<NOTE>

<HEAD1>@{<E>BODMAS</E>@}-er bad~mAyesI</HEAD1>

<U>arNab chak+rabar+tI, inDiyAn sTyATis+TikAl insTiTiuT, kalkAtA</U> <BR/>

<U>siddhArthasha.nkar chaT+TopAdhyAy, abasarprApta shixak, bidhAn~nagar gabha.H hAi skul </U>

<P/>

skulshixAr jagate prAysha.Hi kono kichhu mukhastha rAkhAr janya chaT.h jaldi baTikAr dekhA pAoyA yAy. mogal bAdshAharA

parYabasita han bAbAr jwar sArAnor kAhinIte (''bAbAr haila AbAr jwar sArila \ouSadhe''), kendrashAsita anchal~gulo chapalA

seje didimAr dore dhAk+kA dey (''a didA go Ami chapalA'', kendrashAsita anchal~gulor nAmAbalI mukhastha rAkhAr

ei ToTkATi abashya Adhunik bhArabarSe Ar kAj kare nA). tA, buddhi nA khATiye shar+TkATe mere debAr eisab kAydAte

nAnA bipatti ghaTe. sei yeman ekTi chhAtrer galpa shunechhilAm, ye khAmokhA bh\outabijnAner @{<M>P=I^2R</M>@}

farmulATA guliye fele @{<M>P=R^2I</M>@} kare felta. tAke ke ekjan mane rAkhAr amogh nidAn dila--

<Q>

@{<E>Twinkle twinkle little star,<BR/>

<M>P</M> is equal to <M>I^2R</M>

</E>@}</Q>

shune se bhArI khushi. parIxAr hale yei fer tAr mAthA guliyechhe, amni se mane mane Abrritti shuru karla--

<Q>@{<E>Twinkle twinkle little star<BR/>

How I wonder what you are<BR/>

Up above the world so high<BR/>

<M>P</M> is equal to <M>R^2I.</M>

</E>@}</Q>

tA, eTA nichhak bAnAno ekTA hAsir galpa hala. ebAr ekTA satyikArer galpa balba, yeTA ``shunle pare hAsir cheye kAnnA Ase

beshI!''

<P/>

ei galper nAyk hala @{<E>BODMAS.</E>@} er sange parichay nei, bangadeshe eman skul~paRuyA melA bhAr. tAo ekTu goRA thekei shuru

kari. AmrA yesab sa.nkhyA niye kAj kari, tAder janya chArTe @{<E>operation</E>@} Achhe, yog, biyog, guN, bhAg. erA sakalei

@{<E>binary operation,</E>@} arthA.t duTo sa.nkhyAke miliye ekTA sa.nkhyAte pariNata kare (yadio bhAger belAy ekTu bARti

samasyA Achhe, shUnya diye bhAg karA yAy nA).

@{<E>Binary operation</E>@}-der byabahAr karte hale bale dite hay kon.h duTo sa.nkhyAke

melAno habe. seiTA

bojhAnor janya AmrA bryAkeT byabahAr kari. yeman @{<M>(2+3)-5</M>@}-er madhye duTo @{<E>binary operation</E>@} rayechhe

@{<M>+</M>@} Ar @{<M>-.</M>@} pratiTAr bhoger janya ek joRA kare sa.nkhyAo n\oibedya deoyA Achhe, @{<M>+</M>@}-er janya

@{<M>2</M>@} Ar @{<M>3,</M>@} eba.n @{<M>-</M>@}-TAr janya @{<M>(2+3)</M>@} eba.n @{<M>5.</M>@} kintu ebhAbe pratiTA

@{<E>binary operation</E>@}-er n\oibedyake AlAdA kare p,nAchil tule bhAg karte hale anek anek

bryAkeT ese k,nATAfolAno shajArur mata dekhAbe, yeman

@{<D>(3\times(4+ 5))-((4\times 4)\div 6).</D>@}

e jinis likhteo kaSTa, paRteo kaSTa. sei kAraNe loke ekTA prathA srriSTi karechhe, yAte kichhu kichhu bryAkeT

Ar AlAdA kare nA likhleo AmrA oder upasthiti bujhe nite pAri. Thik yeman mASTAr~mashAi nAm DAkAr samaye yei balen ``choddo'',

amni chhAtrarA bujhe ney ``rol nambar choddo klAse hAjir thAkle sARA dAo,'' serakam Ar ki! sei prathATA ei--

<Q>prathame dekhte habe

sabcheye gabhIre kI kI bryAkeT Achhe (mAne yAder bhitare Ar kono bryAkeT nei). tAr bhitare @{<E>operation</E>@}-guloke b,nAdik

theke DAn~dike paRte thAko, yadi kono guN bA bhAg pAo, ta.txaNA.t seTA kare nAo. eibhAbe chalte chalte ek samaye sab guNbhAg

karA haye yAbe. ebAr fer b,nAdik theke shuru karo, ebAr ekai kAj karbe yog biyog niye. yei puroTA karA haye gela takhan

Ar ei bryAkeTTA rAkhAr darkAr raila nA. ebAr AbAr b,nAdik theke shuru karo.

</Q>

ei hala niyam. er madhye mahAn kono gANitik tattwa nei, sref lekhAr samaye parishram b,nAchAnor

janya ekTA prathA mAtra. yeman, ei niyam mAnle ekTu Ager sajAruTA tAr chAr~joRA bryAkeTer tin joRAi mochan kare ei bhadrasabhya

rUp lAbh karbe--

@{<D>3\times(4+ 5)-4\times 4\div 6.</D>@}

bAki bryAkeTgulor as+titwa ekhano Achhe, kintu likhe bojhAte hachchhe nA.

ArekTA udAharaN dekhi--

@{<D>

6\times5\times4\div8\div5\times7

=

((((6\times5)\times4)\div8)\div5)\times7

</D>@}

ekhAne khAli guN Ar bhAg chhila, tAi b,nAdik theke DAn~dike kare gelei hala. ArekTu jaTil

udAharaN habe--

@{<D>2+3\times4\div (5+4) = 2+((3\times4)\div (5+4)).</D>@}

<P/>

kAydATA khubai kaThin lAgla ki? yAi hok, skulshixAr \oitihya mene er janyao loke shar+TkAT ber

karechhe. erakamai ekTA shar+TkATer nAm hala @{<E>BODMAS.</E>@}

<HEAD2>@{<E>BODMAS</E>@} kI?</HEAD2>

@{<E>BODMAS</E>@} hala ekTA sa.nxipta rUp, ekhAne @{<E>B</E>@} mAne bryAkeT, @{<E>O</E>@} mAne

@{<E>of</E>@}, @{<E>D</E>@} mAne @{<E>division,</E>@} @{<E>M</E>@} mAne

@{<E>multiplication</E>@}, @{<E>A</E>@} hala @{<E>addition</E>@} Ar @{<E>S</E>@} hala giye

@{<E>subtraction.</E>@} ei shar+TkATer bak+tabya hala--prathame bryAkeTer bhitarer kAj karo,

tArpar @{<E>of</E>@}, tArpar bhAg, tArpar guN, tArpar yog eba.n abasheSe biyog.

<P/>

dekhte cham.tkAr, khAli tinTe samasyA. eke eke bali.

<HEAD2>pratham samasyA</HEAD2>

dharo ballAm @{<M>1-2+3</M>@} bAr karte. tumi amni @{<E>BODMAS</E>@} mene prathame yogTA karbe, fale pAbe @{<M>1-5.</M>@}

ebAr biyogTA karle pAbe @{<M>-4,</M>@} yeTA bhul. amni tumi balbe, AhAhA, Ami amukchandra tamuker baite paRechhi, ``yadi

parpar kebalmAtra yog Ar biyog thAke tAhA haile b,nAdik haite Arambha karibe'', ataeb prathame @{<M>1-2=-1</M>@} eba.n

tArpare @{<M>-1+3=2</M>@} haoyA uchit. Thik kathA. kintu tAhale @{<E>BODMAS</E>@}-TA keman~dhArA niyam hala? spaSTa balchhe

@{<E>S</E>@}-er Age @{<E>A,</E>@} athacha seTA sabsamaye karA chalbe nA. a.nker duniyAy erakam niyamer kono sthAnai thAkte

pAre nA. Asale @{<E>D</E>@} eba.n @{<E>M</E>@}-er madhye ki.nbA @{<E>A</E>@} eba.n @{<E>S</E>@}-er madhye kArorai kono

agrAdhikAr nei. shar+TkAT~oyAlArA besh ekTA uchchAraNYogya shar+TkAT bAnAnor neshAy oibhAbe oder sAjiyechhen. kono kono

ajna mAnuS seTAke a.nkashAs+t+rer bedabAkya dhare chhAtrader gelAbAr cheSTA karen.

<HEAD2>dwitIya samasyA</HEAD2>

exuNi ye amuk chandra tamuker bai theke ``b,nAdik haite DAn~dike yAibe'' ballAm, sei kathATA kintu

atyanta darkArI. yeman yadi @{<M>3-1-1</M>@} likhi, tabe kon.h biyogTA Age karchhi, tAr upar nirbhar kare uttarTA badle

yAbe. tAi se biSaye ekTA niyam thAkte habe. eba.n ``b,nAdik haite DAn~dike yAibe''-TAi hala sei niyam. @{<E>BODMAS</E>@}

sei biSaye sampUrNa nIrab.

<HEAD2>trritIya samasyA</HEAD2>

@{<E>BODMAS</E>@}-er madhye @{<E>of</E>@} bale ekTi @{<E>operation</E>@}-er ul+lekh Achhe.

du.Hkher kathA, ei nAme kono @{<E>operation</E>@}

a.nker duniyAy nei. <RED>kathATA shune anekei chamke uThben, kAraN chhelebelA theke AmrA sakalei ei

``@{<E>of</E>@}''-er galpa shune Aschhi. eman ki @{<E>1903</E>@} sAler ekTA prashnapatreo ei ul+lekh pAoyA yAy--

<CIMG web="bodmasqn.png">janAb shAmsul Alamer @{<E>Facebook</E>@} pAtA theke pAoyA</CIMG>

shatAdhik bachhar yAba.t ye jinis chale Aschhe, tAke haThA.t ek fu.tkAre uRiye dile Apatti uThtei pAre. abashya bahudin

dhare chale elei ye kono dhAraNA saThik hatei habe, eman kono kathA nei. satIdAha prathAr \oitihya er chAiteo gabhIratara

chhila. kintu yAi hok, ei ``@{<E>of</E>@}''-er byApArTA ekTu

khule balA darkAr. mAdhyamik skulshixAr ganDIr madhye AmrA yesab sa.nkhyA niye kAj kari, tArA

hala @{<E>real number</E>@} bA bAs+tab sa.nkhyA. a.nker duniyAr sab jiniser matai ederao

yAbatIya dhar+ma, AchAr, AcharaN suspSTa niyame b,nAdhA. sei niyamgulike bale ekekTA

@{<E>axiom.</E>@} eder sampUrNa tAlikA yekono @{<E>real analysis</E>@}-er baitei pAoyA yAy. eder

upar ekbAr chokhbolAlei dekhA yAbe ye ekhAne yog eba.n guNer kathA Achhe, yoger ul+To hisebe

biyoger kathA Achhe, eba.n guNer ul+To hisebe bhAger kathAo Achhe. kintu ``@{<E>of</E>@}'' nAme

kono kichhur kathAi nei. tAhale ``@{<E>of</E>@}''-TA elo kothA theke? tAr spaSTa kAraN jAni nA,

tAi anumAner upar nirbhar kare balchhi.</RED> a.nka

jinisTA @{<E>abstract</E>@},

kichhu @{<E>symbol</E>@} nARAchARAr

niyamkAnun mAtra. bAs+tab jagater bibhinna indriyagrAhya jinis~ke AmrA sei bhASAy prakAsh karte

pAri. yeman--

<Q>

bAbA duTo Apel dila, mA Aro tinTe dila, moT Apel kaTA hala?

</Q>

ei prashnaTA a.nker bhASAy haye yAy @{<M>2+3=5.</M>@} AbAr yadi bali

<Q>dui mAil uttare yAbAr par Arao tin mAil uttare gelAm,

tabe sab miliye kata dUre gelAm?</Q>

erao gANitik rUp sei @{<M>2+3=5.</M>@} arthA.t ''eksange miliye goNA'' eba.n ''ekai dike Aro dUratwa yAoyA'' ei duii a.nker

bhASAy hala yog. kono bAs+tab samasyAy a.nka prayog karAr Age sar+badAi samasyATake prathame a.nker bhASAy anubAd kare nite

hay. seTA sahaj eman dAbI karchhi nA, eba.n yekono shixakerai ekTA guru dAyitwa hala ei anubAd prak+riyATi shekhAno. eba.n

sekhAneo shar+TkATe mArte giye samasyA b,nAdhe. kIbhAbe bali.

<P/>

''@{<E>He became nervous.</E>@}'' bAkyer bangAnubAd hisebe yadi keu lekhe ''se nArbhAs haye paRla'', tabe seTAke sampUrNa anubAd

balA yAy nA, kAraN ''@{<E>nervous</E>@}'' bA ''nArbhAs'' yAi likhi nA kena, oTA bA.nlA shabda nay.

anubAdkArI ekhAne kAje f,nAki diyechhe.

i.nrAjir belAy yerakam a.nke belAteo tAi. anubAdTA sampUrNa karA chAi. ardhekTA kare bAkiTuku ``bojhAi yAchchhe'' bale chAliye

deoyATA Thik nay. Thik sei samasyA thekei @{<E>of</E>@}-er janma. yadi balte chAi--

<Q> ghare tinTe meye Ar

p,nAchTi chhele Achhe, tAder

ardheker madhye chab+bishTi lajens samAn bhAg kare dite hale pratyeke kaTA kare pAbe?</Q>

tabe tAr gANitik rUp habe

@{<D>24\div (*(~[[12]]\times(3+5)~)*).</D>@}

seTA nA kare anekei shar+TkATe mere debAr cheSTA karen, eba.n sambhabata.H sekhAn thekei ``@{<E>of</E>@}''-er u.tpatti--

@{<D>24\div [[12]]\text{ of }(3+5).</D>@}

eTA dekhte sahajatara, eba.n mUl barNanAr sange er sAdrrishyao beshI. eba.n f,nAkiTukur doS DhAkte bAdhya hayei mithye balte

hay ye ``@{<E>of</E>@}''-o Asale a.nkerai ekTA @{<E>operation.</E>@} oTA ye Asale guN ete sandeha nei, kintu oi guNTA bhAger

Age nA karle hiseb milbe nA, ataeb fer ekTA g,nojA chApAte hay @{<E>of</E>@}-TA guN hayeo guNAtIt, bhAgerao Age or sthAn.

eibhAbe natun ekTA @{<E>operation</E>@} bAniye a.nker madhye g,nuje deoyA bipajjanak.

a.nker sUtragulo khub yatna kare t\oirI karA, yAte oder madhye kothAo durakam mAne haye yAbAr suYog nA thAke. tAr janya

bis+tar loke mAthA khATiyechhe. AmrA yadi shar+TkAT karAr janya natun natun niyam DhokAte thAki, tabe tAr madhye f,nAk theke

yAy bis+tar. yeman ekdal srrijanshIl guNI mAnuS ei @{<E>of</E>@}-er janapriyatAy u.tsAhita haye ArekTi

guNAtIt guNer srriSTi kare base Achhen

<Q>@{<M>[[12]](3+5)</M>@} hala @{<M>[[12]]\times (3+5)</M>@}-erai mata, kintu eTio

@{<E>of</E>@}-er mata, bhAger Age karte habe. </Q>

ei jinis ye chhAtrerA shikhbe tAder ki durabasthA tAr jAjwalyamAn ekTi nidarshan samp+rati najare esechhe. soshAl miDiyAy

keu ekjan likhechhilen @{<M>64\div 2(2+2)</M>@} kata habe? mAS+TAr~mashAyrA ei niye dui dale bhAg haye giyechhen, keu

balchhen @{<M>128,</M>@}

AbAr keu balchhen @{<M>8.</M>@} mAS+TAr~mashAyderai yadi ei durgati, tabe t,nAder chhAtrader nA jAni ki abasthA. ei prasange

rabIndranAther gorA upanyAser ekTA jAygA mane paRla. gorAr dAdA mahim chhApoSA mAnuS, meyer biye deoyAr janya hanye haye

pAtra kh,nujchhilen. eman samaye gorAr bandhu binayke dekhe tAke jAmAi karAr bAsnAy tARA karechhen. begatik dekhe binay balechhe

iye dekhun, AmAr paribArer ekTA niyam Achhe, AmrA aghrAN mAse biye kari nA. ei kathAy hatAsh mahimer sakhed uk+ti--''eke to

poRA deshe shubhadin kh,nujei pAoyA yAy nA, tAr upar AbAr ghare ghare prAibheT p,nAji khule basle

kAjkarma chalbe kI kare?''

Ajker yug kampiuTArer yug. eirakam prAibheT p,nAji niye chalte gele sekhAneo h,nochaT khete habe.

AmrA bIjgaNita shekhAr samaye shikhechhi @{<M>2x</M>@} Asale @{<M>2\times x.</M>@} ataeb @{<M>x = 2+2</M>@} hale

@{<M>64\div2x = 64\div 2\times x.</M>@} ebAr AmrA kampiuTAre bA kyAlkulaTare yakhan likhba @{<CODE>64/2*x</CODE>@} takhan

uttarTA moTei milbe nA. takhan AmrA kon.hTA rAkhba, prAibheT p,nAji nAki kampiuTAr? nAki balba, @{<M>2x</M>@} mAne sab

samaye @{<M>2\times x</M>@} nay?

<HEAD2>konTA Thik?</HEAD2>

e lekhATi etaxaN y,nArA dh\oirYya kare paRechhen, t,nAder anekei hayto balben, se to nA hay bujhlAm, kintu sarkArI baiteo

to e biSaye spaSTa nirdesh nei. chintAr biSay b\oi ki! Antar+jAtik Adhunik gaNitshAs+t+ra kI balechhe tAr hAoyA to skuler

pAl ab.hdhi baRa ekTA p,n\ouchhoy nA. sekhAne silebAs~praNetArAi beshI gurutwer adhikArI. shunechhi AmAder

desher anunnata kichhu grAme

sthAnIya dArogAr etai dApaT

ye, sekhAnkAr lokeder kAchhe dArogAr kathAi desher Ain, tAr biruddhe ApIlao chale nA. tabu

baltei hay ye deshe Ain bale satyii AlAdA kichhu ekTA Achhe,

eba.n yadi ei swechchhAchArI dArogAke shAyes+tA kartei hay, tabe sei Asal Ainer dwArastha haoyAi sar+bo.tkrriSTa panthA.

<P/>

AmAder belAy sei Asal Ain hala a.nker sar+bajanas+wIkrrita niyam. kathA hala segulo kothAy pAoyA yAy? skul~pAThya biSay

niye sAdhAraNata.H @{<E>rigourous</E>@}-bhAbe kichhu lekhA pAoyA yAy nA. s\oubhAgyak+rame, AmAder AlochanAy ye niyamgulo

lAgbe segulor @{<E>rigourous</E>@} sa.nskaraN Achhe. tAr kAraN hala kyAlkuleTar eba.n kampiuTAr.

bishwer tAba.t kyAlkuleTar

Ar kampiuTArer madhye progrAm kare DhokAno Achhe yogbiyog guNbhAger sUtragulo. yehetu yantragulor kono buddhi nei, eba.n

(tAder srrijanshIlatAo bipadsImAr anek nIche), tAi oderke bojhAnor janya ye progrAm karA hay, sekhAne bAdhya hayei sab kichhu

kh,nuTiye kh,nuTiye balA thAke, pAThya baiyer mata goTA kayek udAharaN dekhiyei dAy sArA hay nA.

sekhAne ye kAydA Achhe, seTAi hala bar+tamAne a.nker duniyAr swIkrrita kAydA. ei kAydAr duTi stambha--@{<E>precedence</E>@}

eba.n @{<E>associativity.</E>@} prathamTA balchhe bibhinna dharaNer @{<E>operation</E>@}-er madhye konTA Age konTA pare

karte habe. yog biyog guN bhAger madhye yog Ar biyoger <U>samAn</U> @{<E>precedence,</E>@} eba.n guN Ar bhAgerao nijeder

madhye samAn @{<E>precedence,</E>@} eba.n @{<M>\{\times,\div\}</M>@}-er @{<E>precedence</E>@} hala @{<M>\{+,-\}</M>@}-er

cheye beshI. ebAr @{<E>associativity</E>@} balbe, ekai @{<E>precedence</E>@}-er kichhu @{<E>operation</E>@} parpar thAkle

konTA Age habe. b,nAdikerTA Age hale @{<E>left associative</E>@} Ar DAn~dikerTA Age hale @{<E>right associative.</E>@}

tA, AmAder belAy sakalei @{<E>left associative</E>@} (ei kAraNei sei ``b,nAdik haite DAn~dike yAibe'' mantraTA kAj kare).

yadi AmrA eman kono kichhu likhte chAi yeTA er bAire, tabe seTA bryAkeT diye bojhAte habe. bas+tuta.H, yog karA ekAdhAre

@{<E>left associative</E>@} eba.n @{<E>right associative.</E>@} guNao tAi. kintu biyog Ar bhAg kebal~mAtra @{<E>left associative.</E>@}

sei kAraNe nehAt yadi guN Ar bhAger madhye @{<E>precedence</E>@}-er tArtamya An~tei hay, tabe bhAgTAi Age karA uchit (sekhAne

@{<E>BODMAS</E>@}-er sange mil). kintu ekai yuk+tite yog biyoger madhye yadi @{<E>precedence</E>@}-er tArtamya chAi, tabe

biyog Asbe Age (sekhAne @{<E>BODMAS</E>@} h,nochaT khAy).

<P/>

@{<E>Precedence</E>@} Ar @{<E>associativity</E>@}-r kAydAy Aro anek kichhu sAm~lAno yAy. yeman mAinAs chihnaTA khAli ye biyog

bojhAtei byabahrrita hay tAi nay, @{<E>negation</E>@}

bojhAteo kAje lAge, yeman @{<M>-(2+3).</M>@} tAi seTA Asale ekTA AlAdA @{<E>operation,</E>@} yAke bale @{<E>unary minus.</E>@} (ekhAne

@{<E>unary</E>@} mAne yeTA ekTA mAtra sa.nkhyA niye kAj kare). ei @{<E>operation</E>@}-TAr

@{<E>precedence</E>@} sabAr Age. @{<E>BODMAS</E>@}-er prabak+tArA sambhabata.H ei kathATA bhule giyechhilen. Aro ekTA

jinis hala @{<E>power.</E>@} AmrA yakhan @{<M>2^3</M>@} likhi, takhan anek samayei kheyAl kari nA ye oi ``ekTu upare uThiye

lekhA''-TAo Asale ekTA @{<E>binary operation.</E>@} kyAlkuleTar bA kampiuTArer belAy abashya seTA Ar gopan thAke nA, kAraN

@{<E>power</E>@}-TA bojhAnor janya sekhAne kichhu ekTA Tip~tei hay. sei @{<E>operation</E>@}-TA kintu @{<E>right associative,</E>@}

yeman @{<M>2^{3^4}</M>@} mAne Age @{<M>3^4</M>@} kare tArpare seTAke @{<M>2</M>@}-er ghARe chApAte habe.

<P/>

yAi hok, e niye Ar kathA bARAba nA. parisheSe Ain~s+TAiner ekTA kathA ul+lekh kari--

<Q>

ye kono biSaykei bojhAnor samaye yathAsambhab

sahaj kare bojhAno uchita, kintu tAr chAiteo sahaj kare fellei bibhrAT.</Q>

@{<E>BODMAS</E>@} sUtraTio eirakam ekTi atisahajIkaraNer

drriSTAnta. yathAsAdhya bhAlor cheye Aro bhAlo kare bojhAnor hAtchhAnite shixakmashAyrA sARA nA dilei

chhAtrader bhabiSyate mangal habe.

</NOTE>