement form)

evBbvix ¸b:

`kwgK c×wZi gZ ¸‡bi bvgZv g‡b ivLvi `iKvi bvB| evBbvix ïay PviwU ¸bd‡ji bvgZv Rvb‡j h‡_ô|

wb‡¤§ evBbvixi PviwU Ae¯’v nj|-

0 0 = 0 1 0 = 0

0 1 = 0 1 1 = 1

evBbvixfvM:

`kwgK c×wZi gZ evBbvi‡Z fvM Kiv hvq | evBbvix c×wZ‡Z 0 w`‡q fvM Kiv‡K A_©nxb e‡j|

wb‡¤§ evBbvixi PviwU Ae¯’v nj|-

0 0 = A_©nxb 1 0 = A_©nxb

0 1 = 0 1 1 = 1

cÖkœ: 1 evB‡U ev 8 weU †iwR÷v‡i †h mKj wPwýZ ev mvBÛ msL¨v Dcv ’vcb Kiv hvq Zvi e¨vwß KZ?

mgvavb: †iwR÷vi 1 evBU ev 8 we‡Ui nq Zvn‡j mvBb we‡Ui Rb¨ 1 weU Ges gv‡bi Rb¨ 7 weU e¨envi Kiv hv‡e|

m‡ev©”P FbvZœK gvb:

1 0 0 0 0 0 0 0

(10000000)2 = -27 = ( -128)10

m‡ev©”P abvZœK gvb:

0 1 1 1 1 1 1 1

(01111111)2 = -27-1 = (+127)10

gvb wPý weU

gvb

wPý weU

9


myZivs e¨vwß n‡e -128 †_‡K +127 Gi g‡a¨ ; †gvU 256 wU ev 28 wU c„_K gvb (k~b¨mn Dcv ’vcb Kiv hv‡e|

cÖkœ: 2 Gi c~iK c×wZ‡Z 8 wU msL¨v e¨venvi K‡i +8 n‡Z +5 we‡qvM Ki|

mgvavb: +8 I +5 Gi evBbvix h_vµ‡g 00001000 , 00000101|

+5 Gi †b‡Mkb = 11111010 + 1 = 11111011

GLb, +8 = 0 0001000

-5= 1 1111011

+3 0 0000011

wPý weU 0 , myZivs djvdj mivmwi evBbvix MV‡b Av‡Q; djvdj +3 |

KvW (Code)

we wm wW †KvW (BCD Code)

Avjdv wbDgvwiK †KvW (Alpha Numaric Code)

A¨vmwK †KvW (ASCI Code)

B we wm wW AvBwm †KvW (EBCDIC Code)

BDwbK †KvW (Unicode)

cÖkœ: 1 . †KvW wK?

DËi: Kw¤úDUvi wm‡÷‡g eëüZ cÖwZwU eY©, msL¨v ev we‡kl wPý‡K Avjv`v fv‡e wm wc BD ‡K †evSv‡bvi Rb¨ we‡Ui (0 ev 1) wewfbœ

web¨v‡mi mvnv‡h¨ AwØZxq ms‡KZ ‣Zwi Kiv nq| GB AwØZxq ms‡KZ†K †KvW ejv nq| evBbvix †KvW Øviv wWwRUvj B‡jKUªwb· c×wZi

mv‡_ ms‡hvM iÿv Kiv nq | wWwRUvj B‡jKUªwb· c×wZ‡Z cÖ‡qv‡Mi iKg†f‡` mvaviYZ wewfbœ cÖKvi †KvW eënvi Kiv nq| †hgb : wewmwW

†KvW, AKUvj †KvW, †n·v‡Wwm‡gj †KvW, A¨vmwK †KvW, Ge wm weK †KvW, BDwb †KvW BZ¨vw`|

cÖkœ: 2 . wewmwW †KvW I evBbvix msL¨vi g‡a¨ cv_©K¨ wjL|

DËi: wewmwW †KvW I evBbvix msL¨vi g‡a¨ †h mKj cv_©K¨ cwijwÿZ nq Zv wb‡¤§ †Ìqv nj:

cv_©‡Kï welq wewmwW †KvW evBbvix †KvW

(K) msÁv: (K) `kwgK msL¨vi cÖwZwU AsK‡K mgZzj¨ evBbvix

msL¨v Øviv cÖKvk Kiv‡K wewmwW †KvW e‡j|

(K) †h msL¨v c×wZ‡Z ywU AsK eën„Z nq Zv‡K

evBbvix msL¨v c×wZ e‡j|

(L) c×wZ: (L) wewmwW †Kvb msL¨v c×wZ bq| GUv Avm‡j `kwgK

c×wZ hvi cÖwZwU AsK Gi mgZzj¨ evBbvix‡Z Gb‡KvW

Kiv nq|

(L) evBbvix GKwU msL¨v c×wZ hvi wfwË 2 Ges

wPý ¸‡jv nj 0 I 1|

(M) cÖKvk: (M) `kwgK msL¨v‡K wewmwW †Kv‡W cÖKvk Kiv Lye (M) `kwgK msL¨v‡K evBbvix msL¨vq cÖKvk Kiv






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10


mnR| Lye mnR bq|

(N) we‡Ui msL¨v: (N) ‡Kvb msL¨v‡K wewmwW †Kv‡W cÖKv‡ki Rb¨ †ewk

weU jv‡M|

(N) ‡Kvb msL¨v‡K wewmwW †Kv‡W cÖKv‡ki Rb¨ Kg

weU jv‡M|

(O) cÖKvi †f`: (O) wewmwW †KvW cuvP cÖKvi| †hgb: wewmwW 8421,

wewmwW 5421, wewmwW 7421, wewmwW 2421, wewmwW

642-G|s

(O) evBbvix msL¨v `kwgK msL¨vi Dci wbf©i

K‡i|

(P) eënvi: (P) GwU mvaviYZ K¨vjK z‡jUi, wWwRUvj Nwo, wWwRUvj

†fvë wgUv‡i eën„Z nq|

(P) GwU Kw¤úDUvi I Kw¤úDUvi wbqwš¿Z Ab¨vb¨

h‡š¿ eëüZ nq|

cÖkœ: 3. A¨vmwK †KvW I wewmwW †KvW - Gi g‡a¨ cv_©K¨ wbY©q Ki |

DËi : A¨vmwK †KvW I wewmwW †KvW -Gi g‡a¨ ‡h cv_©K¨ cwijwÿZ nq Zv wb‡¤§ †Ìqv nj:

cv_©‡Kï welq A¨vmwK †KvW wewmwW †KvW

(K) c~Y© bvg: (K) A¨vmwK Gi c~Y© bvg n‡jv Av‡gwiKvb ÷¨vÛvW© †KvW

di Bbdi‡gkb B›Uvi‡PÄ|

(K) we wm wW Gi c~Y© bvg-evBbvix †Kv‡WW

wWwm‡gj|

(L) msÁv: (L Kw¤úDUvi wm wc BD Ges MÖnbgyL wbM©gb gyL

miÄv‡gi g‡a¨ Z_¨ cÖwµqvi Rb¨ †h †KvW eën„Z nq

Zv‡K A¨vmwK †KvW e‡j|

(L) `kwgK msL¨vi cÖwZwU AsK‡K mgZzj¨ evBbvix

msL¨v Øviv cÖKvk Kiv‡K wewmwW †KvW e‡j|

(M) cÖKvk: (M) GwU Øviv 27 ev 128 wU AsK , eY© A_ev wewfbœ wPý

†K evBbvix‡Z cÖKv‡ki Rb¨ eënvi Kiv nq|

(M) GwU Øviv 0 n‡Z 9 ch©šÍ `kwU AsK‡K

evBbvix‡Z cÖKv‡ki Rb¨ eënvi Kiv nq|

(N) weU : (N) GwU GKwU 7 we‡Ui ‡KvW | (N) GwU GKwU 4 we‡Ui †KvW|

(O) eënvi : (O) gvB‡µv Kw¤úDUv†i G ai‡bi †KvW eëüZ nq| (O) `kwgK c×wZi msL¨v‡K evBbvix msL¨vq cÖKvk

Kivi Rb¨ eëüZ nq|

cÖkœ 1: eywjqvb A¨vj‡Reiv wK?

DËi: RR© eywj bv‡g GKRb Bs‡iR MwYZwe` 1854 mv‡j eywjqvb A¨vj‡Reiv Avwe®‥vi K‡ib| RR© eywj-Gi †Pv‡L cÖ_g AsK I hyw³i

†hvMm~Î aiv c‡o| AsK I hyw³i †hvMm~Î cÖgv‡Yi Rb¨ hyw³ †MU I c‡i hyw³ eZ©bx ‣Zix K‡ib| hyw³ †MU I c‡i hyw³ eZ©bxi wewfbœ e¨vL¨vi

Rb¨ eywjqvb A¨vj‡Reiv eëüZ n‡Zv| RR© eywj Gi bvg Abymv‡i GB A¨vj‡Reiv†K eywjqvb A¨vj‡Reiv ejv nq|

cÖkœ 2: mZ¨K mvibx Kv‡K e‡j?

DËi: †h mvibxi gva¨‡g wewfbœ †MB‡Ui Kvh©bxwZ‡K cÖKvk Kiv nq Zv‡K mZ¨K mvibx e‡j| eywjqvb A¨vj‡Reivq jwRK dvsk‡b wewfbœ

ai‡bi †WUv BbcyU Kivi Rb¨ BbcyU Pj‡Ki eënvi Kiv nq Ges AvDUcyU cÖ`v‡bi Rb¨ AvDUcyU PjK eënvi Kiv nq| mKj †MB‡Ui Rb¨

11


GKwU AvDUcyU _vK‡jI wewfbœ †MB‡Ui BbcyU GKvwaK n‡Z cv‡i| cÖwZwU eZ©bx c„_K c„_K BbcyU I AvDUcyU Ae¯’v _v‡K| †Kvb eZ©bxi

BbcyU Ges AvDUcy‡Ui wewfbœ Ae¯’v cÖ`k©‡bi Rb¨ e¨eüZ mvibx‡K mZ¨K mvibx ejv nq|

A I B Gi m¤¢ve¨ gv‡bi wf‡Ë‡Z wb‡¤œ GKwU mZ¨K mvibx cÖ`k©b Kiv nj:

BbcyU AvDUcyU

A B X=A+B

0 0 0

0 1 1

1 0 1

1 1 1

wPÎ: m‡Z¨K mviYx

cÖkœ 3: c¨vwiwU weU wK?

DËi: evBbvix †WUv †KvW‡K GK ¯’vb †_‡K Ab¨ ’v‡b †cÖiY Kivi Rb¨ †KvW MÖæ‡ci mv‡_ †h AwZwi³ 1 weU †hvM Kiv nq ZvB c¨vwiwU weU|

†KvW MÖæ‡ci Dcw¯’Z GK (1) Gi msL¨vi Dci wfwË K‡i ïb¨ ev GK †hvM K‡i c¨vwiwU weU MVb Kiv nq| c¨vwiwU we‡Ui `ywU cÖPwjZ c×wZ

Av‡Q| h_v:-

1| B‡fb ev †Rvo c¨vwiwU I

2| AW ev we‡Rvo c¨vwiwU

cÖkœ 4: eywjqb c~iK wK?

cÖkœ 5: eywjqb aªæeK wK?

cÖkœ 5: cÖgvb Ki †h, 1. A+ = 1

2. A + BC = (A+B)(A+C)

3. (A+B)(A+ ) = A

cÖkœ 5: cÖgvb Ki †h, 1. A+ = A+

2. A( +B) = AB

wW-giM¨v‡bi Dccv`¨:

1g Dccv`¨: yB ev Z‡ZvwaK Pj‡Ki †hvMd‡ji Kgwcø‡g›U H mg Í PjK¸wji c„_K c„_K Kgwcø‡g‡›Ui ¸bd‡ji mgvb|






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12


( )= .

2q Dccv`¨: yB ev Z‡ZvwaK Pj‡Ki ¸bd‡ji Kgwcø‡g›U H mg Í PjK¸wji c„_K c„_K Kgwcø‡g‡›Ui †hvMd‡ji mgvb|

( )= +

`yB Pj‡Ki mvnv‡h¨ wW-giM¨v‡bi Dccv‡`ï mZ¨Zv hvPvB:

Pj‡Ki mvnv‡h¨ cÖgvY Ki †h, (i) ( )= . (ii) ( )= +

A B

A+B

. A.B

+

0 0 1 1 0 1 1 0 1 1

0 1 1 0 1 0 0 0 1 1

1 0 0 1 1 0 0 0 1 1

1 1 0 0 1 0 0 1 0 0

( ) = . ( ) = +

Dc‡ii mZ¨K mviYx †_‡K †`Lv hvq †h, A I B Gi mKj gv‡bi Rb¨-

( )= .

( )= +

`yBwU eywjqvb Pj‡Ki Rb¨ wW-giM¨v‡bi m~Î cÖgvwYZ nj

** KvR: wZb Pj‡Ki mvnv‡h¨ wW-giM¨v‡bi Dccv‡`ï mZ¨Zv hvPvB Ki?

jwRK dvskb mijxKiY: jwRK¨vj dvskb¸‡jv jwRK †M‡Ui gva¨‡g ev Í‡e eënvi Kiv| jwRK dvsk‡b jwRK Acv‡iU‡ii msL¨v Kgv‡j

jwRK †M‡Ui msL¨v Kgv‡bv nq| Kv‡RB A_© I mgq evuPv‡Z jwRK dvskb‡K mijxKiY Kiv nq|

eywjqvb Dccv‡`ï mvnv‡h¨ jwRK dvsk‡bi mijxKiY Kiv hvq|

mij Ki: ( ).B.

mgvavb: ( ).B.

= ( ) B. [wW-giM¨v‡bi Dccv‡`ï mvnv‡h¨]

= .C. B.

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= . B.C. [ . B = C. = 0 ]

= .0.0

= 0 [ .0 = 0]

= 0 Ans

mij Ki: ( )+( )

mgvavb: ( )+( )

= ( ) + ( ) [ ]

= ( +(B. )

= ( +(B ) Ans

cÖgvY Ki ‡h, xy+ z = z+xy+yz

R.H.S = z+xy+ yz

= xy.1+( z.1)+ yz.1

= xy(z+ )+ z(y+ )+ yz(x+ ) [X+ =1]

= xyz+ xy + zy+ z + xyz+ yz

= xyz+ xy + zy+ z [A+A=A]

= yz + z +xy +xyz

= z(y + ) + xy +z)

= z+ xy

= L.H.S

L.H.S = R.H.S (Proved)

cÖgvY Ki ‡h, ABC+ BC+A C+ C=C

L.H.S= ABC+ BC+A C+ C






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= BC(A+ )+ C(A+ )

= (BC+ C)

= C (B+ )

= C

= R.H.S


wW-giM¨v‡bi Dccv‡`¨i mvnv‡h¨ cÖgvY Ki:

(i) ( )( ) = 0 (ii) A+ B+ = 1

cÖgvY:

(i) ( )( ) = 0

L.H.S = ( )( )

= ( ) ( )

= ( ) (A.B)

= A. B

=0.0

=0

=R.H.S


(i) A+ B+ = 1

L.H.S = A+ B+

= A+ B+ )

= A+ .1






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= A+

=1

=R.H.S


‡g․wjK †MBU (Basic Gate)

‡MB‡Ui cÖKvi‡f`

Kw¤^‡bkb jwRK †MBU (Combinational Logic Gate)

jwRK †MBU: eywjqvb G¨vj‡Reivi eënvwiK cÖ‡qv‡Mi Rb¨ †h mg Í wWwRUvj B‡iKUªwbK mvwK©U eënvi Kiv nq Zv‡K jwRK †MBU

e‡j|

‡gŠwjK †MBU: †h mg Í jwRK †MBU Ab¨ †Kvb †MB†Ui mvnv‡h¨ QvovB •Zwi Kiv hvq Zv‡ì‡K †g․wjK jwRK †MBU e‡j| †g․wjK jwRK

†MBU wZb cÖKvi|

(i) AND †MBU (ii) OR †MBU (iii) NOT †MBU

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cÖkœ 1: mve©Rbbxb †MBU wK?

DËi : †h mKj †MB‡Ui mvnv‡h¨ †g․wjK †MBUmn Ab¨vb¨ mKj †MBU cÖKvk Kiv hvq ‡m mg Í ‡MBU†K mve©Rbxb †MBU ejv nq| ïaygvÎ

NAND ev NOR ‡MBU w`‡q †g․wjK †MBUmg~n cÖKvk Kiv hvq G Kvi‡Y NAND Ges NOR ‡MBU‡K wWwRUvj jwRK mvwK©‡U mve©Rbxb

†MBU ejv nq|

cÖkœ 2: NAND Ges NOR ‡MB‡Ui me©RbxZv cÖgvY Ki|

A_ev, ïay NAND ev NOR ‡MBU w`‡q †gŠwjK ‡MBU¸‡jvi ev Íevqb †`LvI|

A_ev, ïay NAND ‡MBU w`‡q †gŠwjK ‡MBU¸‡jvi ev Íevqb †`LvI|

A_ev, ïay NOR ‡MBU w`‡q †gŠwjK ‡MBU¸‡jvi ev Íevqb †`LvI|

DËi : AND ‡MBU OR ‡MBU Ges NOT ‡MBU GB wZbwU †gŠwjK †MB‡Ui mgš‡q †h †Kvb jwRK mvwK©U ev hyw³ eZ©bx ‰Zix Kiv hvq|

†Zgwb ïay NAND ev NOR ‡MBU w`‡q †g․wjK †MBU ev Ab¨vb¨ †MBU ev¯Íevqb Kiv hvq e‡j NAND ev NOR ‡MB †K mve©Rbxb

jwRK †MBU e‡j|

ïay NAND ‡MBU Gi mve©RbxbZv : ïay NAND ‡MBU e¨envi K‡i mKj †g․wjK †MBU ev¯Íevqb Kiv hvq| G Kvi‡Y NAND ‡MBU‡K

mve©Rbxb †MBU e‡j| bx‡P NAND ‡MB‡Ui mvnv‡h¨ 3wU †g․wjK †MBU Gi ev Íevqb †`Lv‡bv nj|

NAND ‡MBU w`‡q †g․jK ‡MBU ev¯Íevqb:






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NOT ‡MBU: wP‡Î NAND ‡MB‡Ui BbcyU `ywUB (A) . G‡ÿ‡Î b¨vÛ †MBU GKwU bU †MBU wn‡m‡e KvR K‡i|

wPÎ: NAND ‡MB‡Ui gva¨‡g •Zix NOT ‡MBU wPÎ: NOT ‡MBU OR‡MBU :

wP‡Î wZbwU b¨vÛ †MBU eënvi Kiv n‡q‡Q| wPÎ Abyhvqx ms‡hvM AvDUcyU A+B ‣Zix nq| A_v©r b¨vÛ †MBU w`‡q Ai †MBU Gi KvR Kiv

n‡q‡Q|

wPÎ: NAND ‡MB‡Ui gva¨‡g •Zix OR‡MBU wPÎ: OR‡MBU

AND ‡MBU :

wP‡Î yBwU b¨vÛ †MB‡Ui BbcyU Ggb fv‡e eënvi Kiv n‡q‡Q hv‡Z AvDUcyU AB nq| A_v©r b¨vÛ †MBU w`‡q G¨vÛ †MBU Gi KvR Kiv

n‡q‡Q|

wPÎ: NAND ‡MB‡Ui gva¨‡g •Zix AND ‡MBU wPÎ: AND ‡MBU

ïay NOR ‡MBU Gi mve©RbxbZv : ïay NOR ‡MBU eënvi K‡i mKj †g․wjK †MBU ev¯Íevqb Kiv hvq| G Kvi‡Y NOR ‡MBU‡K

mve©Rbxb †MBU e‡j| bx‡P NOR ‡MB‡Ui mvnv‡h¨ 3wU †g․wjK †MBU Gi ev¯Íevqb †`Lv‡bv nj|

NOT ‡MBU: wP‡Î NOR ‡MB‡Ui BbcyU `ywUB (A) . G‡ÿ‡Î b¨vÛ †MBU GKwU bU †MBU wn‡m‡e KvR K‡i|

18


OR‡MBU :

wP‡Î yBwU bi †MB‡Ui BbcyU Ggb fv‡e eënvi Kiv n‡q‡Q hv‡Z AvDUcyU A+B nq| A_v©r bi †MBU w`‡q Ai †MBU Gi KvR Kiv

n‡q‡Q|

AND ‡MBU : wP‡Î wZbwU bi †MBU eënvi Kiv n‡q‡Q| wPÎ Abyhvqx ms‡hvM AvDUcyU AB ‣Zix nq| A_v©r bi †MBU w`‡q A¨vÛ †MBU

Gi KvR Kiv n‡q‡Q|

Dc‡ii Av‡jvPbv n‡Z †aLv hvq †h, mwVK eënv‡ii gva¨‡‡g b¨vÛ I bi †MBU w`‡q mKj cÖKvi jwRK mvwK©U •Zwi Kiv hvq| G Kvi‡Y b¨vÛ

I bi †MBU‡K mve©Rbxb †MBU e‡j|

cÖkœ 3 : ïaygvÎ NAND ‡MBU eënvi K‡i F=AB+BC+AC jwRK dvskbwUi ev Íevqb jwRK wP‡Îi gva¨‡g ‡`LvI|

cÖkœ 4 : ïaygvÎ NAND ‡MBU w`‡q G·-bi †MBU (X-NOR) Gate ev¯Íevqb Ki|

cÖkœ 1:mve©Rbxb †MBU wK?

DËi : †h mKj †MB‡Ui mvnv‡h¨ †g․wjK †MBUmn Ab¨vb¨ mKj †MBU G cÖKvk Kiv hvq ‡mB mg Í ‡MBU †K mve©Rbxb †MBU e‡j| ïaygvÎ

NAND ev NOR ‡MBU w`‡q †g․wjK †MBU mg~n cÖKvk Kiv hvq G Kvi‡Y NAND Ges NOR ‡MBU‡K wWwRUvj jwRK mvwK©‡U

mve©Rbxb †MBU ejv nq|

cÖkœ 2: NAND ‡MB‡Ui mve©RbxbZv cÖgvY Ki|

y NAND ‡MBU Gi mve©RbxbZv : ïay NAND ‡MBU eënvi K‡i mKj †g․wjK †MBU ev¯Íevqb Kiv hvq| G Kvi‡Y NAND ‡MBU‡K

mve©Rbxb †MBU e‡j| bx‡P NAND ‡MB‡Ui mvnv‡h¨ 3wU †g․wjK †MBU Gi ev¯Íevqb †`Lv‡bv nj-

19


cÖkœ 3 : NOR ‡MB‡Ui mve©RbxbZv cÖgvY Ki|

NOR ‡MBU Gi mve©RbxbZv : ïay NOR ‡MBU e¨envi K‡i mKj †g․wjK †MBU ev¯Íevqb Kiv hvq| G Kvi‡Y NOR ‡MBU‡K mve©Rbxb

†MBU e‡j| bx‡P NOR ‡MB‡Ui mvnv‡h¨ 3wU †g․wjK †MBU Gi ev¯Íevqb †`Lv‡bv nj|






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20


Theorem:

i.

ii.

we‡kl †MBU(Special Gate): X-OR Ges X-NOR ‡MBU `ywU we‡kl †MBU bv‡g cwiwPZ|

X-OR gate: ‡g․wjK †MBU w`‡q GB mvwK©U •Zwi Kiv †M‡jI AND, OR, NOT, NAND Ges NOR ‡MBU Gi b¨vq GwU GKxf~Z

mvwK©U AvKv‡i cvIqv hvq| X-OR ‡MBU Gi wPý I mZ¨K mviYx †`qv nj-

X-OR ‡MBU Gi Bbcy‡U †e‡Rvo msL¨K 1 n‡j AvDUcyU 1 nq Ab¨_vq 0 nq|

X-NOR gate: X-OR ‡MBU Gi AvDUcyU‡K NOT gate w`‡q cÖevwnZ Ki‡j X-NOR gate cvIqv hvq| X-NOR ‡MBU Gi wPý I

mZ¨K mviYx †`qv nj-




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number system and digital device (chapter 3)

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