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ØemleeJevee (MeerIeÇieefCeleeÛeer ieg®efkeâuueer)
efvemeie& cnCepes efJeefJeOe KeefpevÙeebveer vešuesuee vešmeceüeš Deens. efvemeiee&leerue keâener
Iešvee Ùee mJewjheCes Ie[leele lej keâener IešveebceOÙes cee$e efJeefMe° Deeke=âefleyebOe Demelees.
DeeleeheÙeËle Ie[uesuÙee, Ie[le DemeuesuÙee Je heg{s Ie[CeeNÙee ØelÙeskeâ Iešveebceeies keâenerlejer
keâejCe o[uesues Demeles. lÙeeÛee MeesOe IesCÙeeÛee ØeÙelve ceeveJeeves efJeefJeOe ceeOÙeceebletve
kesâuesuee Deens. ÙeeletveÛe efJe%eeve, KeieesueMeeŒe, DeLe&MeeŒe FlÙeeoer MeeŒeebÛeer efveefce&leer
Peeueer. met#ce efJeÛeej kesâuÙeeme Ùee meJe& MeeŒeebÛÙee cegUeMeer SkeâÛe efJe<eÙe Dee{Usue!
lees cnCepes ieefCele! ÙeeÛe keâejCeemleJe keâeue& øesâ[efjkeâ iee@me Ùee Leesj MeeŒe%eeves ieefCeleebme
efJe%eeveeÛeer jeCeer (Queen of Science) Demes cnšues Deens.
efJe%eeve, KeieesueMeeŒe, DeLe&MeeŒe FlÙeeoer MeeŒeebÛÙee Øeieleerle ieefCeleeÛee ceesueeÛee
Jeeše Deens. ØeeÛeerve keâeUer ieefCeleeÛee cegKÙe GheÙeesie JemletbÛeer ieCevee efkebâJee ceespeoeo
keâjCÙeemee"er nesle Demes. Ùeemee"er he[leeàÙeeÛÙee jsIee efkebâJee Fbkeâe meeceüepÙeeleerue
efkeäJehet veeJeeÛÙee iee" ceejuesuÙee oesNÙee Jeehe¤ve mebKÙeelcekeâ ceeefnleerÛeer veeWo "sJeueer
peele. hejbleg ner heæle HeâejMeer ØeYeeJeer veJnleer. heg{s JemletbceOÙes Peeuesueer Jee{ Jee Ieš
ÙeeÛeer veeWo "sJeCes megueYe JneJes Ùeekeâefjlee efÛevnebÛee GheÙeesie keâjCÙeele ÙesT ueeieuee.
lÙeeletve peieeÛÙee efJeefJeOe Yeeieeble jesceve mebKÙee heæle, oMeceeve heæle FlÙeeoer mebKÙee
heæleeRÛee GoÙe Peeuee. Ùee meJe& heæleeRle mejme heæle cnCepes Deemee ¢ee Jewefokeâ
keâeUeÛÙee DeejbYeer nesTve iesuesuÙee JeeÙeJÙe Yeejleele jenCeeNÙee YeejleerÙe ieefCele%eebÛÙee
ØeefleYesletve efvecee&Ce Peeuesueer oMeceeve heæle.
meeNÙee peieeuee oMeceeve heæleerÛes %eeve keâ¤ve osCÙeeÛee ceeve peelees lees meve
598 meeueer pevceuesuÙee yeÇÿeieghle Ùee YeejleerÙe ieefCeleer Je KeieesueMeeŒe%eeuee!
JeÙeeÛÙee 30 JÙee Je<eea cnCepes F. me. 628 meeueer lÙeeves ‘yeÇeÿe mHegâš efmeæeble’ lej
65 JÙee Je<eea ‘Keb[KeeÅekeâ’ ne KeieesueMeeŒeeÛee keâjCe «ebLe efueefnuee. 10 JÙee
Melekeâele Deued ceecetve Ùee KeefueHeâeves yeÇÿeieghleeÛÙee oesvner «eLeebÛeer Dejyeer Yee<esle
Yee<eeblejs keâ¤ve Iesleueer. ‘yeÇeÿe mHegâš’ efmeæebleeuee `efmebo efnbo' Je Keb[ KeeÅekeâeuee
`Deued Deke&âo' DeMeer veeJes osCÙeele Deeueer. Ùee oesve «ebLeebveerÛe Dejyeebvee oMeceeve
heæleerÛes %eeve keâ¤ve efoues. 1202 meeueer efueDeesveeoex Dee@Heâ efhemee THe&â efHeâyeesve@mmeer
Ùee FšeefueÙeve ieefCelep%eeves efueyesj Deye@meer ne «ebLe efuentve oMeceeve heæleerÛee Ùegjesheele
Øemeej meg¤ kesâuee. lejerner peJeUpeJeU 18 JÙee MelekeâeheÙeËle Ùegjesheele oMeceeve

iegCeekeâejeÛÙee Jesieele Jee{ nesCÙeeme ceole nesles. mebieCekeâMeeŒeeceOÙes 10000001
ner efÉceeve mebKÙee cnCepesÛe 1111111 Ùee efÉceeve mebKÙesÛee yegLe iegCekeâ nesÙe.
ÙeeJe¤ve mebKÙesÛes megueYe ceeb[Ceerleerue MetvÙeebÛee meceeJesMe DemeCeejs ¤he mebieCekeâeÛeer
iegCeekeâejieleer Jee{CÙeeme keâMeer keâejCeerYetle Deens ns mhe° nesles. 0 Ùee mebKÙesÛes
MeerIeÇieefCeleeleerue cenòJe ue#eele Ieslee `MeerIeÇieefCele cnCepes MetvÙeeÛeer efkeâceÙee'
Demes cnCeCes cnCepes DeefleMeÙeeskeäleer veener! ÙeeJe¤ve MetvÙeeÛee MeesOe Yeejleeves ueeJeuÙeeÛee
ØelÙeskeâ YeejleerÙeeme hegvne Skeâoe meeLe& DeefYeceeve Jeešsue Ùeele Mebkeâe veener.
Deepe Jewefokeâ ieefCele ns JesieJeeve, megueYe Je ceveesjbpekeâ heæleeRcegUs meJe&$e
ueeskeâefØeÙe nesle Deens. Jewefokeâ ieefCeleeÛÙee megueYe Je ceveesjbpekeâ heæleeRceeies o[uesueer
jnmÙesosKeerue megueYe ceeb[Ceer DeLeJee MetvÙeeÛÙee meeneÙÙeeves mhe° keâjlee Ùesles. Ùee
«ebLeeÛeer JÙeehleer ner ceÙee&efole DemeuÙeecegUs Jewefokeâ ieefCeleeleerue MeerIeÇ heæleer Je lÙeeÛes
keâejCe ÙeeÛee meKeesue DeYÙeeme keâjCes, ÙesLes MekeäÙe veener. heCe heg{s Fb«epeer Ùee JewefÕekeâ
Yee<esletve ØekeâeefMele nesCeeNÙee «ebLeeleerue JewefÕekeâ mebKÙee heæleerceeHe&âle (Global Number
System) Ùee meJe& iees°eRbÛee meKeesue Je MeeŒeMegæ DeYÙeeme, iegCeekeâejeÛeer Skeâ
DeefYeveJe heæle (VJ's Multiplication Method) Je mebKÙesÛee megiegCekeâ efceUJeCÙeeÛÙee
JesieJesieàÙee heæleer JeeÛekeâebheg{s "sJeCÙeeÛee ceePee ØeÙelve Demesue.
ieefCeleele nuueer JeehejuÙee peeCeeNÙee veesšMevehewkeâer keâenerÛe meesUeJÙee MelekeâeheÙeËle
MeesOeues iesues nesles. lÙee DeeOeer ieefCele ns kesâJeU Meyoeble JÙekeäle kesâues peele Demes.
MeyoebÛÙee yeespe[heCeecegUs ieefCeleeÛee Heâejmee efJekeâeme nesT Mekeâuesuee veJnlee. DeeOegefvekeâ
veesšsMevecegUs lep%eebmee"er ieefCele meesÙeerÛes, hejbleg veJeefMekeäÙeebmee"er DeefOekeâ efkeäue°
Peeues Deens. DeeOegefvekeâ veesšsMeve DeefleMeÙe mebef#ehle Deens. ceespekeäÙeeÛe cegUe#ejebceOÙes
ØeÛeb[ ceeefnleer oslee Ùesles. hee§eeòÙe mebieerleeÛÙee veesšsMeveØeceeCesÛe ieefCeleeÛÙee veesšsMeveÛes
keâ[keâ efveÙece Demetve les veesšsMeve pÙee ØekeâejÛeer ceeefnleer efueefKele ¤heele meebieles, leer
Flej keâesCelÙeener heæleerves JÙekeäle keâjCes peJeUpeJeU DeMekeäÙeÛe Deens. lÙeecegUs
DeeOegefvekeâ ieefCeleeuee `MeeŒeebÛeer mebef#ehle heefjYee<ee' Demes mebyeesOelee ÙesF&ue. Øemlegle
«ebLeele cee$e DeeOegefvekeâ veesšsMevecegUs efvecee&Ce nesCeejer efkeäue°lee šeUCÙeemee"er ØelÙeskeâ
met$e ns heefjÛÚsoeÛÙee DeLeJee ef›eâÙeemet$eeÛÙee ¤heeves yeesueer Yee<esletve mecepeeJeCÙeeÛee
ØeÙelve kesâuesuee Deens. ÙeeefMeJeeÙe hegmlekeâeleerue meJe& heæleer Ùee veJeefveefce&le Demetve
DeeJeMÙekeâ lesLes heæleeRceeieerue mebkeâuheveeosKeerue mhe° kesâuesueer Deens. ceeveJeer peerJeveeleerue
efkebâJee efJeefJeOe mheOee& hejer#eebleerue keâ"erCe ØeMve mees[JeCÙeeme ueeieCeejer leke&âMegæ
heæleer meeJe&ef$ekeâ Peeueer veener. oMeceeve heæleerÛee hetCe& mJeerkeâej kesâuÙeeveblejÛe Ùegjesheeleerue
ieefCele yenjues.
oMeceeve heæle ner mebKÙeebÛÙee ceeb[Ceermee"er ueeieCeejer mebef#ehle Je megueYe heæle
Demetve Ùee heæleerle keâesCeleerner mebKÙee 0, 1,2, 3, 4, 5, 6, 7, 8, 9 Ùee one
DebkeâebÛÙee meeneÙÙeeves ceeb[lee Ùesles. oMeceeve heæle Flej mebKÙee heæleeRhes#ee mejme
"jueer leer lÙeeleerue ÙeesiÙe Je mebef#ehle ceeb[CeerÛÙee meeceLÙee&cegUs! pÙeeÙeesies yesjerpe,
Jepeeyeekeâer, iegCeekeâej, Yeeieekeâej Ùee ØeeLeefcekeâ ef›eâÙee Flej mebKÙee heæleeRhes#ee (Goe.
jesceve mebKÙee heæle) megkeâj PeeuÙee. Lees[keäÙeele ``mebKÙesÛes ÙeesiÙe Je megueYe
ceeb[Ceerleerue ¤he'' (Number's proper representation) nerÛe Kejer
ØeeLeefcekeâ ef›eâÙee megkeâj keâjCÙeeÛeer DeLeJee MeerIeÇieCeveeÛeer ieg®efkeâuueer Deens!
Goe. 1245334µ125, 1234µ9,999,998 JejJej heenlee ns iegCeekeâej
[esF&pe[ Jeešleerue. heCe Ùeeleerue 125 Je 9,999,998 Ùee iegCekeâebÛeer ceeb[Ceer
Deveg›eâces 1000/8 Je 10,000,002 DeMeer keâje cnCepes iegCeekeâejebÛeer Gòejs
1556541750 Je 12339997532 DeMeer #eCeeOee&le osTve mJele:Ûe Ûeefkeâle
Jneue! Ùee ef"keâeCeer Skeâ Keyejoejer IÙeeÙeuee nJeer, leer cnCepes mebKÙesÛes ¤he efkebâJee
ceeb[Ceer yeouetve efceUCeejer mebKÙee ner cetU mebKÙesMeer mececetuÙe Demeueer heeefnpes. met#ce
efJeÛeej kesâuÙeeme, MeerIeÇieCeveeÛeer š@^Ûesveyeie& heæle, Jewefokeâ ieefCele efkebâJee mebieCekeâeuee
iegCeekeâej Ùee Øeef›eâÙesle Jesie Øeehle keâ¤ve osCeeje `yegLeÛee iegCekeâ' (Booth's Multiplier)
Ùee meJeeËÛÙee MeerIeÇlesceeieerue jnmÙeosKeerue mebKÙesÛes megueYe ceeb[Ceerleerue ¤he DemeuÙeeÛes
ue#eele ÙesF&ue. meesÙeerkeâefjlee mebKÙesÛÙee megueYe ceeb[Ceerleerue ¤heeme megiegCekeâ (Emultiplier) Demes mebyeesOet. megiegCekeâeÛeer DeÛetkeâ JÙeeKÙee keâjCes lemes keâ"erCe Deens. cetU
mebKÙeshes#ee peemle MetvÙeebÛee meceeJesMe DemeCeeNÙee mebKÙesÛÙee heÙee&Ùeer mececetuÙe ceeb[Ceer
(Equivalent easy representation) DeLeJee ¤heeme megiegCekeâ Demes cnCelee
ÙesF&ue. Ùee JÙeeKÙesvegmeej 1,999,989 Ûee megiegCekeâ 2,0000,011 ns ¤he lej
4,444,444 Ûee megiegCekeâ (4/9)µ106-(4/9) ner ceeb[Ceer nesF&ue.
efÉceeve mebKÙee heæleerle (Binary Number System) 1,111,111 (oMeceeve
127) Ùee mebKÙesves Flej mebKÙesme, heejbheeefjkeâ iegCeekeâej heæleerves, iegCeeÙeÛes cnšues
lej 6 JesUe yesjpesÛeer Øeef›eâÙee keâjeJeer ueeieles, heCe neÛe iegCeekeâej pej cetU iegCekeâeÛes
meeshes ¤he cnCepesÛe 1,111,111 Ûes meeshes ¤he · 10000001 ne iegCekeâ
ceevetve kesâuÙeeme yesjpesÛeer Øeef›eâÙee keâceer JesUe (ÙesLes Heâkeäle 1 JesUe) keâjeJeer ueeieles Je

efJeÛeejMekeäleer Je MeerIeÇieCeve#ecelee Jee{JeCes ne Ùee «ebLeeÛee cegKÙe nslet Deens. lÙeecegUs
JeeÛekeâebme ns hegmlekeâ jbpekeâ Jeešsue Ùeele Mebkeâe veener. DeeJeMÙekeâlee Deens leer Heâkeäle
[esUmeheCes JeeÛeCÙeeÛeer, mejeJeeletve jbpekeâlee DevegYeJeCÙeeÛeer, keâe? Je keâmes? Ùee
MeesOeebÛeer peveveer DemeCeeNÙee oesve ØekeâejÛÙee ØeMveebveer ØelÙeskeâ ueIegheæleeRceeieerue
mebkeâuhevee (Concept behind shortcuts) MeesOeCÙeeÛee ØeÙelve keâjCÙeeÛeer!
mejlesMesJešer Úboceeuesleerue ns heg<hejlve JeeÛekeâebme meceefhe&le.
–efJeúue yeeueepeer peeOeJe

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Manoranjak Sheeghraganit

  • 1. ØemleeJevee (MeerIeÇieefCeleeÛeer ieg®efkeâuueer) efvemeie& cnCepes efJeefJeOe KeefpevÙeebveer vešuesuee vešmeceüeš Deens. efvemeiee&leerue keâener Iešvee Ùee mJewjheCes Ie[leele lej keâener IešveebceOÙes cee$e efJeefMe° Deeke=âefleyebOe Demelees. DeeleeheÙeËle Ie[uesuÙee, Ie[le DemeuesuÙee Je heg{s Ie[CeeNÙee ØelÙeskeâ Iešveebceeies keâenerlejer keâejCe o[uesues Demeles. lÙeeÛee MeesOe IesCÙeeÛee ØeÙelve ceeveJeeves efJeefJeOe ceeOÙeceebletve kesâuesuee Deens. ÙeeletveÛe efJe%eeve, KeieesueMeeŒe, DeLe&MeeŒe FlÙeeoer MeeŒeebÛeer efveefce&leer Peeueer. met#ce efJeÛeej kesâuÙeeme Ùee meJe& MeeŒeebÛÙee cegUeMeer SkeâÛe efJe<eÙe Dee{Usue! lees cnCepes ieefCele! ÙeeÛe keâejCeemleJe keâeue& øesâ[efjkeâ iee@me Ùee Leesj MeeŒe%eeves ieefCeleebme efJe%eeveeÛeer jeCeer (Queen of Science) Demes cnšues Deens. efJe%eeve, KeieesueMeeŒe, DeLe&MeeŒe FlÙeeoer MeeŒeebÛÙee Øeieleerle ieefCeleeÛee ceesueeÛee Jeeše Deens. ØeeÛeerve keâeUer ieefCeleeÛee cegKÙe GheÙeesie JemletbÛeer ieCevee efkebâJee ceespeoeo keâjCÙeemee"er nesle Demes. Ùeemee"er he[leeàÙeeÛÙee jsIee efkebâJee Fbkeâe meeceüepÙeeleerue efkeäJehet veeJeeÛÙee iee" ceejuesuÙee oesNÙee Jeehe¤ve mebKÙeelcekeâ ceeefnleerÛeer veeWo "sJeueer peele. hejbleg ner heæle HeâejMeer ØeYeeJeer veJnleer. heg{s JemletbceOÙes Peeuesueer Jee{ Jee Ieš ÙeeÛeer veeWo "sJeCes megueYe JneJes Ùeekeâefjlee efÛevnebÛee GheÙeesie keâjCÙeele ÙesT ueeieuee. lÙeeletve peieeÛÙee efJeefJeOe Yeeieeble jesceve mebKÙee heæle, oMeceeve heæle FlÙeeoer mebKÙee heæleeRÛee GoÙe Peeuee. Ùee meJe& heæleeRle mejme heæle cnCepes Deemee ¢ee Jewefokeâ keâeUeÛÙee DeejbYeer nesTve iesuesuÙee JeeÙeJÙe Yeejleele jenCeeNÙee YeejleerÙe ieefCele%eebÛÙee ØeefleYesletve efvecee&Ce Peeuesueer oMeceeve heæle. meeNÙee peieeuee oMeceeve heæleerÛes %eeve keâ¤ve osCÙeeÛee ceeve peelees lees meve 598 meeueer pevceuesuÙee yeÇÿeieghle Ùee YeejleerÙe ieefCeleer Je KeieesueMeeŒe%eeuee! JeÙeeÛÙee 30 JÙee Je<eea cnCepes F. me. 628 meeueer lÙeeves ‘yeÇeÿe mHegâš efmeæeble’ lej 65 JÙee Je<eea ‘Keb[KeeÅekeâ’ ne KeieesueMeeŒeeÛee keâjCe «ebLe efueefnuee. 10 JÙee Melekeâele Deued ceecetve Ùee KeefueHeâeves yeÇÿeieghleeÛÙee oesvner «eLeebÛeer Dejyeer Yee<esle Yee<eeblejs keâ¤ve Iesleueer. ‘yeÇeÿe mHegâš’ efmeæebleeuee `efmebo efnbo' Je Keb[ KeeÅekeâeuee `Deued Deke&âo' DeMeer veeJes osCÙeele Deeueer. Ùee oesve «ebLeebveerÛe Dejyeebvee oMeceeve heæleerÛes %eeve keâ¤ve efoues. 1202 meeueer efueDeesveeoex Dee@Heâ efhemee THe&â efHeâyeesve@mmeer Ùee FšeefueÙeve ieefCelep%eeves efueyesj Deye@meer ne «ebLe efuentve oMeceeve heæleerÛee Ùegjesheele Øemeej meg¤ kesâuee. lejerner peJeUpeJeU 18 JÙee MelekeâeheÙeËle Ùegjesheele oMeceeve iegCeekeâejeÛÙee Jesieele Jee{ nesCÙeeme ceole nesles. mebieCekeâMeeŒeeceOÙes 10000001 ner efÉceeve mebKÙee cnCepesÛe 1111111 Ùee efÉceeve mebKÙesÛee yegLe iegCekeâ nesÙe. ÙeeJe¤ve mebKÙesÛes megueYe ceeb[Ceerleerue MetvÙeebÛee meceeJesMe DemeCeejs ¤he mebieCekeâeÛeer iegCeekeâejieleer Jee{CÙeeme keâMeer keâejCeerYetle Deens ns mhe° nesles. 0 Ùee mebKÙesÛes MeerIeÇieefCeleeleerue cenòJe ue#eele Ieslee `MeerIeÇieefCele cnCepes MetvÙeeÛeer efkeâceÙee' Demes cnCeCes cnCepes DeefleMeÙeeskeäleer veener! ÙeeJe¤ve MetvÙeeÛee MeesOe Yeejleeves ueeJeuÙeeÛee ØelÙeskeâ YeejleerÙeeme hegvne Skeâoe meeLe& DeefYeceeve Jeešsue Ùeele Mebkeâe veener. Deepe Jewefokeâ ieefCele ns JesieJeeve, megueYe Je ceveesjbpekeâ heæleeRcegUs meJe&$e ueeskeâefØeÙe nesle Deens. Jewefokeâ ieefCeleeÛÙee megueYe Je ceveesjbpekeâ heæleeRceeies o[uesueer jnmÙesosKeerue megueYe ceeb[Ceer DeLeJee MetvÙeeÛÙee meeneÙÙeeves mhe° keâjlee Ùesles. Ùee «ebLeeÛeer JÙeehleer ner ceÙee&efole DemeuÙeecegUs Jewefokeâ ieefCeleeleerue MeerIeÇ heæleer Je lÙeeÛes keâejCe ÙeeÛee meKeesue DeYÙeeme keâjCes, ÙesLes MekeäÙe veener. heCe heg{s Fb«epeer Ùee JewefÕekeâ Yee<esletve ØekeâeefMele nesCeeNÙee «ebLeeleerue JewefÕekeâ mebKÙee heæleerceeHe&âle (Global Number System) Ùee meJe& iees°eRbÛee meKeesue Je MeeŒeMegæ DeYÙeeme, iegCeekeâejeÛeer Skeâ DeefYeveJe heæle (VJ's Multiplication Method) Je mebKÙesÛee megiegCekeâ efceUJeCÙeeÛÙee JesieJesieàÙee heæleer JeeÛekeâebheg{s "sJeCÙeeÛee ceePee ØeÙelve Demesue. ieefCeleele nuueer JeehejuÙee peeCeeNÙee veesšMevehewkeâer keâenerÛe meesUeJÙee MelekeâeheÙeËle MeesOeues iesues nesles. lÙee DeeOeer ieefCele ns kesâJeU Meyoeble JÙekeäle kesâues peele Demes. MeyoebÛÙee yeespe[heCeecegUs ieefCeleeÛee Heâejmee efJekeâeme nesT Mekeâuesuee veJnlee. DeeOegefvekeâ veesšsMevecegUs lep%eebmee"er ieefCele meesÙeerÛes, hejbleg veJeefMekeäÙeebmee"er DeefOekeâ efkeäue° Peeues Deens. DeeOegefvekeâ veesšsMeve DeefleMeÙe mebef#ehle Deens. ceespekeäÙeeÛe cegUe#ejebceOÙes ØeÛeb[ ceeefnleer oslee Ùesles. hee§eeòÙe mebieerleeÛÙee veesšsMeveØeceeCesÛe ieefCeleeÛÙee veesšsMeveÛes keâ[keâ efveÙece Demetve les veesšsMeve pÙee ØekeâejÛeer ceeefnleer efueefKele ¤heele meebieles, leer Flej keâesCelÙeener heæleerves JÙekeäle keâjCes peJeUpeJeU DeMekeäÙeÛe Deens. lÙeecegUs DeeOegefvekeâ ieefCeleeuee `MeeŒeebÛeer mebef#ehle heefjYee<ee' Demes mebyeesOelee ÙesF&ue. Øemlegle «ebLeele cee$e DeeOegefvekeâ veesšsMevecegUs efvecee&Ce nesCeejer efkeäue°lee šeUCÙeemee"er ØelÙeskeâ met$e ns heefjÛÚsoeÛÙee DeLeJee ef›eâÙeemet$eeÛÙee ¤heeves yeesueer Yee<esletve mecepeeJeCÙeeÛee ØeÙelve kesâuesuee Deens. ÙeeefMeJeeÙe hegmlekeâeleerue meJe& heæleer Ùee veJeefveefce&le Demetve DeeJeMÙekeâ lesLes heæleeRceeieerue mebkeâuheveeosKeerue mhe° kesâuesueer Deens. ceeveJeer peerJeveeleerue efkebâJee efJeefJeOe mheOee& hejer#eebleerue keâ"erCe ØeMve mees[JeCÙeeme ueeieCeejer leke&âMegæ
  • 2. heæleer meeJe&ef$ekeâ Peeueer veener. oMeceeve heæleerÛee hetCe& mJeerkeâej kesâuÙeeveblejÛe Ùegjesheeleerue ieefCele yenjues. oMeceeve heæle ner mebKÙeebÛÙee ceeb[Ceermee"er ueeieCeejer mebef#ehle Je megueYe heæle Demetve Ùee heæleerle keâesCeleerner mebKÙee 0, 1,2, 3, 4, 5, 6, 7, 8, 9 Ùee one DebkeâebÛÙee meeneÙÙeeves ceeb[lee Ùesles. oMeceeve heæle Flej mebKÙee heæleeRhes#ee mejme "jueer leer lÙeeleerue ÙeesiÙe Je mebef#ehle ceeb[CeerÛÙee meeceLÙee&cegUs! pÙeeÙeesies yesjerpe, Jepeeyeekeâer, iegCeekeâej, Yeeieekeâej Ùee ØeeLeefcekeâ ef›eâÙee Flej mebKÙee heæleeRhes#ee (Goe. jesceve mebKÙee heæle) megkeâj PeeuÙee. Lees[keäÙeele ``mebKÙesÛes ÙeesiÙe Je megueYe ceeb[Ceerleerue ¤he'' (Number's proper representation) nerÛe Kejer ØeeLeefcekeâ ef›eâÙee megkeâj keâjCÙeeÛeer DeLeJee MeerIeÇieCeveeÛeer ieg®efkeâuueer Deens! Goe. 1245334µ125, 1234µ9,999,998 JejJej heenlee ns iegCeekeâej [esF&pe[ Jeešleerue. heCe Ùeeleerue 125 Je 9,999,998 Ùee iegCekeâebÛeer ceeb[Ceer Deveg›eâces 1000/8 Je 10,000,002 DeMeer keâje cnCepes iegCeekeâejebÛeer Gòejs 1556541750 Je 12339997532 DeMeer #eCeeOee&le osTve mJele:Ûe Ûeefkeâle Jneue! Ùee ef"keâeCeer Skeâ Keyejoejer IÙeeÙeuee nJeer, leer cnCepes mebKÙesÛes ¤he efkebâJee ceeb[Ceer yeouetve efceUCeejer mebKÙee ner cetU mebKÙesMeer mececetuÙe Demeueer heeefnpes. met#ce efJeÛeej kesâuÙeeme, MeerIeÇieCeveeÛeer š@^Ûesveyeie& heæle, Jewefokeâ ieefCele efkebâJee mebieCekeâeuee iegCeekeâej Ùee Øeef›eâÙesle Jesie Øeehle keâ¤ve osCeeje `yegLeÛee iegCekeâ' (Booth's Multiplier) Ùee meJeeËÛÙee MeerIeÇlesceeieerue jnmÙeosKeerue mebKÙesÛes megueYe ceeb[Ceerleerue ¤he DemeuÙeeÛes ue#eele ÙesF&ue. meesÙeerkeâefjlee mebKÙesÛÙee megueYe ceeb[Ceerleerue ¤heeme megiegCekeâ (Emultiplier) Demes mebyeesOet. megiegCekeâeÛeer DeÛetkeâ JÙeeKÙee keâjCes lemes keâ"erCe Deens. cetU mebKÙeshes#ee peemle MetvÙeebÛee meceeJesMe DemeCeeNÙee mebKÙesÛÙee heÙee&Ùeer mececetuÙe ceeb[Ceer (Equivalent easy representation) DeLeJee ¤heeme megiegCekeâ Demes cnCelee ÙesF&ue. Ùee JÙeeKÙesvegmeej 1,999,989 Ûee megiegCekeâ 2,0000,011 ns ¤he lej 4,444,444 Ûee megiegCekeâ (4/9)µ106-(4/9) ner ceeb[Ceer nesF&ue. efÉceeve mebKÙee heæleerle (Binary Number System) 1,111,111 (oMeceeve 127) Ùee mebKÙesves Flej mebKÙesme, heejbheeefjkeâ iegCeekeâej heæleerves, iegCeeÙeÛes cnšues lej 6 JesUe yesjpesÛeer Øeef›eâÙee keâjeJeer ueeieles, heCe neÛe iegCeekeâej pej cetU iegCekeâeÛes meeshes ¤he cnCepesÛe 1,111,111 Ûes meeshes ¤he · 10000001 ne iegCekeâ ceevetve kesâuÙeeme yesjpesÛeer Øeef›eâÙee keâceer JesUe (ÙesLes Heâkeäle 1 JesUe) keâjeJeer ueeieles Je efJeÛeejMekeäleer Je MeerIeÇieCeve#ecelee Jee{JeCes ne Ùee «ebLeeÛee cegKÙe nslet Deens. lÙeecegUs JeeÛekeâebme ns hegmlekeâ jbpekeâ Jeešsue Ùeele Mebkeâe veener. DeeJeMÙekeâlee Deens leer Heâkeäle [esUmeheCes JeeÛeCÙeeÛeer, mejeJeeletve jbpekeâlee DevegYeJeCÙeeÛeer, keâe? Je keâmes? Ùee MeesOeebÛeer peveveer DemeCeeNÙee oesve ØekeâejÛÙee ØeMveebveer ØelÙeskeâ ueIegheæleeRceeieerue mebkeâuhevee (Concept behind shortcuts) MeesOeCÙeeÛee ØeÙelve keâjCÙeeÛeer! mejlesMesJešer Úboceeuesleerue ns heg<hejlve JeeÛekeâebme meceefhe&le. –efJeúue yeeueepeer peeOeJe