Number system and digital device (Chapter 3)
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1. The document discusses converting between different number systems including binary, decimal, octal and hexadecimal. 2. Methods are provided for converting integers and decimals from one base to another by breaking down the numbers into place values and recombining in the target base. 3. Examples are given of converting specific numbers between bases such as decimal to binary and vice versa.
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ICT Private Program; Ashek Mahmud khan; 01725-402592;
( )= .
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‡`¨i 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‡`¨i mZ¨Zv hvPvB Ki?
jwRK dvskb mijxKiY: jwRK¨vj dvskb¸‡jv jwRK †M‡Ui gva¨‡g ev¯Í‡e e¨envi 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‡`¨i mvnv‡h¨ jwRK dvsk‡bi mijxKiY Kiv hvq|
mij Ki: ( ).B.
mgvavb: ( ).B.
= ( ) B. [wW-giM¨v‡bi Dccv‡`¨i mvnv‡h¨]
= .C. B.](https://image.slidesharecdn.com/numbersystemanddigitaldevice-160119180231/85/Number-system-and-digital-device-Chapter-3-12-320.jpg)
![13
ICT Private Program; Ashek Mahmud khan; 01725-402592;
= . 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
h‡kvi weÁvb I cÖhyw³ wek¦we`¨vj‡qi
Computer Science & Engineering
wefv‡Mi Qv·`i, GKv`k I Øv`k †kÖYxi
Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó
Gi Private Program…….
†hvMv‡hvM: 01725-402592](https://image.slidesharecdn.com/numbersystemanddigitaldevice-160119180231/85/Number-system-and-digital-device-Chapter-3-13-320.jpg)
Number system and digital device (Chapter 3)
- 1. 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 wkÿvgvbweK 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 Qv·`i, GKv`k I Øv`k †kÖYxi Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó Gi Private Program……. †hvMv‡hvM: 01725-402592
- 2. 2 ICT Private Program; Ashek Mahmud khan; 01725-402592; 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 Qv·`i, GKv`k I Øv`k †kÖYxi Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó Gi Private Program……. †hvMv‡hvM: 01725-402592
- 3. 3 ICT Private Program; Ashek Mahmud khan; 01725-402592; 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 fMœvsk8 ¸bdj cÖvß fMœvsk c~Y© msL¨v 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
- 4. 4 ICT Private Program; Ashek Mahmud khan; 01725-402592; 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 +022 +121 +020 = 8 + 0 + 2 + 0 = (10)10 fMœv‡ki †ÿ‡Î: (.101)2 =12-1 + 02-2 + 12-3 = 1/2 + 0 + 1/8 fMœvsk16 ¸bdj cÖvß fMœvsk c~Y© msL¨v 0.5016 .0016 8.00 0.00 .00 .00 8 0 GLv‡b wfwË 2 A_v©r n Gi gvb 2 h‡kvi weÁvb I cÖhyw³ wek¦we`¨vj‡qi Computer Science & Engineering wefv‡Mi Qv·`i, GKv`k I Øv`k †kÖYxi Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó Gi Private Program……. †hvMv‡hvM: 01725-402592
- 5. 5 ICT Private Program; Ashek Mahmud khan; 01725-402592; = 0.5 + 0.125 = (0.625)10 cÖkœ: (11011.101)2msL¨vwU‡K `kwg‡K iƒcvšÍi Ki| mgvavb: (11011.101)2 = 124 + 123 + 022 + 121 + 120 + 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 + 181 + 380 + 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 +5161 + D 160 + 416-1 + 816-2 = 11256 + 80 + 13 + 0.25 + 0.03125 = 2909.28125 = (2909.28125)10 myZivs (B5D.48)16 = (2909.28125)10 h‡kvi weÁvb I cÖhyw³ wek¦we`¨vj‡qi Computer Science & Engineering wefv‡Mi Qv·`i, GKv`k I Øv`k †kÖYxi Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó Gi Private Program……. †hvMv‡hvM: 01725-402592
- 6. 6 ICT Private Program; Ashek Mahmud khan; 01725-402592; 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 +081 +280 +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. 7 ICT Private Program; Ashek Mahmud khan; 01725-402592; 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 h‡kvi weÁvb I cÖhyw³ wek¦we`¨vj‡qi Computer Science & Engineering wefv‡Mi Qv·`i, GKv`k I Øv`k †kÖYxi Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó Gi Private Program……. †hvMv‡hvM: 01725-402592
- 8. 8 ICT Private Program; Ashek Mahmud khan; 01725-402592; 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 gvbwPý weU gvbwPý weU
- 9. 9 ICT Private Program; Ashek Mahmud khan; 01725-402592; 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 e¨eüZ 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 e¨envi 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‡¤§ †`Iqv nj: cv_©‡K¨i 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 e¨en„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 h‡kvi weÁvb I cÖhyw³ wek¦we`¨vj‡qi Computer Science & Engineering wefv‡Mi Qv·`i, GKv`k I Øv`k †kÖYxi Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó Gi Private Program……. †hvMv‡hvM: 01725-402592
- 10. 10 ICT Private Program; Ashek Mahmud khan; 01725-402592; 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) e¨envi: (P) GwU mvaviYZ K¨vjKz‡jUi, wWwRUvj Nwo, wWwRUvj †fvë wgUv‡i e¨en„Z nq| (P) GwU Kw¤úDUvi I Kw¤úDUvi wbqwš¿Z Ab¨vb¨ h‡š¿ e¨eüZ 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‡¤§ †`Iqv nj: cv_©‡K¨i 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 e¨en„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¨ e¨envi Kiv nq| (M) GwU Øviv 0 n‡Z 9 ch©šÍ `kwU AsK‡K evBbvix‡Z cÖKv‡ki Rb¨ e¨envi Kiv nq| (N) weU : (N) GwU GKwU 7 we‡Ui ‡KvW | (N) GwU GKwU 4 we‡Ui †KvW| (O) e¨envi : (O) gvB‡µv Kw¤úDUv†i G ai‡bi †KvW e¨eüZ nq| (O) `kwgK c×wZi msL¨v‡K evBbvix msL¨vq cÖKvk Kivi Rb¨ e¨eüZ 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 e¨eüZ 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 e¨envi Kiv nq Ges AvDUcyU cÖ`v‡bi Rb¨ AvDUcyU PjK e¨envi Kiv nq| mKj †MB‡Ui Rb¨
- 11. 11 ICT Private Program; Ashek Mahmud khan; 01725-402592; 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| h‡kvi weÁvb I cÖhyw³ wek¦we`¨vj‡qi Computer Science & Engineering wefv‡Mi Qv·`i, GKv`k I Øv`k †kÖYxi Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó Gi Private Program……. †hvMv‡hvM: 01725-402592
- 12. 12 ICT Private Program; Ashek Mahmud khan; 01725-402592; ( )= . 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‡`¨i 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‡`¨i mZ¨Zv hvPvB Ki? jwRK dvskb mijxKiY: jwRK¨vj dvskb¸‡jv jwRK †M‡Ui gva¨‡g ev¯Í‡e e¨envi 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‡`¨i mvnv‡h¨ jwRK dvsk‡bi mijxKiY Kiv hvq| mij Ki: ( ).B. mgvavb: ( ).B. = ( ) B. [wW-giM¨v‡bi Dccv‡`¨i mvnv‡h¨] = .C. B.
- 13. 13 ICT Private Program; Ashek Mahmud khan; 01725-402592; = . 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 h‡kvi weÁvb I cÖhyw³ wek¦we`¨vj‡qi Computer Science & Engineering wefv‡Mi Qv·`i, GKv`k I Øv`k †kÖYxi Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó Gi Private Program……. †hvMv‡hvM: 01725-402592
- 14. 14 ICT Private Program; Ashek Mahmud khan; 01725-402592; = BC(A+ )+ C(A+ ) = (BC+ C) = C (B+ ) = C = R.H.S L.H.S = R.H.S (Proved) 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 L.H.S = R.H.S (Proved) (i) A+ B+ = 1 L.H.S = A+ B+ = A+ B+ ) = A+ .1 h‡kvi weÁvb I cÖhyw³ wek¦we`¨vj‡qi Computer Science & Engineering wefv‡Mi Qv·`i, GKv`k I Øv`k †kÖYxi Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó Gi Private Program……. †hvMv‡hvM: 01725-402592
- 15. 15 ICT Private Program; Ashek Mahmud khan; 01725-402592; = A+ =1 =R.H.S L.H.S = R.H.S (Proved) ‡g․wjK †MBU (Basic Gate) ‡MB‡Ui cÖKvi‡f` Kw¤^‡bkb jwRK †MBU (Combinational Logic Gate) jwRK †MBU: eywjqvb G¨vj‡Reivi e¨envwiK cÖ‡qv‡Mi Rb¨ †h mg¯Í wWwRUvj B‡iKUªwbK mvwK©U e¨envi 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‡`i‡K †g․wjK jwRK †MBU e‡j| †g․wjK jwRK †MBU wZb cÖKvi| (i) AND †MBU (ii) OR †MBU (iii) NOT †MBU
- 16. 16 ICT Private Program; Ashek Mahmud khan; 01725-402592; 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: h‡kvi weÁvb I cÖhyw³ wek¦we`¨vj‡qi Computer Science & Engineering wefv‡Mi Qv·`i, GKv`k I Øv`k †kÖYxi Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó Gi Private Program……. †hvMv‡hvM: 01725-402592
- 17. 17 ICT Private Program; Ashek Mahmud khan; 01725-402592; 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 e¨envi 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 e¨envi 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 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| 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. 18 ICT Private Program; Ashek Mahmud khan; 01725-402592; OR‡MBU : wP‡Î `yBwU bi †MB‡Ui BbcyU Ggb fv‡e e¨envi 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 e¨envi 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 e¨env‡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 e¨envi 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 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-
- 19. 19 ICT Private Program; Ashek Mahmud khan; 01725-402592; 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| h‡kvi weÁvb I cÖhyw³ wek¦we`¨vj‡qi Computer Science & Engineering wefv‡Mi Qv·`i, GKv`k I Øv`k †kÖYxi Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó Gi Private Program……. †hvMv‡hvM: 01725-402592
- 20. 20 ICT Private Program; Ashek Mahmud khan; 01725-402592; 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- h‡kvi weÁvb I cÖhyw³ wek¦we`¨vj‡qi Computer Science & Engineering wefv‡Mi Qv·`i, GKv`k I Øv`k †kÖYxi Avewk¨K welq ÒZ_¨ I †hvMv‡hvM cÖhyw³Ó Gi ICT Private Program……. †hvMv‡hvM: 01725-402592