Omugereeso gw'Enamba(Number Theory)Jn Prae r P

Omugereeso gwa Namba (Number Theory) (Introducing an Educationist To Number Theory in Luganda) Bya Charles Muwanga Email:akiinitiative@ymail.com

Omugereeso gwa Namba

(i) Sessomo mu Luganda (the discipline in Luganda)

Ebigambo biri bisatu nga byonna birimu omulamwa gwe gumu ogwa sessomo lino.

• Ekibalangulo (Mathematics) • Okubala (Short for ekibalangulo) • Ekibalo (Also short for ekibalangulo )

(ii) Amatabi g’ekibalangulo ky’abato ag’etaagisa.

(a) Omugereeso gwa namba (number theory). Omugereeso gw’ennamba gukwanjulira ennamba ez’enjawulo nga eza kyegabanya (even numbers) oba eza kigaanira (odd numbers). Omugereso gw’ennamba gwe gusobozesa ekibalirizo

(b) Ekibalirizo (arithmetic)

Ekibalirizo kyogeza nnamba. Omuntu yenna akola okubalirira okw’amangu agatta, ayawuza, akubisa emirundi, oba okwawuza ng’akozesa ennamba, Eno y’ensonga ettabi lino liyitibwa ekibalirizo (arithmetic).

Obutaba nga kibalirizo, algebra ayogeza byombi ennamba n’ennukuta. Eky’okulabirako 2+y =5

(d) Omukutule = Omuwendo omukutulemu (Fraction)

(e) Omutonnyeze=omuwendo ogw’akatonnyeze (decimal number). Yadde nga oyinza okukiyita “disimoolo”, “omutonnyeze” kya Luganda nnyo okusinga “disimoolo”.

(f) Essomankula (geometry).

Lino ssomo lya nkula za kibalo (study of mathematical shapes) era eriyitibwa essomampimo. Essomapimo ssolo lya mpimo (dimensions) za nkula za kibalo.

(iii) Emiramwa egyetaagisa Omusomesa okunnyonnyola omuyizi eky’okukola mu kubala:

• Okubalangula (to calculate)

• Okubaza (short for okubalanguza or okubalangula)

• Okusonjola (to simplify, to define)

• Okubalanguza (to solve)

• Ekibalanguzo (Formula)

• Omubalanguzi (mathematician)

• Omweyoleko (mathematical expression)

• Omuwendo gw’ekifo (place value)

• Endagamuwendo (digits)

• Emigaramiro (rows)

• Emisimbalaala (columns)

• Ekikunizo (mathematical question/problem)

• Okusonjola (to simplify)

• Omweyoleko (mathematical expression)

• Omuwendo (value or number)

o Ennamba oba namba (number)

• Enkuumakifo/Enkyusibwo (placeholder or variable)

• Omusittale/layini/olukoloboze (line)

• Omugereko (set)

• Omusengeko (pattern)

(iv) Emiramwa emirala gye wetaaga

• Ziro / Zeero (zero)

• Endagamuwendo (digits)

• Namba/ennamba (number)

• Omuwendo (value)

• Omuwendo gw’ekifo (Place value).

Abamu bagamba nti tetwetaagisa kwewola kigambo “number” kubanga mu luganda kiba “muwendo. Bano beerimbye!! Bw’oba oyagala okugamba “place value” okyogera otya mu Luganda? “Muwendo gw’ekifo” kubanga omuwendo mu luganda kye kitegeeza “value”, obuzito bw’ennamba. Omuwendo n’olwekyo kitegeeza “buzito bwa namba oba buzito bwa digito ez’enjawulo ezikola ennamba”. Obutalemerako nnyo era number oyinza okukiyita “omuwendo” okusinziiira ku mbeera gy’okikozesezaamu.

• Embaza mapeesa (abacus)

Nga omuzadde oba omusomesa, olina okubeera waggulu w’omuto gw’oyigiriza mu kumanya emiramwa egyetaagisa. Mu lungereza omuzadde oba omusomesa aba munnyonnyofu mu miramwa egyetaagisa nga solve, calculate. Simplify, mathematics, math, omubalanguzi, formula, operations, variable, units, n’ebirala.

Okusobola okuyamba omuto wo mu luganda n’olungereza kikwetaagisa ggwe nga omuzadde oba omusomesa okumanya emiramwa gino mu luganda n’olungerza okwewala okusikattira n’okunanaagira nga obadde oyamba okuteekateeka omwana wo mu kubala okwabasooka.

Essomeso 2 :Omugereeso gwa Namba

Omugereeso gw’ennamba (Number theory) kitundu kinene ku ttabi ly’ekibalangulo eriyitibwa ekibalirizo (arithmetic). Omugereeso kye ki ?

Omugereeso (theory) kyekuusiza ku kikolwa ky’oluganda “okugereesa”, kyokka kiva mu kugattika ebigambo by’oluganda “omubalanguzio by’agereesa”(What a mathematician theorises) .

“Omugereeso gwa namba” (number theory) gutandika na kuyiga namba, obutoffaali obuzimba omugereeso gwa namba ne sessomo ly’ekibalangulo lyonna okutwalira awamu.

“Omugereeso gwa namba”(numbertheory) gugereesa na ndagamuwendo(digits oba ziyite nigina(units) kkumi era digito zino gy’emiramwa egy’ekibalo (math concepts) omuzadde n’omusomesa yenna gy’asookerako mu kutendeka omwana okubala nga muto. Endagamuwendo ziri kkumi era zino bwe butaffaali obuzimba namba endala zonna, enntono ennyo ez’emikutule (fractions) oba emitonnyeze (decimals), entono eziri mu nsusuuba, n’ennene ennyo eziri mu butabalika.

Endagamuwendo kye ki?

Digito 0 1 2 3 4 5 6 7 8 9

Obutoffaali obuzimba sessomo ly’ekibalangulo, obutaffaali obuzimba namba ze ndagamuwendo(digits). Namba ziri buli wamu mu kibalangulo era ziyinza okusengekebwa mu biti oba emitendera egy’enjawulo, ebisonjozo n’ebinnyonnyozo eby’enjawulo.

. “Ekibalangulo” for Mathematics

According to the Enkuluze y’Oluganda ey’Ennono (Lugnda Cultural Dictionary) , ekibalangulo is an archaic Luganda word associated with sharpening(metallic weapons). So the etymological meaning of ekibalangulo is rooted in the act of “sharpening” . Today the mathematics is considered to be the subject that sharpens the mind. In fact, a student who performs well in mathematics finds other disciplines of study easy because his mind is sharpened.

Ekibalangulo (mathematics) in short form is “ ekibalo” or okubala for “math”. The good thing we have in the mathematical word “ekibalangulo” over other suggested words is that from the same concept we are able, with ease, to form other necessary concepts that refer to mathematical calculations. These include:

a) Okubaza (to solve, to strategise)

b) Ekibazo ( solution to a math problem)

c) Okubalangula ( to calculate)

d) Okubalanguza (to calculate)

e) Ekibalanguzo (formula)

f) Obubalanguzo (mathematical operations)

g) Omubalanguzi (mathematician)

You can note the advantage we get by refering to mathematics as ekibalangulo from which we derive several; other required concepts for any discourse in mathematics. Any way in short you can still call it okubala or ekibalo in ordinary language.

Ekibalirizo(Arithetic)

Obutafaanana nga aligebbula ayogeza byombi , namba n’ennyukuta, ekibalirizo(arithmetic) kyogeza namba na nambiso byokka. Ekibalirizo (arithmetic) kyetaagisa okuwa namba amannya n’obubonero. Amannya ga namba enzijuvu mu luganda gali “emu, bbiri, satu, nnya, ttaano, mukaaga, n’okweyongerayo. Amannya gano gayitibwa mannya agabala oba namba za kibazo (namba ezibala).

Obubonero bwa digito ekkumi ezizimba buli namba, omuli:  0,1,2,3,4,5,6,7,8,9  buyitibwa “nambiso”(numerals) kyokka mu kubwatuukiriza tuziyita “namba”. Ekibaririzo (arithmetic) kikozesa bubonero okubala. Kyetaagisa okuwa obubonero buno amannya. Erinnya ly’akabonero akabala liyitibwa “namba”.

Amannya g’obubonero buno mu luganda gali : “zero, emu, bbiri, satu, nnya, ttaano, mukaaga, n’okweyongerayo mu entakoma. Ezo ze namba ez’enjawulo. Amannya gano gayitibwa mannya agabala oba namba za kibazo (namba ezibala).

Obubonero obukiikirira namba bulagibwa nga: "1, 2, 3, 4, 5, 6, 7” n’okweyongerayo mu ntakoma. Obubonero buno buyitibwa nambiso (numerals). Waliwo ne nambiso z’ekirooma” (Roman numerals) nga I,II,III,IV n’okweyongerayo. 'Okiraba nti mu nambiso z’ekirooma temuli kabonero ka zeero.

Zeero(0) nayo etwalibwa okuba namba enzijuvu.Erina ekinnyonnyozo(property) nti singa egattibwako namba yonna endala, namba eyo tekyuka. Ekyokulabirako 5+0 = 5. Omuntu atandika sessomo ly’ekibalangulo alina kusookera ku kuyiga: o Okubala, okusoma, n’okuwandiika namba enzijuvu okutandika n’ezisooka ekkumi, n’azzaako ekikumi, nadda ku lukumi,n’okweyongerayo.

o Okunokoolayo omuwendo gw’ekifo(place value) buli digito ya namba mweri. o Waliwo akakwate akali wakati wa namba, obungi, n’omuwendo gw’ekifo mu namba enzijuvu okutuuka ku lukumi. o Waliwo enkozesa y’omulamwa gw’ensuusuuba n’amakumi mu nsengekera ya namba ey’omuwendo gw’ekifo. o Bw’ova awo n’oyiga omuwendo gw’ekifo buli digito w’eri okutuuka ku mutwaalo. o Okuva awo alina okumanya omuwendo gw’ekifo ogwa namba enzijuvu awamu n’ogw’emitonnyeze (decimals) egirina ebifo by’emitonnyeze (decimals places) ebibiri awamu n’engeri namba enzijuvu n’ebigana gye zikwataganamu n’emikutule (fractions). o Bw’ova kw’ekyo aba alina okuyiga okusoma n’okuwandiika namba enzijuvu eziri mu bukadde. o Okusengeka n’okugeraageranya namba enzijuvu n’emitonnyeze okutuuuka ku bifo by’emitonyeze bibiri. o Okuzingirako namba (rounding off numbers) enzijuvu ku kumi, ku kikumi, lukumi, mutwalo, kakadde, kawumbi okutuuka ku butabalika (the nearest ten, hundred, thousand, ten thousand, or hundred thousand). o Okusengeka ennamba (0rdering numbers) o Enziringanya y’ennamba (Number sequencing)

Okuyiga bino kisobozesa omuntu atandika ekibalangulo okugenda mu maaso:

(i) Okuyiga okubala obungi bw’ebintu. (ii) Okuyiga okugatta n’okwawuza.Kino balina okukikola n’okwegezaaamu kungi okutuuka nga buli muwendo bagukuba budinda. (iii) Okwegezaamu n’okubala mu bibinja n’enkubiso z’ebibinja (multiples of groups). Eky’okulabirako okubala ebintu ne 5, 10, 15, n’okweyongerayo (iv) Okubalanguza okuyita mu kukozesa emisimbalaala (columns) naddala ng’oyagala okugatta oba okwawuza namba ennene ezirina digito eziwera.Wano tulina okukozesa obumanyirivu bw’emiwendo gy’ekifo kya buli digito. Ojja kukizuula nti bwe twogera obwogezi namba tubuuka emiwendo gy’ebifo (place values) ebiba bitakiikiriddwa namba, gamba, mu namba nga 4, 006; eno esomwa nga enkumi nnya mu lukaaga. Bwe tuwandiika awatali digito ekifo tukiteekamu ziro (0). Essomeso 3: Ennamba n’ennambiso

              (Numbers & Numeracy)

Sooka omanye bino

(a) Ekikolwa eky’okubala (counting) kyetaagisa obubonero n’amannya gabwo.

(b) Namba ly’erinnya ly’obubonero obukozesebwa okubala mu ttabi ly’ekibalangulo eriyitibwa “ekibaririzo” (arithmetic). Obubonero buyitibwa “nambiso’ (numerals).

(c) “Namba” ye namunigina(unit) omubalanguzi gye yeyambisa ng’akola okubalangula kwe okwa buli kika mu sessomo ly’ekibalangulo.

“Omugereeso gw’Ennamba” (Number Theory)

There are two words crucial for grasping the basic concepts in number theory:

(i) The symbols are called numerals e.g 0, 12, 2, 3,… (ii) The names of the numerals are the numbers like zero, one , two, three, …

We have borrowed the two concepts , that is , adopted and adapted them to Luganda phnology and orthography” as ennamba , the name for number and ennambiso , the symbol for numeral. The Luganda word “omuwendo” refers to the value in a number or ennambiso or a digit . Users should be careful because “omuwendo” in most cases refers to “a value” as distinct from number. Ennambiso (numeral) is a symbol but a number is the name of a symbol . The building blocks of numbers are the digits(figures) , called digito in Luganda .

Values and numbers are almost synonyms but the value depends on the place of a particular digit in a number. This means that the digits have different values depending in the places where they are found in a number. In Luganda , we can talk of “omuwendo gw’ekifo” , to mean place value in a number .

“Omugereeso” is the Luganda word for “mathematical theory”. So , when we want to say “number theory” in Luganda we say “omugereeso gw’ennamba” or “omugereeso gwa namba”.

In fact , every topic in the mathematical field of study can be referred to as “omugereeso”. For example we can talk of “omugereeso gw’essomampimo” to mean the “theory of geometry” or “omugereeso gw’emigereko” to refer to “set theory).

Omugereeso gwa namba (number theory) gutandika na kuyiga namba, obutaffaali obuzimba omugereeso gwa namba ne sessomo ly’ekibalangulo lyonna okutwalira awamu. Namba kye ki?

Namba kintu ekibalanguzo ekikozesebwa okubala, okulaga, n’okupima. Ensonjola ya namba etwaliramu namba nga ziro, namba eza kiddannyuma (namba za negatiivu) , namba ez’omugerarenyo(rational number), namba ezitali za mugerageranyo, ne namba enzibuwavu, okunokoolay ezimu .

Omulamwa “namba” ke kataffaali akasookelwako mu kuyiga okwa sayansi n’ekibalangulo kyokka ekibuuzo “namba kye ki?” tekyasobola kuddibwamu okutuuka mu 1884 A.D. Gottlob Fregé [1], eyavumbula okusengeka ensonga z’ekibalangulo(mathematical logic) kyokka ansa ye teyamanyibwa mu nsi okutuuka omungereza omubalanguzi era omukugui w’ensengeka y’ebirowoozo Bertrand Russell, eyagezaako okunnyonyola buli ekigenda mu maaso mu kibalangulo ng’akyesigamya ku mulamwa gw’emigereko(sets), n’atuuka okuttuukiriza omulamwa gwa “namba”.

Omulamwa gwa namba gwekuusiza ku mulamwa gw’omugereko (set).Omugereko taminologiya eringa taminologiya ya “punkuti(akatonnyeze) mu essomampimo kubanga omulamwa guno si musonjole. Mu ngeri y’emu omugereko gunyonyolwa biteeberezo(axioms) bya mugereeso gwa migereko . Mu ngeri etali ntongole omugereko gusonjolwa nga ekikung’anyo ky’ebintu ebigere era ebyeyawudde eby’ekikakkyonna kwe tusobola okusalirawo oba ekintu ekimu oba ekirala memba yagwo.

Kino kitegeeza nti okulaga omugereko. Olina okulaga ebintu byonna ebigirimu, ebiyitibwa bammemba nga nga olaga buli memba mu kikung’anyo oba nga onnyonnyola bamemba ki abalimu.

Njawulo ki eri wakati w’endagamuwendo( digit) ne Namba?

Endagamuwendo(digit) ke “akataffaali akazimba namba” (the building block of a number).

Digito 1 2 3 4 5 6 7 8 9 Ziro Namba 1 * Namba 2 ** Namba 7 ******* Namba 15 *************** Namba 10 ********** Nmba 5 ***** Namba 20 ******************* Namba 25 ************************ Namba 30 ***************************** Namba 34 ********************************* Namba 29 **************************** Namba 18 *******************

Nga namba bwe buli obutffaali obuzimba sessomo ly’ekibalangulo, obutaffaali obuzimba namba ze ndagamuwendo (digits). Namba ziri buli wamu mu kibalangulo era ziyinza okusengekebwa mu biti oba emitendera egy’enjawulo, ebisonjozo n’ebinnyonnyozo eby’ejawulo.

Omugereeso gwa namba gutandika n’ebintu kkumi era gino gy’emiramwa omuzadde n’omusomesa yenna gy’asookerako mu kutendeka omwana nga muto .Digito ziri kkumi era zino bwe butaffaali obuzimba namba endala zonna, entonno eziri mu nsusuuba n’ennene ennyo eziri mu butabalika.

Endagamuwendo ekkumi ezizimba buli namba: 0,1,2,3,4,5,6,7,8,9.

Okunokoolayo omuwendo gw’ekifo buli ndagamuwendo mweri.

Akakwate akali wakati wa namba , obungi,n’omuwendo gw’ekifo mu namba enzijuvu okutuuka ku lukumi.

Enkozesa y’omulamwa gw’ensuusuuba n’amakumi mu nsengekera ya namba ey’omuwendo gw’ekifo.

Bw’ava awo n’ayiga omuwendo gw’ekifo buli digito weeri okutuuka ku mutwaalo. Okuva awo alina okumanya omuwendo gw’ekifo ogwa namba enzijuvu awamu n’ogwa emitonnyeze (decimals) egirina ebifo by’emikumito/emitonnyeze (decimals places) ebibiri awamu n’engero namba enzijuvu n’emikumito gye gikwataganamu n’emikutule/namba enkutulemu/emitunduwazo (fractions). Bw’ava kw’ekyo aba alina okuyiga okusoma n’okuwandiika namba enzijuvu eziri mu bukadde.

Okusengeka n’okugeraageranya namba enzijuvu n’emitonnyeze okutuuuka ku bifo by’emitonyeze bibiri.

“Okuzingirako namba” (rounding off numbers) enzijuvu ku kumi , ku kikumi,lukumi, mutwalo, kakadde, kawumbi okutuuka ku butabalika (the nearest ten, hundred, thousand, ten thousand, or hundred thousand.).

Okuyiga bino kikusobozesa okuzzaako okuyiga okubala obungi bw’ebintu.

(ii) Okuyiga okugatta n’okwawuza .Kino balina okukikola n’okwegezaaamu kungi okutuuka nga buli muwendo bagukuba budinda.

Okwegezaamu n’okubala mu bibinja n’enkubiso z’ebibinja(multiples of groups). Ekyo’kulabirako okubala ebintu n’ 5, 10, 15. n’okweyongerayo . okubalanguza okuyita mu kukozesa obusimbalaala(columns) naddala ng’oyagala okugatta oba okwawuza namba ennene ezirina digito eziwera.Wano tulina okukozesa obumanyirivu bw’emiwendo gy’ekifo kya buli digito. Ojja kukizuula nti bwe twogera obwogezi namba tubuuka emiwendo gy’ebifo(place values) ebiba bitakiikiriddwa namba ,gamba, mu namba nga 4,006 ; eno esomwa nga enkumi nnya mu lukaaga. Bwe tuwandiika awatali digito ekifo tukiteekamu ziro(0).

“Okubala”(to count) kuba kusengeka namba ezibala oba namba eza kibazo(counting numbers), zino nga ze namba enzijuvu(whole numbers) oba namba kennyini(real numbers) nga 0,1,2,3,4,5,6,7,8,9,10,11,12,……………okweyongerayo ewatali kkomo(into infinity).

Ebyafaayo bya sessomo ly’ekibalangulo bitandika na byafaayo bya namba na kubala okukozesa namba okusinziira ku buwangwa obw’obulabufu nga Bugereeki, Abarooma, Bukyayina , Buyindi , Bulaaya n’ebitundu ebimu mu afirika ey’abaddugavu.

Abagezigezi ababalanguzi batandika okwebuuza ku:

• Ki ekibeerawo nga omuntu abala kiddamaaso okuva ku ziro oba okuva ku emu? • “Ki ekibeerawo nga omuntu abala kiddanyuma okuyita mu ziro ? • Emikutule (namba enkutule) bwe giba namba, giyinza okugattibwa, okwawulibwa, okugabizibwamu oba okukubisibwamu?

Ebibuuzo nga bino byaleetawo obutakkaanya mu babalanguzi era abo abaatandikanga okubuuza ebibuuzo ebyo ne basekererwa nga abamu babayita bagwi ba ddalu oba bbulwabikolwa kyokka ne bivaamu okutuuka ku kituufu. Baatandika okukuba empawa n’okukola ebigambululo (statements) nga:

• Namba za pozitiivu zivaako namba za negatiivu. • Kyebiriga zivaako emirandira gya kyebiriga • Kyesatuza zivaako emirandira gya kyesatuza

Entandikwa y’ekibalangulo mu kuyiga kubala .Okubala kuba kusoma na kuwandiika namba enzijuvu: i. Okuva ku ziro okutuuka ku kumi

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

ii. Okuva ku kumi okutuuka ku kikumi 10, 20, 30, 30, 40, 50, _____, _____, _____ 100

iii. Okuva ku kikumi okutuuka ku lukumi 100, 200, 300, 400, _____, _____, _____, 1000 “Okubala” (to count) kuba kusengeka namba ezibala (counting numbers), zino nga ze namba enzijuvu (whole numbers) oba namba kennyini (real numbers) nga 0,1,2,3,4,5,6,7,8,9,10,11,12,………okweyongerayo ewatali kkomo(into infinity). Ebyafaayo bya sessomo ly’ekibalangulo bitandika na byafaayo bya namba na kubala okukozesa namba okusinziira ku buwangwa obw’obulabufu ngamu Bagereeki, Abarooma, Bukyayina, Buyindi, Bulaaya n’ebitundu ebimu mu afirika ey’abaddugavu. Abagezigezi ababalanguzi abasooka, basooka okwebuuza ku:  "Ki ekibeerawo nga omuntu abala kiddanyuma okuyita mu zeero?  "Buwanvo” (disitansi) ki obuliwo okuyita mu nsittalo ya kyebiriga?  Emikutule (emiwendo emikutuleremu) bwe giba namba, giyinza okugattibwa, okwawulibwa, okugabizibwamu oba okukubisibwamu? Ebibuuzo nga bino byaleetawo obutakkaanya mu babalanguzi abasooka era abo abatandikanga okubuuza ebibuuzo ebyo ne basekererwa ng’abamu babayita bagwi ba ddalu oba bambulwabikolwa kyokka ne bivaamu okutuuka ku kituufu. Baatandika okukuba empawa n’okukola ebigambululo(statements) nga:  Namba za pozitiivu zivaako namba za negatiivu?  Kyebiriga zivaako emirandira gya kyebiriga? Okubala Namba n’Omuwendo gw’Ekifo

“Okubala”(to count) kuba kusengeka namba ezibala era eziyitibwa eza kibazo (counting numbers), zino nga ze namba enzijuvu (whole numbers) oba namba kennyini (real numbers) nga 0,1,2,3,4,5,6,7,8,9,10,11, 12,……okweyongerayo ewatali kkomo (into infinity).

Okubala ppaka ku kkumi:

1 2 3 4 5 6 7 8 9 10 Emu Bbiri Satu Nnya Ttaano Mukaaga Musanvu Munaana Mwenda kkumi

Okubala ng’obusaamu

 “Okubala ng’obusaamu” (Skip Counting) kitegeeza okubala ne namaba etali 1. Okuyiga okubala ng’obusaamu kikuyamba:  Okubala ebintu ebingi amangu  Okuyiga okukubisaamu (emirundi) Ebyokulabirako:

Okubala ng’obusaamu 2 okola bw’oti:

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... n’okweyongerayo

Okubala ng’obusaamu 10

10, 20, 30, 40, 50, 60, 70, 80, 90, 100,n’okweyongerayo

Okubala ng’obusaamu 5

5, 10, 15, 20, 25, 30, 35, 40, 45, 50, n’okweyongerayo

Okubala ng’obusaamu 3

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, n’okweyongerayo.

Okubala ng’obusaamu 4 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...

Okusengeka namba (Ordering Numbers)

Okuteeka namba mu nsengeka tandikira ku esembayo obutono okutuuka ku esingayo obunene. Kino kye kiyitibwa ensengeka elinnya; nga oli bw’alinnya olusozi.

	Eky’okulabirako: Teeka  15, 6, 7 and 9 mu nsengeka erinnya:

Ansa : 6, 7, 9, 15

Olumu oyinza okwagala namba zikole ekya kikontana nga zitandikira ku esinagyo obunenene okutuuka ku esembayo obutono; kino kiyitibwa nsengeka ekka; nga bwokka wansi w’olusozi.

	Ekyokulabirako e:  Teeka  15, 6, 7 and 9 mu nsengeka ekka.

Ansa: 15, 9, 7, 6

Omuwendo gw’Ekifo mu Namba (Place Value in a number)

Mu kibalangulo bwe twogera ku muwendo gw’ekifo tuba tutegeeza omuwendo gw’ekifo mu namba. Mu namba buli kifo kirina omuwendo(value). Waliwo ebintu bitaano ebyetaagisa okutegeera omulamwa gw’omuwendo gw’ekifo.

(i ) Okuyiga buli namba bw’eyitibwa awamu, engeri gye zidding’ana n’engeri gye zikozesebwa okubala obungi, n’okukozesa namba mu ngeri ez’enjawulo mu bulamu obwa bulijjo mu kwogera, okusoma n’okuwandiika. Omuzadde alina okulaba nga omwana ayiga okubala ebintu n’okutegeera bimeka oba meka buli linnya lya namba kye litegeeza. Abazadde n’abasomsa wano balina okulaba ng’abaana bafuna okwegezaamu ekimala nga babala namba okusooka nga bakoma ku 10 bwe bamala ne beyongerayo okumanya nti namba eziri mu kkumi zonna zirina ennyingo “kumi”, eziri mu abiri zonna zirina ennyingo “abiri”, n’okweyongerayo awatali kusooka kwebuuza lwaki kiri kityo.

(ii)Okukozesa namba okugatta n’okwawuza mu bulamu obwa bulijjo. Okugatta n’okwawuza ebisookerwako omwana abiyigira mu bulamu obwa bulijjo nga azannya n’ebintu ebiri mu bungi okutandika n’ebifaanagana. Kyetaagisa nnyo abaana okufuna okwegezaamu okunnyonnyoka engeri y’okugattamu emigogo gya namba za digito emu ku emu awamu n’okwawuza.

(iii) Obubondo bwa namba (group numbers). Muno mulimu okubala ebintu mu bubondo n’okumanyiira okubalira mu bubondo (practice with groupings), nga okubala ebintu mu misengeko egya ttaano ttaano awatali kuddingana buddinganyi 7, 14, 21, n’okweyongerayo. Muno mulimu n’okusobola okusoma n’okuwandiika ebintu mu bubondo awatali miramwa gya muwendo gwa kifo naye lwa kuba wayiga okuwandiika namba. Wegezeemu n’obubondo (ebibinja) bwa namba nga obubondo obwa 2, obwa 3, obwa 5, oba obwa 10.

(iv) Okukozesa obwongo ng’okozesa namba ng’ogoberera emitendera emisengeke (logical steps). Ensengekera ya namba gye tukozesa  esinziira  ku kumi kubanga ekozesa obubonero kumi bwokka obuyitibwa digito (digits). Nga tukozesa obubonero buno ekkumi tuyinza okuwandiika namba yonna.

Endagamuwendo ekkumi

Endagamuwendo(digits) ze tukozesa kati ziyitibwa “Namba za Kiyindi-Kiwarabu” ("Hindu-Arabic Numerals) era zifaanana bwe zti:

0 1 2 3 4 5 6 7 8 9

Zino oyinza okuzikozesa ku bwazo okubala okutuuka ku mwenda. Okubala kkumi oba okweyongerayo, tuba tutandika n’omusimbalaala omulala, omusimbalala ogwa “amakumi”, olwo ne tuwandiika amakumi ameka ge tulina, ne tuddirira namunigina meka.

Namba “13 “etegeeza erimu ekkumi limu ne namunigina(ensusuuba) 3Kino era kiyinza okuwandiikibwa nga (1 × 10) + (3 × 1). Ekyokulabirako:"75" kitegeeza amakumi musanvu ne namunigina 5 oba(7 × 10) + (5 × 1)

Bwe tuba n’emiramwa egisukka mu 99, tuba tutandika omusumbalala omulala. Wano twetaaga okulaga ebikumi bimeka, amakumi ameka ne namunigina meka. Mu namba 194 tuba tulina ekikumi kimu, amakumi mwenda, ne namunigina nnya. Kino era tuyinza okukiwandiika nga (1 × 100) + (9 × 10) + (4 × 1). Mu namba "735" oba okiraba nti mulimu ebikumi 7, amakumi 3 ne namunigina 5,era ekiyinza okulagibwa nga (7×100 )+ (3×10 )+ (5×1).

Okutwalira awamu buli lw’oyagala okulaga namba esingako obunene, tuba tugattako omusimbalala gumu (one column) ku kkono era ne tumanya nti kino kitegeeza emirundi 10 okusingako omusimbalala oguli ku ddyo. Kati okiraba nti wetuteeka digito wa mugaso nnyo.

Ziro Kiba kitya nga tulina ekkumi limu awatali namunigina yonna? Tulaga nti tetulina namunigina nga tuteeka ziro mu kifo ekyo. Makumi namunigina 1 O Mu namba "10", tulina okuteekamu ziro awatali ekyo kumi(10) eba erabika nga emu(1).

Amannya ga buli musimbalaala Gano ge mannya ga buli musimbalaala (column): Bukadde Mitwalo Nkumi Bikumi Makumi Namunigina/nsusuuba

Eky’okulabirako : Mu taabulo y’omuwendo gwa ekifo eno wansi wandiika namba emitawo kumi na munaana, mu enkumi ssatu mu bina mu ataano mu mukaga.

Emitwalo Nkumi bikumi makumi namunigina 180,000 3000 400 50 6

Manya omutwalo = enkumi kkumi Emitwalo kkumi = enkumi kikumi Emitwalo kumi na munaana = enkumi kikumi mu kinaana

Namba 11,327

Awo oba okiraba nti mu “mugereeso gwa namba”, buli kifo kirina omuwendo gwe kikiikirira. Ekikunizo:

Bala emipiira mu buli mugalamiro owandiike ennamba entuufu mu kabokisi ku kkono.

Emipiira Ennamba

	1
 	…………
  	……………
   	4
    	…………..
     	……………..
      	………………
       	8
        	……………
         	…………….

(iii)Omuwendo gw’ekifo (Place value): Embaza Mapeesa (Abacus) Wano ziro “nkuumabusangiro” (Posotion holder)

Ebikunizo: 1. Wandiika ennamba wansi w’embaza mapeesa zino:

2. Laga emiwendo gy’ebifo mu nnamba zino: a) 215 = ebikumi ____ amakumi _____ ensusuuba_____ b) 230 = ebikumu_____amakumi______ensusuuba______ c) 804 = ebikumi _____amakumi______ ensusuuba_____ d) 100 = ebikumi______ amkumi______ensusuba______ e) 399 = ebikumi______amakumi_____ensusuuba______

3. Wandiika ebigambo bino nga ennamba: i) Ebikumi bibiri mu taano = _____________________ ii) Ebikumi bibiri mu asatu = ____________________ iii) Lunaana mu nnya = _________________________ iv) Ebikumi bisatu mu kyenda mu munda = __________ v) Kikuumi = _______________________________

4. Okugatta n’okwawuza Okugatta (addition)

 +  0= 1 + 0
 +  = 1 +1
+  +   = 1 + 1 + 1
+  +  +   = 1 + 1 + 1 + 1
	+ + + +  = 1 + 1 + 1 + 1 + 1

		1 + 1 + 1      +       1 + 1 + 1 + 1

3 + 4 = 7

1 + 1       +   	1 + 1 + 1 
               2 	+	3	=

Ebikunizo by’okugatta eby’emigalamiro (addition using rows):

Okugatta okw’obusimba (Vertical addition) 1. Okugatta Nk Bi Ma Ns Nk Bi Ma Ns 1, 5 2 3 7, 6 3 8 + 5 3 + 2, 0 2 1 1, 5 7 6 9 6 5 9

2. Okwawuza Ma Ns Ma Ns

 5		6				9		4
     -   2		3			       - 7		2

3 3 2 2

Ebikunizo: a) 7 2 8 0 c) 6 4 5 2 + 1 3 1 0 + 3 1 4 1 _______________ ____________________ b) 5 4 d) 8 4

       - 3	3				         - 5        1
   ___________				___________

e) 4 2 6 f) 4 4 6

   	-	2	6				-	3	2

___________ ___________

g) 1 7 2 h) 2 1 7 + 1 1 + 8 1 ______________ ___________

Essomeso 4: Emisengeko n’Enziring’anya y’ennamba

          (Patterns and Sequencing of Numbers)

Omwana ayiga okubala namba ensengeke mu ngeri ez’enjawulo , omuli : (i) Okuyiga Okubala (Counting) okutuuka ku kumi 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (ii)Okuyiga okubala ennamba za Kibalirampuyibbiri (Yintegya = Intergers)

(iii)Okubala mu bbiri bbiri (Counting in 2s).

(iv) Okubala mu bibinja ebya bina bina (counting in fours)

Ekibinja 1 ekya 4 =4

Ebibinja 2 ebya 4 = 4 + 4 =8

Ebibinja 3 ebya 4 = 4 +4 +4 =12

Ebibinja 4 ebya 4 = 4 + 4 + 4 + 4 =16

Ng’omaze okunnyonnyoka emisengeko egyo, n’enziringanya yagyo namba ki ezijjuza mu mabanga ag’obukolaloze 4, 8, 12, ______, ______, _____ (v) Okubala mu bibinja ebya taano taano (counting in fives)

Ekibinja 1 ekya 5 =5

Ebibinja 2 ebya 5 = 5 + 5 =10

Ebibinja 3 ebya 5 = 5 + 5 +5 =15

Ebibinja 4 ebya 5 = 5 + 5 + 5 + 5 =20

Jjuza mu bukoloboze: 5, 10, 15, ____, ____, ____, ____ (vi) Okubala mu bibinja ebya kumi kumi (counting in fives)

Ekibinja 1 ekya 10=10

Ebibinja 2 ebya 10 = 10 + 10=20

Ebibinja 3 ebya10 = 10 + 10 + 10=30

10, 20, 30, ____50, ____60, ____, ____

Ekikunizo: Namba ki ezijjuza mu bukoloboze? 2, 4, ____ 8, 10 ____, 14 ____ _____

3. Okubala mu bibinja bisatu bisatu (counting in threes)

	Ekibinja 1 ekya 3 = 3
 	Ebibinja 2 ebya 3 = 3 + 3 = 6
  	Ebibinja 3 ebya 3 = 3 +3 +3 = 9
   	Ebibinja 4 ebya 3 = 3 + 3 + 3+3 = 12
    	Ebibinja 5 ebya 3 = 3 + 3 + 3 + 3 + 3 + 3 = 15
     	Ebibinja 6 ebya 3 = 3 + 3 + 3 + 3 + 3 + 3 = 18

Ekikunizo: Namba ki ezijjuza mu bukoloboze? 3, 6, 9, _____, _____ 18, 21, 24

Essomeso 5: Okulambulula Ebika bya Namba (Unfolding the Types of Numbers in Luganda) Ebika by’ennamba Ez’omusingi gwa kumi(Types of Base Ten Numbers) Ebika by’ennamba Manya : (a) Nambiso(numerals) buba bubonero obulaga namba z’obutonde era namba ezibala nga 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ………………….. (b) Namba(Numbers) ge mannya g’ennambiso/amannya g’obubonero nga emu , bbiri, satu, nnya,ttaano,mukaaga …n’okweyongerayo mu entakoma (c) Endagamuwendo(Digits) – bwe bubonero ekkumi 0, 1, 2, 3, 4, 5, 6, 7, 8, ne 9, obweyambisibwa okulaga emiwendo gy’ekifo oba okutondeka namba ez’omusingi ogwa kumi. Namba zennyini (Real Numbers) Namba ez’Omugerageranyo (Rational numbers) Namba ezitali za mugerageranyo (Irrational numbers) Namba ezibala Counting numbers 1,2,3,4,5,6,7,8…….. Zino ze namba zennyini zonna ezitasobola kuwandiikibwa nga mikutule. Zirimu: Pai (P)i erina akabonero (π).

π = 3.14159265...

Buno buba bwetoloovu bwa ntoloovu erina olusekkati olwa nigina 1 Namba z’obutonde Natural numbers 1,2,3,4,5,6,7,8 ……… Namba enzijuvu Whole numbers 0,1,3,4,5,6,7,8………. Kibalirampuyibbiri Integers ..., -2, -1, 0, +1, +2 , ... Namba za kiddamaaso Positive numbers +1, +2 ………. Namba za kiddannyuma Negative numbers …….., -2, -1 Namba eza kyegabanya Even numbers Namba eza kigaanira Odd numbers Namba eza ndagakifo Ordinal numbers Namba eza ndagalinnya Nominal numbers

Oluganda olubalanguzi (mathematical Luganda) has the following categories of numbers of the decimal base system: a) Namba eza kibazo cardinal numbers b) Namba ezibala counting numbers c) Namba eza ndagakifo ordinal numbers. d) Namba eza ndagalinnya nominal numbers e) Namba enzijuvu Whole numbers f) Yintegya: integers g) Namba ez’ensibo (ez’obutonde) natural numbers h) Namba za kyegabanya Even numbers i) Namba eza kigaanira odd numbers j) Emikutule /Emwenkutule Fractions k) Namba ez’omugerageranyo Rational Numbers l) Namba ezitali za mugerageranyo Irrational Numbers m) Namba Zennyini Real numbers n) Namba Ez’omuteeberezo Imaginary Numbers o) Ennamba enzibuwavu complex numbers p) Namba za kyebiriga square numbers q) Namba za kyesatuza cube numbers Namba z’omusingi ogwa kumi ez’enjawulo Omugereeso gwa namba mu lungereza kivvuunulwa “number theory”. “Namba” ke kabonero oba obubonero obukozesebwa okubala oba obubonero bw’ekibazo nga .Mu njogera y’oluganda olwa bulijjo , abamu bakozesa omulamwa gw’omuwendo okutegeeza namba naye ekituufu emiramwa gino gyombi girina amakulu ga njawulo. Namba kaba kabonero ka kubala ate omuwendo (value) y’endagabungi ya namba ez’enjawulo. Kino kitegeeza nti buli namba eba n’omuwendo (value) gwa njawulo okuva ku ndala. Eky’okulabirako omuwendo oguli mu nnamba 7 gwa wansi okusinga ku muwendo oguli mu namba mwenda ate omuwendo oguli mu ziro guli waggulu okusinga oguli mu negatiivu kkumi.

(i) Ziro

Eno eba nkuumakifo. Endowooza ya ziro(0) , yadde nga kati buli muntu aginnyonnyoka bulungi , yalwawo okukozesebwa abantu ab’edda. Kyokka oluvannyuma abantu abagezigezi baatandika okwebuuza nti bwe waba tewali kya kubala , ekyo okibaza otya ? Eky’okulabirako bwe bakuwa ebisawo bibiri ekimu nga kirimu sente ate ekirala nga kikalu , temuli sente , ekyo omutali sente n’emu okiraga otya mu buwandiike obwa obubonero oba mu njogera ey’ekibalangulo(mathematically speaking) ? Abantu bwe baali bakatandika okuwandiika namba ennene nga 60 basanga obuzibu okulaga enjawulo wakati wa 6 ne 60 kubanga awatali kabonero ka kubala aka ziro mukaaga ne nkaaga bilabika nga bye bimu mu mpaandiika. Ababalanguzi (mathematicians) kye baakola kwe kuteekawo “enkuumakifo”( placeholder), akabonero ak’enjawulo okulaga nti “ mu kifo ekyo tewali digito ” yonna. 909 N’olwekyo namba 909 kitegeeza :ebikumi mwenda(lwenda), tewali digito ya makumi ate waliwo ensusuuba mwenda Endowooza ya ziro(zero) wano we yatandikira era oluvannyuma ababalanguzi ne batandika okujibalira mu namba era kati oyinza okulowooza mu ngeri egamba nti nabadde ne yiika z’ettaka 75 ne ntundako yiika 75 , kati nina yiika ziro, ekitegeeza nti kati sirina yiika n’emu. Kiba kitya nga tulina ekkumi limu awatali namunigina yonna ? Tulaga nti tetulina namunigina nga tuteeka ziro mu kifo ekyo. Makumi namunigina 1 0 Mu namba "10" , tulina okuteekamu ziro awatali ekyo kumi(10) eba erabika nga emu(1).

Tukozesa ekintu kye kimu okulaga amakumi , ebikumi, enkumi, emitwalo, obukadde, obuwumbi, obutabalika awatali namba yonna eteekebwawo. Eky’okulabirako 506 kitegeeza ebikumi bitaano , amakumi ziro , ne namunigina 6 . Amannya ga buli musimbalala Gano ge mannya ga buli musimbalala (column): Bukadde Mitwalo Nkumi Bikumi Makumi Namunigina/nsusuuba Eky’okulabirako : Mu taabulo y’omuwendo gw’ekifo eno wansi wandiika namba emitwalo kumi na munaana , mu enkumi ssatu mu bina mu ataano mu mukaga . Emitwalo Nkumi bikumi makumi namunigina 180,000 3000 400 50 6

Manya omutwalo = enkumi kkumi Emitwalo kkumi = enkumi kikumi Emitwalo kumi na munaana = enkumi kikumi mu kinaana

Waliwo namba ennene n’entono . 1 , 2,3, ……56,57,…..345…16,993….n’okweyongerayo zonna namb. Namba nga 11,327 erimu digito ttano nga esembayo y’eyitibwa namunigina oba ensusuuba. Awo oba okiraba nti mu “mugereeso gwa namba”, buli kifo kirina omuwendo gwe kikiikirira.

Mu nsengekera ya namba ey’omitonnyeze/ekkumiko (decimal number system), omuwendo gw’ekifo (digit) gusinziira ku kifo kya digito mu nnamba . Buli kifo kirina omuwendo (value) gwa kkumi kubisaamu ekifo ekiri ku ddyo wakyo. Namba eyetongodde eyawulibwamu ebibinja bya digito satu satu, ebiragibwa n’akawummuza(comma) .

Omuganda namba agisoma bubwe. Nnamba ennene nga 95,394,173, 879 erimu digito kumi n’emu esomebwa nga:

95, 394, 170, 3 8 7 9 Obuwumbi Obukadde Emitwalo Mu enkumi Bikumi (lunaana) Makumi (nsanvu) Nsusuuba (mwenda)

Omuwendo gw’akifo gwe mutima gw’ensengekera ya namba (number system). Kyokka kyazuulibwa nga walina okubeerawo akabonero ka tewali (nothing). Akabonero ka “tewali kintu” kano ye ziro (0). Ziro yeteekebwa mu kifo ky’omuwendo gwonna nga tewali digito ndala yonna egenda mu kifo ekyo.

Eky’okulabirako namba 1,000 mu bigambo etegeeza lukumi lumu, awatali bikumi, awatali makumi yadde ensuusuuba. Wano oba okiraba nti singa tewaaliwo kabonero ka “tewali kintu” (nothing) kano aka ziro, ensengekera ya namba eriwo teyandikoze.

Okutandika n’ekifo eky’ensusuuba ku ddyo, buli muwendo gwa kifo (place value) gukubisibwamu (gulinnyisibwa) n’omuwendo gw’obufunza obwa 10 (increasing powers of 10) ogulinya waggulu. Jjukira:

Buwumbi Bukadde Mitwalo Nkumi Bikumi Makumi Nsusuuba

Omuwendo gw’ekifo ogwa namba enzijuvu(whole number Place Value) gwe muwendo oguweebwa ekifo kya digito mu namba . Mu nsengekera ya disimoolo/ey’ekkumiko (decimal system), omuwendo gwa digito gusinziira ku kifo kya digito eno mu namba. Buli kifo kirina omuwendo gwa 10 ng’okubisizzaamu ekifo ekiguli ku ddyo. Namba eyetongodde obulungi eyawulibwaamu ebibinja(emizingo) bya digito ssatu ,akawummuza . Namba zikolebwa digito ate nga buli digito erina omuwendo ogw’enjawulo okusinziira ku wa w’eri(ekifo kyayo) mu namba . Omulamwa gw’omuwendo gw’ekifo (place value) guyinza okukozesebwa ku namba enzijuvu n’emikumito/emitonnyeze(decimals) . Bw’oba otandika okuyiga ekibalangulo wetaaga taabulo eno wansi eraga omuwendo gw’ebifo mu namba ezirina digito ennyingi : Buwumbi (1,000,000,000) Bukadde (1,000,000) Busiriivu (100,000) Mitwalo (10,000) Nkumi (1000) Bikumi (100) Makumi (10) Ensuusuuba (1) 7 3 4 8 6 Endowooza y’omuwendo gw’ekifo gwe mutima gw’ensengekera ya namba (number system). Akabonero akalaga ekyangaala ke kasookerwaako era kayitibwa ziro(0). Ziro ekuuma ekifo ky’omuwendo omugere, singa waba tewali digito yonna egenda mu kifo ekyo. (i) Ekyokulabirako namba 1000 mu bigambo etegeeza olukumi lumu, tewali bikumi, tewali nkumi ate tewali nsuusuuba (ones). Singa tewaaliwo kabonero ka “ahaa” oba “tewali kintu kyonna” (ziro) , ensengekera ya namba eno teyandisobodde kukozesebwa mu kubaza . nnnnnnnnnnnnnn Ensonjola ya namba etwaliramu ebika by’ennamba eby’enjawulo nga bye bino mu Luganda : • Zeero (Zero) • Namba ezibala (counting numbers) • Namba ezinjuvu ( Whole numbers) • Emikutule (Fractions) • Namba eza ndagakifo (ordinal numbers) • Namba eza ndagalinnya (nominal numbers) • Namba eza kyegabanya (even numbers) • Namba eza kigaanira (odd numbers) • Emitonnyeze (Decimal numbers) • namba eza kiddannyuma (namba za negatiivu), • namba eza kiddamaso (namba eza pozitiivu • Nmba eza kibalira mpuyibbiri (yintegya) • namba ez’omugeragerenyo (rational number), • namba ezitali za mugerageranyo (irrational numbers), • Namba zennyini (rea numbers) • namba enzibuwavu (complex numbers), • namba ez’omuteeberezo (imaginary numbers)

Namba eza kyegabanya ne Kigaanira: (Even and Odd Numbers)

Ennamba Kigaanira oba Kyegabanya 1 Kigaanira 2 Kyegabanya 3 Kigaanira 4 Kyegabanya 5 Kigaanira 6 Kyegabanya 7 Kigaanira 8 Kyegabanya 9 Kigaanira 10 Kyegabanya

Namba zisonjolwa okusinziira ku bika byazo. Ekika kya namba ekisooka ky’ekika kye watandikirako okuyiga; ntegeeza namba ez’ensibo era eziyitibwa ez’obutonde oba namba ezibala (eza kibazo) ezifaanana bwe ziti:

1, 2, 3, 4, 5, 6, ...

Ekika kya namba ekiddirira namba ezibala oba namba ez’ensibo ze namba enzijivu, zino nga ze namba ez’ensibo oba ezibala nga ziriko ne ziro: 0, 1, 2, 3, 4, 5, 6, ...

Bw’ova ku namba enzijuvu odda ku namba eziyitibwa entegere, kino nga kiva mu kya lungereza “entegers”. Entegere zirimu zero, namba ez’ensibo ne namba eza kiddannyuma (eza negatiivu):

...-7, –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6,-7 ...

Bw’ova ku yintegya, odda ku namba ez’omugerageranyo (rational), oba namba enkutulemu (omukutulo) ezitwalibwa nga omugabizo gw’entegere (divisions of integers). Mu ngeri endala namba enkutulemu etondekebwawo nga ogabiza namba entegere emu n’endala.

Okiraba nti buli kika kya namba ekiddako kiba kirimu ekika oba ebika ebiba bisooka waggulu. Enzijuvu (the wholes) ziba ez’ensibo nga zirimu ziro. Jjukira nti oyinza okufuula entegere okuba namba enkutulemu nga oteeka 1 wansi wayo. Eky’okulabirako entegere 7 era eri mu nkula eya namba enkutulemu 7/1).

Bw’omala okuyiga ku namba enkutulemu (emikutule), oba oddirirra ekisonjolo ekirala ekya namba: zino ze namba ezitasobola kuwandiikibwa nga nkutulemu. Jjukira nti namba enkutulemu era z’oyinza okuyita ez’omugerageranyo ziyinza okuwandiikibwa nga ez’enkomerero (terminating) oba emitonnyeze egy’eddingana (repeating decimals nga 0.5, 0.76, oba 0.333333....). Ku luuyi olulala, namba zonna eziyinza okuwandiikibwa nga emitonnyeze egitedding’ana (non-repeating), oba egitekomya (non-terminating) ziyitibwa zitali za mugerageranyo (rrationals). Eky’okulabirako gwe mulandira gwa kyebiriga ogwa 2 oba namba eya paayi (“3.14159okuva mu essomampima.

Namba ez’omugerageranyo n’ezitali bika bya namba eby’enjawulo. Bw’oteeka awamu ebika bino ebibiri ebya namba ez’emigerageranyo n’ezitali ofuna namba kennyini (“real” number). Okujjako nga onnyonnyose namba ez’omugombagano(complex numbers ), namba ezirimu “i” nga 4 – 3i), buli namba gye wali olabye ebadde namba kennyini. Lwaki ziyitibwa namba kennyini ? Waliwo ezitali namba zennyini? Kin kituufu,zino ze ziyitibwa ez’omuteeberezo (“imaginary” numbers); ze zikola namba ez’omugombagano era kino ennukuta “i” ky’etegeeza.

Ekibuuzo ekisinga okubuuzibwa ku bika bya namba kiwulikika nga “namba yennyini ya mugerageranyo oba namba etali ya mugerageranyo namba kennyini obe si emu ku byombi? Okujjako nga omanyi ebikwata ku bintu eby’omugombagano, buli kintu kye wali okoze mu kibalo kikozesezza namba kennyini. Mu butuufu okujjako nga namba erimu “i”, eba namba yennyini. Mu nnyanjula y’ebika bya namba ka tutandike ne ziro bwe tuti:

(i)Ziro

Abantu bwe baali bakatandika okuwandiika namba ennene nga 60 baasanga obuzibu okulaga enjawulo wakati wa 6 ne 60 kubanga awatali kabonero ka kubala aka ziro mukaaga ne nkaaga bilabika nga bye bimu mu mpaandiika. Ababalanguzi (mathematicians) kye baakola kwe kuteekawo “enkuumakifo” (placeholder), akabonero ak’enjawulo okulaga nti “mu kifo ekyo tewali digito” yonna.

909 N’olwekyo namba 909 kitegeeza:ebikumi mwenda(lwenda), tewali digito ya makumi ate waliwo ensusuuba mwenda

Endowooza ya ziro(zero) wano we yatandikira era oluvannyuma ababalanguzi ne batandika okujibalira mu namba era kati oyinza okulowooza mu ngeri egamba nti nabadde ne yiika z’ettaka 75 ne ntundako yiika 75, kati nina yiika ziro, ekitegeeza nti kati sirina yiika n’emu.

(ii) Ennamba za Kibazo

Namba ya Kibazo (Cardinal number/counting number) y’ebala obungi (quantity) era ye namba eraga ebintu bimeka ebiri awo oba ebiri omwo. Eky’okulabirako ekugamba nti kimu, bibiri, bisatu, bina, bitaano, kuminakimu, kikumi kimumimwabiri, nokweyongerayo.

Mu kubala, tuyinza okukozesa namba 1, 2, 3, 4, 5, 6, 7, 8, 9 ,10, 11, 12,… nokweyongerayo. Abantu babadde bakozesa namba okubala enkumi n’enkumi z’emyaka egiyise era okubala kintu kya buttonde. Omuntu ayinza okwogera ku byalina ng’agamba nti alina abaana kumi na bataano, n’eddundiro eririko embuzi abiri mu nnya, abakyaala munaana, n’okweyongerayo. Namba za kibazo (namba ezibala) ze namba ezikozesebwa okubala ng’ezo waggulu: era namba zino zibadde zikozesebwa abantu ova emabega emyaka nkumu.

Namba ya kibazo n’olwekyo ekugamba “emirundi emeka” kyokka era zino ziyitibwa “namba za ndagabungi” (counting numbers,) kubanga ziraga obungi. Omuntu bw’aba alaga ebintu by’agenda okugula mu katale akozesa namba za ndagabungi bw’ati”:

 Enkoko 5  Amatooke 10  Kiro z’omukyeere 3  Kiro za sukaali 6  Embuzi 1 N’omusomesa nga ayigiriza omwana okubala akozesa namba za kibazo bw’ati: ziro 0 emu 1 bbiri 2 ssatu 3 , n’okweyongerayo

Namba ya kibazo teba ya “mukutulo/mutunduwazo (fraction) yadde “omutonnyeze/omukumito” (decimal). Eba namba nzijuvu (whole number).

(iii) Namba za Ndagakifo (ezisengeka)

 Namba endagakifo oba edding’anya (Ordinal Number) ye namba erondobamu okusinziira ku kifo ekintu mwe kisengekeddwa mu lunyiriri(ist) bweti “. 1ka, 2li, 3tu, 4na, 5no, 6ga, 7vu, 8na, 9da, 10mi, 11mu, n’okweyongerayo. Namba za ndagakifo bwe ziba zoogera ku kintu ng’ojjeko 1 eyatulwa ekisooka, eziddako zaatulwa na “kya”……oba “ekyo….”.okugeza kya kubiri oba eky’okubiri (2li) kyokka bwe ziba ziraga kifo kya muntu zitandika na “wa….”.eky’okulabirako wa kubiri, wa kusatu, wa kuna, n’okweyongerayo. Ate olunaku luba lwa kusooka, lwa kubiri

Ekyokulabirako:

Mu kkala za musoke“. Kkala eya bbululu ekwata ekifo kya 6ga. Omwana yakoze bulungi; yabadde wa 8na mu kibiina kyonna. Ku lusooka (1ka) wajja kubaawo okuwandiisa abaana ku ssomero lya Twekembe Kindergarten. N’olwekyo namba za endagakifo zikugamba ensengeka oba ekifo ky’ekintu ekiri mu musengeko (set).

(iv) Namba z’endagalinnya

“Namba ya ndagalinnya” (nominal number), era gy’oyinza okuyita namba ya ndagamuntu oba ndagakintu, ewa ekintu/omuntu erinnya –gamba nga namba ya ssimu, omuzannyi mu ttiimu, abakulembeze ab’ensikirano abalina amannya agafaanagana, abakulembeze b’eddiini.Ebyokulabirako:  Omujoozi namba 7  Muteesa 2  Paapa Bendikito XVI  Ssimu namba 071000000  Ekifo namba 23  Mmeeza namba 9

(v) Namba enzijuvu ne Yintengya

Omugereeso gwa namba” (number theory) era gulimu: (a) Namba enzijuvu (whole numbers): 0,1,2,3,4,5,6,7,8,9, ..... Kati ka tugatte ziro ku namba za kibazo(counting numbers) tukole omusengeko omupya ogwa namba. Wano we wava erinnya eppya erya namba eziyitibwa “ Namba Enzijuvu”(Whole Numbers) ezikola omusengeko (set) bwe guti : {0, 1, 2, 3, 4, 5, 6, ...n’okweyongerayo}. Eno wanso ye layini ya namba enzijuvu:

Namba eza Negatiivu

Ebyafaayo by’ekibalangulo byafaayo bya babalunguzi kwebuuza bibuuzo n’okunoonya ansa zabyo. Ekibuuzo ekimu eky’okwebuuza kiri: “singa odda oludda olumu oyinza okudda mu lwa kikontana (opposite way)?”

Omubalanguzi ayinza okubala ekiddamaaso (forwards) bwati: 1, 2, 3, 4,5, 6 ... naye kiri kitya ng’abala ekiddannyuma (backwards) bwati : …6,5, 4,3,2,2,0 ?

Eky’okuddamu kiri nti aba abalira mu namba za negatiivu (negative numbers).

Kati wano omubalanguzi ayinza okubala ekiddamaaso n’ekiddannyuma mu entakoma nga bw’ayagadde. Naye namba ddala eba etya eya negatiivu? Kino kibaawo nga namba eri wansi wa ziro. Eky’okulabirkao mu kupima tempulikya oyinza okusonjola digiri ziro eza Selisiyaasi (zero degrees Celsius (0° C) ekisonjolwa ng punkuti (point) amazzi we gakwatira kyokka singa amazzi geyongera okunnyogoga ennyo okuva ku tempulikya eya ziro Selisiyaasi, tuba tukiraga nga tempulikya eza negatiivu.

Oyinza n’okwogera ku magoba aga negatiivu (negative profits) kubanga singa osuubula ebijanjaalo bya mitwalo kumi obitunde ofunemu amagoba, kyokka olw’okuba minzaani yo efunye obuzibu nga ebala bubi(bingi okusinga ku kituufu) n’omaliriza okutunda ng’okunganyizzaawo emitwalo musanvu wano oba mu magoba ga negatiivu kubanga ku sente ze wasize kweyawuddeko emitwalo esatu, ekitegeeza nti mu kifo ky’okukola amagoba yafiiliddwa bufiilwa emitwalo esatu ku sente ze yasigamu.Namba za kirumirampuyibbiri/Namba Yintegya (Integers) :...., -5,-4,-3,-2,-1,0,1,2,3,4,5,6,7, .... Singa namba eza negatiivu tuziteeka wamu ne namba enzijuvu (whole numbers), tufuna omusengeko gwa namba empya(new set of numbers) eziyitibwa entegere (integers). Yintegya ziba zisengekeddwa bwe ziti: {...,-5,-4 -3, -2, -1, 0, 1, 2, 3,4,5,….}

Yintegya zirimu ne ziro, namba za kibazo, ne namba za kibazo eza negatiivu(the negative of the counting numbers), kin one kikola endaga ya namba eyeyongerayo mu ntakoma (indefinitely) ku buli ludda nga mu layini ya namba eno:

Yintegya (Integers) ze namba ezitali nkutulemu kyokka nga zirimu eza negatiivu n’eza pozitiivu awamu ne ziro. Yintegya zikolebwamu omusengeko (set) ku layini ya namba (number line) bw’eti:

Mu layini ya yintegya, ziro(0) teba pozitiivu yadde negatiivu. Kyokka namba nga 1/3, 5/7, 0.009. 1.1, 7.7 oba 8.0009922 si yintegya kubanga nkutulemu

(vi) Namba ez’ensibo Waliwo n’eziyitibwa Namba ez’Ensibo oba namba ez’obutonde (natural Numbers). Omulamwa gwa namba ez’ensibo gutegeeza: Namba eza kibazo: {1, 2, 3, 4,5,6,7,8 ,……..} Oba namba enzijuvu: {0, 1, 2, 3, 4,5,...} Wano oba okilaba nti wakyaliwo okukuba empawa ku kuba nti ziro nayo namba ya “nsibo” oba nedda.

(vii) Namba za Kyegabanya (Even numbers)

Namba za kyegabanya (Even numbers) zisobola okugabanyizibwamu emirundi ebiri awatali nsuusuuba/nfikko (remainder). Enamba za kigabanya zikomekkereza ne digito eya, 0,2,4,6,8 oba 10. N’olwekyo namba 0,2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 namba za kigabanya.

(vii) Namba za kigaanira (Odd numbers).

Obutaba nga namba za kyegabanya, namba za kigaanira (odd numbers) tezisobola kugabanyizibwamu mu bwenkanya bibinja bibiri awatali kufikkiza/nfikkira (remainder). Enamba za kigaanira ziba zikomekkereza n’enfikkizo omuli digito (digits) zino: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31…….Zino zonna namba za kigabanya.

Ebisonjozo bya namba eza kigaanira n’eza kyegabanya

(a)Okugatta:  Kigaanira + KIgaanira = Kyegabanya  Kigaanira + Kyegabanya = Kigaanira  Kyegabanya +Kyegabanya =Kyegabanya (b)Okwawuza :  Kigaanira - KIgaanira = Kyegabanya  Kigaanira - Kyegabanya = Kigaanira  Kyegabanya - kyegabanya = Kyegabanya ( c ) Okukubisaamu  Kigaanira x kigaanira = Kigaanira  KIgaanira x Kyegabanya =Kyegabanya  Kyegabanyax kyegabanya = Kyegabanya (d) Okugabiza:  Kigaanira gabiza kigaanira(Kigaanira / Kigaanira = Kigaanira  Kyegaabanya gabiza kigaanira(Even / Odd) = Kyegabanya  Kyegabanya gabiza kyegabanya (Even / Even) = Kyegabanya

(viii)Emikutule (Fractions)

Singa oba n’omukyungwa gumu ng’oyagala ogugabane kyenkanye n’omuntu omulala (ow’okubiri) oba olina okugusalamu ebitundu bibiri ebyenkanankana. Wano oba otondeseewo namba ey’ekika ekirala kubanga oba nga adidde namba 1 n’ogigabanyamu namba endala 2 okufuna ekiyitibwa kimukyakubiri (1/2). Kye kimu ekyandiibaddewo singa obadde olina emiyembe mukaaga (6) nga oyagala ogigabanyize kyenkanyi mu ombi basatu (3). Buli omu aba afuna ebitundu mukaaga bya kusatu ekitegeeza mukaaga ebigabizemu ssatu ekiwandiikibwa nga 6/3) buli omu ky’afuna.Ekika kya namba kino kye kiyitibwa omukutule (fraction) oba emikutule (fractions).

(ix) Namba entunduwavu (Rational Numbers) nga 5/1, 1/2, 1.75, -97/3... Namba entunduwavu (rational number) ye namba yonna eyinza okuwandiikibwa ng’ekitundu /omugerageranyo (ratio) gw’entegere ebbiri mu ngeri ey’okugerageranya. Kino kitegeeza nti namba enzijuvu eba ya kitundu bwe tuba tuyinza okugiwandiika ng’enkutulemu (fraction) nga kinnawaggulu (numerator) ne kinnawansi (denominator) zombi ntegere (integers).

Mu sayansi wa namba, ekigambo “namba entunduwaze” kiva mu kigambo “kitundu kya namba” oba omugerageranyo (ratio) kubanga namba entunduwaze (rational numbers) ze zisobola okuwandiikibwa mu butunduwavu(in the ratio form) a/b, nga a ne b namba ntegere(integers).

Buli namba entegere eba ntunduwaze (rational numba) kubanga eyinza okuwandiikibwa ng’enkutulemu bw’eti n/1. Ekyokulabirako 1 = 1/1 , 2 = 2/1, 3=, 3/1 n’okweyongerayo. Kyokka namba nga 1/2, 1/3, ¼, 64444288/968463, ne -1/8 nazo ntunduwaze kubanga nkutulemu ezirina kinawaggulu ne kinnawansi nga ntegere.

N’olwekyo omugereko (set) gwa namba entunduwaze (rational numbers) gwonna guba gwefaananyirizaako bwe guti: (5/9, -7, 2.15, -103/13, n’endala. Namba yonna eyinza okuwandiikibwa nga omukutule eyitibwa Namba ntunduwaze(Rational Number). Kino kitegeeza singa x ne y ze namba entegere (integers), kitegeeza x/y namba ntunduwaze.

Eky’okulabirako: singa x ye 7 ate nga y ye 2, kitegeeza x/y = 7/2 = 3.5 nga namba entunduwaze. Embaranguza eno w’etakolera wokka nga y eri ziro kubanga okugabanya ne ziro tekirina nsonjola. Manya: Mu namba z’entunduwaze : x/y : x ne y ziba namba ntegere, naye y teyinza kuba ziro. N’olwekyo kimu kya kibiri (½) eba namba ntunduwaze ne 2 nayo ntunduwaze kubanga oyinza okugiwandiika nga 2/1. Okutwalira awamu namba z’emigerageranyo zirimu:

 Yintegya zonna

 Emikutule gyonna

Waliwo namba ezitali ntunduwaze (numbers that are not rational number). Zino tuziyita namba ezitali ntunduwavu (Irrational).

(x) Namba ezitali ntuduwavu (Irrational Numbers)

Namba ezitali ntunduwavu (Irrational numbers) ze namba eziyinza okuwandikibwa ng’emikumito (decimals) naye zombie nga za mikutulo (fractions). Omulandira gwa kyebiriga (square root) ogwa 2 si mutunduwavu kubanga tegusobola kuwandiikibwa nga kitundu (ratio) kya yintegya 2. Mu butuufu waliwo namba endala nyingi ezitali ntunduwavu. Emu ku zzo ye Pi (π).Namba endala ezitali ntunduwavu mulimu omulandira gwa kyebiriga ogwa 3, ogwa 5, pi ,ne e,.; Pi = 3.14159... ate E = 2.71828....

Paayi (Pi) eba namba etali ntunduwavu kubanga tesobola kulagibwa nga kitundu oba nkutulemu (fraction) eya yintegya ebbiri; terina muwendo gwa mutonnyeze mugere( it has no exact decimal equivalent). Kino kye kimu n’omulandira gwa kyekubisa gwa 2 ogutasobola kuwandiikibwa nga mukutulo.

Manya:

 Namba ezitali ezitali ntunduwavu za mugaso kubanga zetaagibwa:  Okuzuula disitansi y’ebisittalo (diagonal distance) ebisaza mu kyebiriga.  Okubalanguza emibalanguzo (calculations) egy’enjawulo egy’ebikulungirivu n’ebyebulungirivu (nga tukozesa π).

(xi) Namba Zennyini

Namba Zennyini (Real Numbers) zirimu:  Namba entunduwavu ne  Namba ezitali ntunduwavu  Mu namba zennyini x : x eyinza okuba entunduwavu oba etali ntunduwavu. Namba kennyini ye namba ze tutera okukozesa eza pozitiivu oba negatiivu, ennene oba entono, enzijuvu oba ez’emikutule oba ez’emitonnyeze nga 1, 15.82, -0.1, 3/4, n’okweyongerayo… Ziyitibwa namba zennyini kubanga si za muteeberezo (not Imaginary)

(xii) Namba Ez’omuteeberezo (Imaginary Numbers).

Eno ye namba yonna nga yebirigiddwa (when squared) ekivaamu kiba mu negatiivu.Singa webiriza namba kennyini yonna oba ofuna eya pozitiivu oba ziro.Eky’okulabirako : 2×2=4, ne (-2)×(-2) nakyo kivaamu =4.

Kakati tuyinza tutya okwebiriza namba yonna ne tufuna negatiivu? Bwe tukuba ekifaananyi nti tuyinza, tuba tuyinza era kivaamu nti namba nga eyo erabika nga etasoboka, erina emigaso era yeyambisibwa okubaza obukunizo bwennyini. Namunigina ya namba eyomuteeberezo ye √(-1) (omurandira gwa kyebiriga ogwa kyawuza 1(minus one) era akabonero kayo ke i, oba j. Namba z’omoteeberezo nga yebiriziddwa (squared) etuwa ekivaamu nga kiri mu negatiivu.

(xiii) Namba enzibuwavu (Complex Numbers)

Singa oteeka wamu namba kennyini ne namba ez’omuteeberezo(imaginaery )ofuna eyitibwa namba enzibuwavu (Complex Number). Eky’okulabirako: 3 + 2i 27.2 – 11.05i

Namba ey’enzibuwavu ebaako ekitundu kya namba kennyini n’ekya namba ey’omuteeberezo e naye ga buli kitundu kiyinza okuba ziro. N’olwekyo namba kennyini eyinza okuba namba enzibuwavu n’ekitundu kyennyini ekya 0. 7i eba namba enzibuwavu kubanga yenkanankana 0 + 7i. N’olwekyo namba enzibuwavu ebaamu namba kennyini ne namba ey’omuteeberezo n’emigattiko gyazo gyonna.

Eky’okulabirako:  laga omulandira gwa kyebiriga ogwa -9?

Eky’okuddamu: = √(9 × -1) = √(9) × √(-1) = 3 × √(-1) = 3i Kino si kikulu! Omurandira gwa kyebiriga ogwa -9 guba mulandira gwa kyebiriga ogwa +9 nga okubisizzaamu emirundi i. Okutwalira awamu: √(-x) = i√x

kasita tuba nga akanyukuta ka “i” ako akatono tukataddewo okutujjukiza nti tukyetaaga okukubisaamu ne √-1 tuba tusigala ku mulamwa. Nga tweyambisa akanyukuta i, era tusobola okutuuka ku mulamwa omupya. Eky’okulabirako: Balanguza x2 = -1 Singa tuba tukozesa namba kennyini tewali kye tusobola kukolawo, kyokk kati okubaako kye tukolawo. Ansa eri x = √-1 = i

Namba ez’ensibo (Namba ez’obutonde) Namba ez’obutonde era eziyitibwa namba ez’nsibo ze zisinga okuba ez’omugaso era ze zino ze namba zetusinga okukozesesa mu bulamu obwa bulijjo mu kibalangulo. Namba ez’ensibo zikiikiridddwa akabonero ‘N’. N = {1,2,3,….}

Namba ez’obutonde era ziyitibwa namba ezibala(counting numbers) kubanga ze zeyambisibwa okubala ebintu.

Namba enzijuvu (Whole Numbers ) Namba 0, 1, 3,3, 4 … ze ziyitibwa namba enzijuvu. Ennyukuta ' W' y’eraga omugereko gw’ekkung’aanyo lya namba enzijuvu lyonna.Tulaba nti W= {0, 1, 2…}

Okiraba nti namba ez’obutonde zonna ate era namba nzijuvu naye 0 si namba ya butonde.

Mu namba enzijuvu temubaamu namba nkutulemu oba emikutule are temuli mitonnyeze yadde namba eza negatiivu.

Namba 0, 1, -1, 2, -2 … ziba yintegya nga ku zino, 1, 2, 3, … ziba za pozitiivu ate -1, -2, -3,… yintegya za negatiivu. Yintegya ziragibwa na ‘ Z '.

Yintegya zezibalirwamu ziro yadde emitonnyeze oba emikutule. Ebinnyonnyozo bya namba Ekinnyonnyozo ekya empanyisabifo (Commutative Property) Okugatta n’okukubisaamu mibalanguzo egirina ebisonjozo eby’empanyisabifo. Mu kugatta : a + b = b + a Kino kitegeeza nti mu kugatta empanyidabifo tekyusa alizaati(ansa). Ate era mu kukubisaamu: a * b = b * a

Kino kiraga nti ne mu kukubisaamu empanyisabifo tekyusa alizaati.

Ekinnyonnyozo eby’enkwanaganya (Associative property) Ate era okugatta n’okukubisaamu birina ebinnyonnyozo eby’enkwanaganya (associative properties). Okugatta: a + ( b + c) = (a + b) + c Kino kitegeeza nti mukugatta okukwanaganya namba emu n’endala tekikyusa alizaati Okukubisaamu : x* (y *z) = (x * y) * z Kino nakyo kiraga nti mu kukubisaamu okukwanaganya namba emu n’endala tekikyusa alizaati. Ekinnyonnyozo ekya Kibunyisa (Distributive property) Ekinnyonnyozo ekya kibunyisa kisonjola akakwate akali wakati w’okugatta n’okukubisaamu. Kikkiriza okukubisaaamu okubuna wonna ku buli namba ey’okugattibwa. Okukubisaamu: a(b + c) = ab + ac Kino kitegeeza nti okubunyisa okukubisaamu ku buli namba tekikyusa alizaati. Namba eza kigaanira ne namba eza Kyegabanya Ensengekera ya namba enzijuvu eri mu biti bibiri, namba eza kyegabanya (even numbers) ne namba eza kigaanira (odd numbers). Singa yintegya eba egabizibwamu 2 (oba nga yintegya nkubise ya 2), namba eno eba namba ya kyegabanya olwo namba ezisigadde ne ziba za kigaanira (namba ezitasobola kugabizibwamu bbiri awatali kuleka nfikkizo). Okutwalira awamu, namba ezikomekkereza ne 0, 2, 4, 6, 8 zitwalibwa okuba namba za kyegabanya. Eky’okulabirako 16, 34, 792, 58 ate mu namba eza kigaanira mulimu 17, 21, 101, 133, 3.

Ziro eba namba ya kyegabanya kubanga etuukiriza ebigoberero bya namba za kyegabanya. Ziro eba yintegya ate nga nkubise ya 2. Ebikunizo: (i) Lag anti namba 85 ya kigaanira oba ya kyegabanya. Ansa: Namba 85 ya kigaanira kubanga nambiso ekomekkereza eri 5. (ii) Lag anti namba 67 ya kigaanira oba ya kyegabanya. Namba 67 ya kigaanira kubanga eno nambiso (numeral) ekomekkereza 7. (iii) Lag anti namba 19 ya kyegabanya oba ya kigaanira. Namba 19 ya kigaanira kubanga eno nambiso ekomekkereza na 9. (iv) Laga nti namba 136 ya kigaanira oba ya kyegabanya. Namba 136 ya kyegabanya kubanga eno nambiso

ekomekkereza na 6. (v) Laga nti namba 2348 ya kigaanira oba ya kyegabanya.. Olw’okuba nimbiso ya 2348 ekomekkereza na namba 8, kinokitegeeza nti namba eno ya kyegabanya. (vi) Laga nti namba 1254 ya kigaanira oba ya kyegabanya. Olw’okuba nambiso 1254 ekomekkereza na 4 kitegeeza nti eno namba ya kyegabanya.

Ebisonjozo bya namba za Kyegabanya ne Kigaanira (i) Okugatta Kigaanira + Kigaanira = Kyegabanya Kigaanira +Kyegabanya = Kigaanira Kyegabanya + Kyegabanya =Kyegabanya

(ii) Okwawuza: Kigaanira - Kigaanira = Kyegabanya Kigaanira - Kyegabanya = Kigaanira Kyegabanya - Kyegabanya = Kyegabanya

(iii) Okukubisaamu: Kigaanira x Kigaanira = Kigaanira Kigaanira x Kyegabanya = Kyegabanya Kyegabanya x Kyegabanya = Kyegabanya

(iv) Okugabiza: Kigaanira / Kigaanira = Kigaanira Kyegabanya/ Kigaanira = Kyegabanya Kyegabanya / Kyegabanya = Kyegabanya Ebikunizo 1. Ku nkalala zino luliwa olw’amannya (namba)? Luliwa olw’obubonero(nambiso) ? a) Emu, bbiri, satu, nnya, ttaano, mukaaga, musanvu, munaana, mwenda b) 1,2,3,4,5,6,7,8 ,9… 2. Wandiika akabonero ka buli limu ku mannya gano: (i) Kikumi (ii) Lukumi (iii) Emu (iv) Satu (v) Kumi na ttaano 3. Mu kibalangulo ky’okubala (mathematics of counting), erinnya ly’akabonero liyitibwa: a) Lukoloboze b) Katonnyeze c) Namba d) Nambiso 4. Mu kibalangulo ky’okubala, akabonero k’erinnya (namba) kayitibwa: a) Kitonnyeze (dot) b) Nambiso c) Mugendo d) Kiserengeto Ensulike oba namba eya kasulike (Inverse and reciprocal)

Omulamwa ogw’ensulike oba namba ya kasulike y’ennamba guva mu kusulika mukutule. Ensulike ya namba kivvuunulwa nnga the reciprocal of a number.

Buli namba enzijuvu etegeeza kino 1 = 1/1, 2 = 2/1, 3 = 3/1 . Ensulike ya 1 oba 1/1 esigala 1 oba 1/1, ensulike ya 2 oba 2/1 eri ½ ate eya 3 oba 3/1 eba 1/3.

Namba 1/x, ekubisidwamu x ewa enkubise (product) 1, era eyitibwa nsulike (a reciprocal).

Mu kibalangulo ensulike (reciprocal) eya namba x (x/1), eragibwa nga 1/x oba x−1.

Ensulike ey’omukutulo (fraction) x/y eba y/x. Ekyokulabirako, ensulike (reciprocal) ya 7 eba 1/7), ate ensulike ya 0.25 n’ebeera 1/0.25 (1gabanyaamu 0.25) = 4.

Namba eya Kifuulannenge (Inverse number)

Namba eya kifuulannenge (inverse number) oba ensulike (reciprocal number) eya namba etali ziro oba enzibuwavu (complex number) eyinza okulagibwa ne a-1 .

(a)Namba bbiri ziba namba za kifuulannenge buli emu ku yinnaayo singa ekitondeko kyazo kiba kikola 1.

(b)Singa a (≠0) eba eweereddwa nga bc, kitegeeza nti namba ya kifuulannenge yayo eba bc-1 =cb.

(c)Okutwalira awamu, bw’oyogera ku kintu ekya kifuulannenge , kitgeeeza: Okuzza ku ntandikwa oba okuzza ekintu awasooka

Mu sessomo ly’ekibalangalo , twogera ne ku mukwataganyo ogwa kifuulannenge (inverse function) oba namba eza kifuulannenge (inverse numbers). Ekifuulannenge kaba kakwate akalimu enkyuso bbiri nga bwe wabaawo okulinnyisa enkyuso emu kiviirako okukendeera (okukka) kw’enkyuso endala. Eky’okulabirako volima ya ggaasi eri mu muugerageranyo ogwa kifuulannenge ku kanyigirizi ka.

Bwe ziba erementi za kikung’anyo (set elements) kifuulannenge ow’engattiso ekya erementi x kiba -x, ate ekifuulannge ky’enkubiso ekya erementi eya x eba 1/x

Mu kibalangalo, ekifuulannenge kitegeeza ekintu ekya kikontana (opposite) nga mu nziring’ana ey’ekiddannyuma. Kifuulannenge kitegeeza omugogo gwa erementi (a, b) nga ekivaamu eba erementi eyefaanayiriza nga: ( b,a )

Kasulike gwe muwendo oguba gulagiddwa ekifuulannengemu ngeri ey’okusulika.Kino era kiyitibwa kifuulannenge ey’enkubiso (multiplicative inverse)

Kiddannyuma (negatiivu) ogwe muwendo oguba gulagiddwa mu negatiivu.Kino era kiyitibwa kifuulannenge ey’omugattiso (additive inverse). Mu migendaganyo (in proportions) kifuulannenge abaamu ennyingo nga emu bw’erinnyisibwa kiviirako endala okukka wansi.

Kifuulannenge era kitegeeza “Engattiso eya kifuulannenge” (additive inverse), ekitegeeza omugogo gwa namba nga omugatte gwazo guba ziro. Eky’okulabirako engattiso eya kifuulannenge eya -345 eri 345 kubanga -345+345 = 0

Enkubiso eya kifuulannenge kitegeeza kasulike oba omugogo gwa namba ogumu nga ekitondeko kyazo kivaami 1. Ekyokulabirako kasulike ya 5/7 eri 7/5 era bw’okubisa mu namba zino ( 5/7 x 7/5 ) ofunaa 1 kubanga 5/7 x 7/5 = 1.

Kikontana y’ekintu (opposite)k’ akakwate akaba wakati w’ebikontana nu sayansi ez’entabaganya akayitibwa embeera eya mugobansonga (dialectics). Laba ebintu bya kikontana bino. 1. Kinontana w’omuwala aba mulenzi 2. Kikontana w’ekiwanvu kiba ba kimpi 3. Kikontana w’emisanaa kiba kiro.

Ebikunizo

1. Zuula ensulike ya 1/3 ? A - 1/3 B -3 C 1/9 D 3

2. Zuula ensulike ya 5½? A 10½ B -11/2 C 11/2 D 11

3. Zuula ensulike ya 7/11 A -7/11 B 11/7 C 7 D 11

Okugabiza mu Ziro

Mu mazima okugabiza mu ziro tekirina busonjole Ekyokulabirako: singa bakugamba okugabanya ebitabo kumi mu kuraasi etariimu bayizi, kino kisobok? Tekisoboka.Tosobola kugabanya mu baana ziro mu ngeri y’emu tosobola kugabiza na ziro. Singa ogabiza 10 ne 2 ofuna 5 kubanga 5 x 2 ofuna 10 Naye 10 singa ogezaako okugabiza 10 ne ziro oba tolina ky’amakulu ky’oyuukako kubanga 0 emirundi kumi tekirina nsonjola yonna

Bw’okubisa mu 0 ofuna 0

Awo oba okiraba nti okugabiza ne ziro tekisoboka. Ebibalo nga 1/0 ne 0/0, 0 ×(1/0)= 0 tebiriiyo mu ngeri y’emu ebibalo nga: 0×1 = 0, 0×2 = 0, 0x 3=0, n’okweyongerayo bwe bitaliiyo.

Ebyo byonna tusobola kubisonjola.Okutwalira awamu okugabiza oba okukubisa mu ziro tekikola makulu. Waliwo enjawulo wakati w’emilamwa kifuulannenge ne kasulike? Kifuulannenge (inverse) ne kasulike (reciprocal) miramwa egikozesebwa ennyo mu sessomo ly’ekibalangulo.

Kasulike eba nkubiso eyakifuulannenge (multiplicative inverse) eya namba, ekiragibwa nga 1/a oba 1/n.ekisonjolwa nga namba singa eba ekubisiddwamu namba yennyini evaamu ekitondeko ky’emu (1). Singa olina namba yennyini ky’okola kugabiza mu namba en one 1 olwo ofune kifuulannnenge yayo.Kifuulannenge yayo eno y’eba eyitibwa “kasulike (reciprocal)

Kyokka yadde emiramwa gino gikwataganye nnyo ddala, waliwonenjawulo wakati wa kifuulannnenge ne kasulike ze tugenda okwekenneenya. Bwe guba mukutule, okutuuka ku kasulike yagwo kiba kyangu kubanga kye wetaaga okukola kwe kukyusaganya kinnawansi ne kinnawaggulu. Omulamwa gwa kasulike guyamba nnyo mu sessomo ly’ekibalangulo kubanga gugonza ebikunizo by’ekibalo bingi ddala. Eky’okulabirako:

7/(1/3) kifuulibwa 7 x 3 = 21. Mu kifo ky’okugabiza 7 ne 1/3, tukubisa 7 ne kifuulannenge ya 1/3,eno nga ye 3.

Nga ojjeeko enkubiso eza kifuulannnge(multiplicative inverses) nga bwe tulabye waggulu, waliwo n’engattiso eza kifuulannenge(additive inverses) ezetaaga okugattibwa ku namba ebaddewo okufuna omugatte ogwa ziro(si 1 nga bwe kiri mu nkubiso eya kifuulannenge). N’olwekyo namba bw’eba p, engattiso yayo eya kifuulannenge eba –p ne kiba nti p+ (-p) = 0.

Omukwataganyo ogwa kasulike (reciprocal function) guba nga “kayipabola (hyperbola). Ggulaafu ya kayipabola eba y = 1/x. Osaana okimanyi nti kifuulannenge eyinza okuba ey’omukwataganyo gwonna, kasita eba nga ekola ekyo kubanga gulaafu okuba eya kifuulannenge, erina okutuukiriza engezeso ya layini ey’omugalamiro (the horizontal line test). Etera okulagibwa nga f^-1(x). Omukwataganyo ogwa kifuulannenge ggwo gudda ku ludda olw’omukwataganyo ogwa kikontana f(x).`1

Weetegereze:

(i) Ekifuulannenge ky’okugatta kiba kwawuza

Okugatta kukutwala mu maaso ate okwawuza kukuzza ku luda olwa kikontana, emabega gy’ova. Ekyokulabirako: 30 + 5 = 35 kiyinza okuzzibwa ennyuma ne 35 - 5 = 20 (okudda we twatandikidde). Ekyokulabiako 35-5 = 30 nagwo guyinza okuzzibwa ennyuma nga tugatta 30+ 5 =35 (okudda ku kyasoose).

(ii) Kifuulannenge y’okukubisa kuba kugabiza

Okukubisaamu kuyinza okulambululwa (okuzzibwayo oba okusumululwa) okuyita mu kugabizaamu.Eky’okulabirako:5x7 = 35 kino kiyinza okuzzibwa emabega okuyita mu kugabizaamu bwoti: 35/7 = 5 oba 25/5 = 7.

N’olwekyo okugabiza kusumululwa na kukubisaamu. Eky’okulabirako: 15 / 3 = 5 kiyinza okuzzibwa obuto nga tukubisa 5 × 3 = 15 Naye si na ziro kubanga tekisoboka kugabiza na 0. ekyokulabirako: 2,438 × 0 = 0 kino tekisobola kuzzibwayo na 0/0 =. Kisoboka?

Kasulike (Reciprocal).

Mu ngeri ennyangi kino kiragibwa nga: 1/namba oba emu gabizaaamu namba Singa namba eba 13 eba kye kimu ne 13/.Kasulike yayo n’eba 1/13

N’olwekyo okufuna kasulike ya yintegya oba namba enzijuvu ky’okola kwe kugabiza namba eyo ne 1, ekifaanana n’okusulika oba okuvuunika namba eyo nga kinnawansi yayo etwalibwa okuba 1. Eky’okulabirako kasulike ya 3 eri 1/3 (kimu kya kusatu). Laba wano mu taabulo:

Namba Nga omukutule Kasulike 2 2/1 ½ 3 3/1 1/3 4 4/1 ¼ 6 6/1 1/6 7 7/1 1/7 8 8/1 1/8 9 9/1 1/9 10 10/1 1/10 11 11/1 1/11 20 20/1 1/20 30 30/1 1/30 33 33/1 1/33 100 100/1 1/100

Jjukira nti buli namba enzijuvu eba ya mugerageranyo (rational / fractional). Awo okiraba nti kiringa kusulika oba kuvuunika namba ey’omukutule, kinnawansi ne kidda waggulu ate kinnawaggulu ne kidda wansi. Namba Kasulike oba ensulike yayo 2 ½ 3 1/3 1,000 1/1,000

Mu mikutule gyennyini (egitali namba enzijuvu), omukutule gwonna gusulike, ekitundu ekibadde wansi kidde waggulu ate ekibadde waggulu ne kidda wansi. Ekyo ky’ekiyitibwa kasulike. Eky’okulabirako . Kasulike oba ensulike ya 5/7 eba 7/5

Okwekyusaganya (flipping/swapping)

Singa onoonya ensulike ya kasulike oba odda we watandikidde.Eky’okulabirako:  Ensulike ya 6 eba 1/6  Ensulike ya of 1/6 eba 6 4 (ezzeeyo ku 6)

Jjukira nti “kasirike” oba “ensulike” kyekuusiza ku kya lulattini “reciprocus” ekitegeeza okuddayo (returning) okwefaananyiriza okugenda ku ssomero ate n’oddayo eka olwegggulo.

Manya:  Buli namba erina kasulike okujjako ziro kubanga 1/0 tekirina nsonjola.  Bw’okubisa namba n’ensulike yayo oba ofuna 1  Okufuna kasulike w’omukutule, kyusaganya kinnawaggulu ne kinnawansi.  Okuzuula kasulike y’omukutule ogw’omugattiko, olina kusooka ku gufuula mukutule ogutatuukiridde, olyoke ogukyusaganye.

Ebyokulabirako: 3 × (1/3) = 3 x 1/3 = 1

1000 × (1/1000) = 1000 x 1/1000 = 1 Mu butuufu kino kikulaga engeri y’okusonjolamu kasulike nga ogamba nti: Kasulike kitegeeza namba gy’okubisaamu omuwendo okufuna 1. Olw’okuba kirimu okukubisa mu namba, kiyitibwa kifuulannenge oba enkubiso eya kifuulannenge ("Multiplicative Inverse").

Omukutule Kasulike(Ensulike) yayo 3/8 8/3 5/6 6/5 1/3 3/1 = 3 19/7 7/19

Kasulike y’Omukutule ogw’Omugattiko

Singa obuuzibwa kasulike ya: 5 1/2 ?, ky’okola:

(i). Gufuule omukutule ogutatuukiridde: 5 1/2 = 11/2 (ii) Gusulike oba gufuulennenge: 2/11 Ansa eri: 2/11

Okukubisa omukutule ne kasulike yagwo

Kino kye kiyitibwa enkubiso eya kifuulannenge (multiplicatyive inverse). Bw’okol enkubiso eya kifuulannenge, kiba kitgeeza nti buli lw’okubisa omukutule ne kasulike yagwo ofuna 1. Eby’okulabirako: 3/5 × 5/3 = 1

1/4 × 4 = 1

Omugendaganyo ogwa kifuulannenge (Inverse Proportion)

Abasajja 20 bazimba ennyumba mu naku 6. Omuwendo gwabwe bwe gulinnyisibwa okutuuka ku basajja 30 batwala ennaku 4 okuzimba ennyumba y’emu.Singa omuwendo gw’abasajja guba 40 bamala ennaku 2 okuzimba ennyumba eno.

Namba y’abasajja Ennaku 20 6 30 4 40 3

Oba okiraba nti namba y’abasajja bwe yeyongera, ebiseera ebitwalibwa okuzimba ennyumba nga bikendeera mu mugerageranyo gwe gumu (in the same ratio).

Mu ngeri endala, singa okwongeza obungi obumu kivaako okukka mu bungi obulala, tugamba nti eobungi buno bwombi bulina akakwate aka kifuuannenge.

Okwongera okutangaaza, singa emiwendo ebiri x ne y giba mu mugendaganyo ogwa kifuulannenge (in inverse proportion), kitegeeza ekitondeko kyagyo kiba mu ntakyuka (constant). Kino kitegeza nti: xy=c, nga c= entakyuka.

Mu ky’okulabirako waggulu tulaba nti:

20 x 6 = 120

30 x 4 = 120 40 x 3 = 120

Kiraga nti buli kitondeko kiri mu ntakyuka oba kye kimu. N’olwekyo bwe tuba nga tubalanguza emiwendo (quantities) ezikwatagana kifuulannenge, tusobola okukozesa ekigoberero kino:

Namba y’abasajja Namba y’ennaku 20 6 = 30 4

20 x 6 = 30 x 4

Okutwalira awamu:

Namba y’abasajja Namba y’ennaku A C B D Ac= bd a/d = b/c

Ekyokulabirako:

Payipu nnya ziyinza okujjuza tanka mu dakiika 70.Kitwala bbanga ki okujjuza tanka ne payipu 7?

Ekibazo:

Namba ya Payipu Ebiseera mu dakiika 4 70 7 (zeyongedde) ? (bisse) Ac= bd a/d = b/c

Nga tukukozesa ekigoberero ky’omugendaganyo ogwa kifuulannenge, tufuna

  x=4x70

 x  = Dakiika 40

Ekigana (Percent)

Okukyusa ekigana okukizza mu mitonnyezekyetaagisa okugabizibwamu 100.

Ebyokulabirako:

1% = 1/100 = 0.01 5% = 5/100 = 0.05 10% = 10/100 = 0.1 35% = 35/100 = 0.35 50% = 50/100 = 0.5 100% = 100/100 = 1 230% = 230/100 = 2.

Okuva mu kigana okudda mu mutonnyeze.

1% = 1/100, n’olwekyo okujja mu kigana okudda mu mutonnyeze, omutonnyeze gulina okukubisibwamu 100% . Ebyokulabirako: 0.01 = 1/100 = 1% 0.05 = 5/100 = 5% 0.3 = 30/100 = 30% 0.35 = 35/100 = 35% 3.5 = 350/100 = 350%

Essomeso 6: Emisimba = Emisingi gya Namba (Number bases)

Namba zirina ensengeka mwe zisimbira(in which they are rooted). Ensengeka zino ze ziyitibwa “emisimba” .Omusimba kiggwayo:”omusingi gwa namba”. Emisimba (Emisingi gya namba) egya bulijjo girina amannya ag’enjawulo kyokka emisingi gya namba egisinga okukozesebwa mulimu: (a)Omusingi gwa namba ogwa kumi(number base ten) . Guno gukozesebwa ennyo abantu mu bulamu obwa bulijjo (b)Omusingi gwa namba ogwa bbiri (number base two), (c)Omusingi ogwa kumi n’omukaaga (hexadecimal) n’omunakkumi.

Omunabbiri n’omusingi ogwa kumi n’omukaaga gisinga kweyambisibwa mu sayansi wa kompyuta.

Omusingi gwa namba (omusimba) eba “nsengekera ya namba” (number system) esimbira mu digito ez’enjawulo.

 Emisingi gya namba number bases  Omusingi gwa namba ogwa bbiri number base two  Omusingi gwa namba ogwa Satu base three  Omusingi gwa namba ogwa nnya base four  Omusingi gwa namba ogwa ttaano base five  Omusingi gwa ogwa mukaaga base six  Omusingi gwa namba ogwa musanvu base seven  Omusingi gwa namba ogwa munaana base eight  Omusingi gwa namba ogwa mwenda base nine  Omusingi gwa namba ogwa kkumi base ten

N’okweyongerayo. Omusingi gwa namba ogwa bbiri gusimbira mu digito bbiri (0 ne 1), omusingi gwa namba ogwa satu gusimbra mu digito satu (0,1, ne 2), omusingi gwa namba ogwa nnya gusimbira mu digito nnya (0,1,2 ne 3) ate omusingi gwa namba ogwa kumi gusimbira mu digito kumi (0.1.2.3.4.5.6.7.8. ne 9).

Mui bufunze “omusungi gwa namba” ka nguyite “omusiimba”, Buli musingi gwa namba kw’egyo gyendaze waggulu era guyinza okuwandiikibwa mu bufunze nga wano wansi:

Emisingi gya Namba(Number Systems or number bases) Mu bufunze English short form 2 Omusungi gwa namba ogwa bbiri Omusimba ogwa bbiri (binary)

3 Omusingi gwa namba ogwa satu Omusimba ogwa satu (ternary)

4 Omusingi gwa namba ogwa nnya Omusimba ogwa nnya (quaternary)

5 Omusingi gwa namba igwa ttaano Omusimba ogwa ttaano (Quinary) 6 Omusingi gwa namba ogwa mukaaga Omusimba ogwa mukaaga (Senary) 7 Omusungi gwa naba ogwa musanvu Omusimba ogwa musanvu (Septenary) 8 Omusingi gwa namba ogwa munaana Omusimba ogwa munaana (octal)

9 Omusingi gwa namba ogwa mwendas Omusomba ogwa mwenda (nonary) 10 Omusingi gwa namba ogwa kkumi Omusimba ogwa kumi (decimal)

Emisingi gya namba okuva ku gw’ekkumi n’ogumu okweyongera waggulu oyinza okugisoma nga si mifunze bwoti: Omusingi gwa namba ogwa kumi n’emu Undenary Omusingi gwa namba ogwa kumi na bbiri duodecimal

Omusingi gwa namba igwa kumi na satu hexadecimal

Omusingi gwa namba ogwa kumi na nnya vigesimal

Omusingi gwa namba ogwa kumi na ttaano sexagesimal

Omusingi gwa namba ogwa kumi na mukaaga

Nga omusomesa oba omuzadde laba nga onnyonnoyoka bino:  Mu nsengekera y’omuwendo gw’ekifo, buli musimbalaala (column) gubaamu digito ezikiikirira obufunza bwa namba ey’ekikolo obw’omudding’anwa.  Mu nsengekera y’omunekkuni, ekikolo (ensingi) kiba 10.  Namba meka ezikozesebwa mu Nsengekera ya Namba? Ensengekera ya namba ey’omunekkumi (omusingi gwa namba ogwa kkumi) gwe tukozesa buli lunaku guzimbirwa “obutaffaali” buno kkumi {0,1,2,3,4,5,6,7,8,9} ekintu ekiguyisa omusingi ogwa -10.  Digito ez’omunebbiri zisobola kuba 0 oba 1, ekiziyisa omunebbiri (omusingi ogwa-2). Yadde nga tusobola okuba n’emisingi gya namba (emisiimba) egy’enjawulo mingi, wano essira tugenda kulissa ku misiimba esatu egisinga okukozesebwa mu bulamu ne mu sayansi. Emisingi gya namba gino esatu gye:

(i) Omusimba ogwa kkumi (Omusingi gwa namba ogwa kkumi) (ii) Omusimba ogwa bbiri (Omusingi gwa namba ogwa bbiri) (iii) Omusimba ogwa kumi na mukaaga. Mu lufuutifuuti, omunnakkumi gwe guyitibwa “base 10” ate omunabbiri ne guba “binary base”. Omusingi gwa namba ogwa 16 gwe guyitibwa “hexadecimal” Omusingi gwa namba ogwa kumi. Guno gusimbiora mu digito kumi.  Mu nsengekera y’omuwendo gw’ekifo, buli musimbalaala (column) gubaamu digito ezikiikirira obufunza bwa namba ey’ekikolo obw’omudding’anwa.  Mu nsengekera y’omunakkuni, ekikolo (ensingi) kiba 10.  Namba meka ezikozesebwa mu Nsengekera ya Namba? Ensengekera ya namba ey’omunakkumi (omusingi gwa namba ogwa kkumi) gwe tukozesa buli lunaku guzimbirwa “obutaffaali” buno kkumi {0,1,2,3,4,5,6,7,8,9} ekintu ekiguyisa omusingi ogwa 10.  Digito ez’omsingi gwa namba ogwa bbiri zisobola kuba 0 oba 1, ekiziyisa omunebbiri (omusingi ogwa-2). Mu kibalangulo ekikolo , enfunzo , entobo oba omusimba ye namba ya digito ez’enjawulo ensengekera y’okubala gy’ekozesa okukiikirira namba. Eky’okulabirako, omusingi gwa namba ogusinga okukozesebwa ensangi zino y’ensengekera ey’omunekkumi (decimal system). Ensengekera eno ekozesa digito kumi, okuva ku 0 okutuuka ku 9. Digito eyinza okuba namba yonna eri waggulu wa ziro(singa eba ziro kitegeeza tewali digito yonna). Oluusi okulaga nti namba eri mu musingi gwa namba ogw’enjawulo n’omusimba ogwa kumi , omusingi gwa namba mw’eri gulagibwa wansi wayo ku ddyo nga mutono okusinga namba yagwo. Eky’okulabirako kitegeeza 23 mu musingi gwa namba ogwa 8 (kino nga kiba 19 mu musingi gwa namba ogwa 10). Kirowoozebwa nti omunakkumi gwalondebwa okukozesebwa mu bulamu obwa bulijjo kubanga omuntu aba n’engalo kkumi. Weetegereze: Emisingi gya Namba(Number Systems or number bases) Mu bufunze English short form 2 Omusungi gwa namba ogwa bbiri Omusimba ogwa bbiri(Binery) Omunebbiri 3 Omusingi gwa namba ogwa satu Omusimba ogwa satu(ternery) Omunessatu 4 Omusingi gwa namba ogwa nnya Omusimba ogwa nnya(Qauternaery) Omunennya 5 Omusingi gwa namba igwa ttaano Omusimba ogwa ttaano(Quinary) Omunettaano 6 Omusingi gwa namba ogwa mukaaga Omusimba ogwa mukaaga (SEnery) Omukeega 7 Omusungi gwa naba ogwa musanvu Omusimba ogwa musanvu(Septenery) Omuswnvu 8 Omusingi gwa namba ogwa munaana Omusimba ogwa munaana(Ocvtal) Omuneena 9 Omusingi gwa namba ogwa mwendas Omusomba ogwa mwenda(nonary) Omunenda 10 Omusingi gwa namba ogwa kkumi Omusimba ogwa kumi(decimal) Omunekkumi

Ebikunizo: 1. Wandiika bino mu lungereza: A. Omusimba : ………………………………. B. Omusimba ogwa bbiri:……………………………….. C. Omusimba ogwa satu:………………………………. D. Omusimba ogwa nnya:……………………………….. E. Omusimba ogwa ttaano:……………………………… F. Omusimba ogwa mukaaga:……………………………….. G. Omusimba ogwa musanvu:…………………………………… H. Omusimba ogwa munaana:…………………………………… I. Omusimba ogwa mwenda:…………………………………… Londako ansa entuufu okuva mu bino: decimal base, nonary base, octol base, binary base, terrnary base, septenary base, quaternary base, quinary base. 2. Wandiika ebigambo bino mu bujjuvu:

3. Wandiika emisingi gya namba gino (emisimba) mu lungereza: (i) Omusingi gwa namba ogwa bbiri . Omusimba: …………………ogwa bbri……………. (ii) Omunabbiri:……………………………….. (iii) Omunassatu:………………………………. (iv) Omunannya:……………………………….. (v) Omunattaano:……………………………… (vi) Omunekaaga:……………………………… (vii) Omusenvu:………………………………….. (viii) Omuneena:…………………………………… (ix) Omunenda:………………………………….. i) Londa ku bino wansi: sexadecimal, unenary, duodecimal, hexadecimal, vigesimal 4. “Musingi ki ku gino ogwa namba” ze tukozesa buli lunaku? (a) Munabbiri (musingi gwa namba ogwa bbiri) (b) Musingi gwa namba ogwa munaana (c ) Musingi gwa namba ogwa kkumi(Omunakkumi) (d) Musingi gwa namba ogwa mukaaga 5. Omusingi gwa namba ogwa kkumi (base ten) gulimu digito kumi. Mugereko ki ku gino ogulaga digito zino ekkumi ? i) 1, 2,3,4,5,6,7,8, 9, ii) 0,1,2,3,4,5,6, 7, 8, 9 iii) -1, 0. 1,2,3,3,4,5,6,7,8 iv) 1,2,3,4,5,6,7,8,9,10 6. Ensengekera y’omusingi gwa namba ogwa digito kkumi (base ten number system) gutandika na digito eya ziro(0) ne gusembyayo digito eya mwenda(9). Bw’ova ku digito eya mwenda weyongerayo otya?Londako ekituufu. i) Tuwandiika 1 okutegeeza ekkumi limu ne tuzzaako ziro okulaga ensusuuba ziro ne tufuna 10, ekitegeeza ekkumi limu n’ensusuuba ziro. ii) Tetweyongerayo; tukoma awo nga tubala

7. Mu nsengekera ya namba ey’omusingi ogwa (digito) ekkumi bino bitegeeza ki? Sazaamu ekitali kituufu: a) 11 kitegeeza: omuzingo gwa digito kkumi gumu n’ensusuuba emu b) 25 kitegeeza: emizingo gya digito kkumi ebiri n’ensusuuba ttaano c) 78 kitegeeza:emizingo gya digito kkumi nsanvu n’ensusuuba munaana. d) 100 Kitegeeza: emizingo gya digito kkumi, kkumi n’ensusuuba ziro 8. Misingi gya namba ki egisinga okukozesebwa mu sayansi wa kompyuta? (a) Munnakumi n’omusingi gwa namba ogwa emu (b) Munnabbiri (binary) n’omusingi gwa namba ogwa kumi na mukaaga (c) Musingi gwa namba ogwa kumi na mukaaga nn’omunassatu (d) Musingi gwa namba ogwa munaana n’ogw’okubiri 9. Mu nsengekera y’omunnakkumi (base-ten system), ki ekitali kituufu ku bino: (a) Omunakkumi gulina digito kumi. (b) Omunakkumi gulina digito mwenda ( c ) Digito esooka ey’omunnakkumi eri 0 ate esembayo eri 9 (d) Mu munnakkumi bw tuva ku 9 tuwandiika 10 naye akabonero (nambiso) ekiikirira ekkumi limu n’ensuuba ziro 10. Mu kinambiso (arithmetic) tusookera ku kulaga bubonero olwe ne tubuwa amannya. Obubonero buno buyitibwa: a) Njawuzo (range) b) Nkubise (multiple) c) Nambiso (numerals) d) Nkubisa (coefficient)

11. Amanya g’obubonero ge gayitibwa: a. Nambiso b. Namba c. Obukomera d. Ebikunizo 12. Saza ku kitali kituufu ku ziro (0) a) Ziro nkuumakifo (placeholder) b) Ziro si namba nzijuvu c) Ziro namba nzijuvu d) Ziro nayo yintegya Ebikunizo: 1. “Omugereeso” kigambo ekirambulula ebirowoozo bya kakensa ku mulamwa ogw’obuyivuwavu. Sazaamu ekitali kituufu ku “mugereeso”. Omugereeso kuba: a) Kugereesa oluusi tekwesigamizibwa ku nkola ya sayansi b) Tegwesigamye ku nkakaso ya nkomeredde c) Gwesigamye ku nkola ya sayansi d) Kugereesa kwa bassekalowooleza okuteesigamye ku nkakaso 2. Njawulo ki eri wakati wa digito ne namba? Saza ku bitali kituufu. a) Namba bwe butaffaali obuzimba sessomo ly’ekibalangulo ate digito ne buba obutaffaali obuzimba namba (the building block of a number) b) Digito bwe butaffaali obuzimba nnyukuta ate namba bwe butaffaali obuzimba essomo ly’ennimi 3. Ziro egasa ki mu kinambirizo (arithmetic). Saza ku kikatali kituufu. a) Bw’oba obala n’omalayo digito ekkumi, okusobola okweyongerayo olaga ekkumi 1 n’ensusuuba 0, ekiragibwa nga 10. N’olwekyo ziro nkuumakifo (place holder) b) Ziro ye namba yokka eraga ekifo ekitaliimu kintu kyonna c) Ziro eba namba enzijuvu d) Ziro si nkuumakifo 4. Ekigambo ‘ensengekera” kyekuusiza ku kigambo: a) Okusenguka b) Okusengeka c) Okusegeka d) Okusigika 5. Ekigambo “ensengekera” (system) kiva mu bigambo: a) Okusengeka eri b) Okusenguka eno c) Ensengeka ey’emalirira d) Ensengeka eyematira 6. Saza ku kitali kituufu ku “nsengekera y’omunnakkumi” (base-ten system): a) Omunakkumi guba ne digito za namba kkumi okuva ku 0 okutuuka ku 9 b) Bwe tuwandiika 10 kiba kitegeeza ensusuuba emu c) Bwe tuwandiika 10 kino kitegeeza "ekumi 1 ne ensusuuba 0". Zino ziba digito bbiri; tetulina digito emu ya namunigina ekiikirira “kumi”. d) Bwe tuwandiika 34 kiba kitegeeza amakumi asatu n’ensusuuba nnya. 7. Waliwo emisengeko gya namba (number patterns) egy’enjawulo. Wandiika amannya g’emisengeko gy’emigereko gya namba (sets) gino mu luganda. A= 1,2,3,4,5,6,7,…………………………………(counting number) B=… –5 , -4, -3, -2, -1 , 0, 1 , 2, 3 , 4 , 5…Zino …….(integers) C= 1,3,5,7,9,11… Zino namba za…………………(Odd numbers) D= 2,4,6,8,10,12…Zino namba za ……………….(Even numbers) E=4,9,16,15,36,….Zino namba za ……………….(Square numbers)

F= 8, 2, 54,75, 216….Zino namba  za……(Triangular numbers)

8. 9. Yintegya kitegeeza ki? a) Namba za kirumira mpuyemu b) Namba za kibalirampuyibbiri c) Namba za kirumirampuyisatu d) Namba za kirumirampuyikumi

10. Namba zino ziyitibwa yintegya.Endaga ya yintegya ng’eno eyitibwa etya? Saza ku kitali kituufu.

(a) Kyesimba ya namba (b) Olukoloboze lwa namba (c) Layini ya namba (d) Olugalamiro lwa namba (e) Endaga ya namba

11. Yintegya eya pozitiivu eyitibwa etya? a) Yintegya eya kiddawansi b) Yintegya eya kiddawaggulu c) Yintegya eya kiddamaaso d) Yintegya eya kigaanira 12. Yintegya eya negatiivu eyitibwa etya? a) Yintegya eya kiddamaaso b) Yintegya eya kiddannyuma c) Yintegya eya kiddawansi d) Yintegya eya kiddawaggulu 13. Kiri wa ku bino ekitali kituufu ku ziro: (a) Ziro terina makulu mu kibalo (b) Ziro nkuumabusangiro (position holder) (c) Ziro etegeeza nti mu kifo kya namba ekyo tewali digito yonna. (d) Ziro yadde nkuumakifo naye nayo etwalibwa nga namba

14. Sazaamu ebitali bya namba ya kibazo(counting number): a) Ali mu kibinja kya 5no (b) omujoozi namba ttaano (c) Omujoozi ogw’okutaano (d) Kiro ya sukaali emu (e) Kiro ya sukaali ey’okusatu 15. Nga oyigiriza omwana okubala otandik na namba za: (a) Ndagakifo (b) Ndagalinnya (c) Kibazo (ezibala) (d) Mpuyissatu 16. Ku zino ziri wa namba eza ndagakifo: (a) 1 , 2 , 3 , 4, 5 , 6, 7….. (b) 1ka, 2li, 3tu, 4na, 5no , 6ga, 7vu, 8na, 9da, 10mi , 11mu (c) Omujoozi namba 2, ennyumba namba 13, Paapa Francisi I,, Mwanga II, essimu namba 070,000,001 17. Zino namba ki? Sazaamu ekikyamu. (i) 1, 2, 3, 4, 5, 6, 7…..Zino “namba za kibazo”(counting numbers) (ii) 1ka, 2li, 3tu, 4na, 5no… Zino namba za “ndagakifo”(ordinal numbers) (iii) Omujoozi namba 2, ennyumba namba 13, Paapa Francisi I,, Mwanga II, essimu namba 070,000,001. Zino namba za “ndagalinnya”(nominal numbers) (iv) 1, 4, 9, 16,.. .Zino namba za kigaanira 18. Namba ki eziyitibwa namba eza “kiddannyuma”?

 (i) Namba za pozitiivu
 (ii) Namba za negatiivu

19. Namba ki eziyitibwa namba eza “kiddamaaso”? (i) Namba za pozitiivu. (ii) Namba za negatiivu 20. Kuba “olukoloboze lwa namba”(number line) okulaga: (i) Namba za kiddannyuma ne (ii) Namba eza kiddamaaso 21. Namba eza kiddannyuma (negatiivu) ne namba za kiddamaaso (pozitiivu) ziyitibwa zitya zombi wamu? Sazaamu ekikyamu. (i) Namba za kirumirampuyibbiri (ii) Yintegya (integers). (iii) Namba enkutulemu

22. Ku zino saza ku namba eya kigaanira: (a) 15, (b) 4 (c) 24 (d) 6 23. Ku zino saza ku namba eya kyegabanya: (a) 10 (b) 11 (c) 23 (d) 101 24. Ku zino saza ku namba ez’omugerageranyo: (a) omurandira gwa kyebiriga ogwa 5 , (b) Pi(π).:Pi = 3.14159. (c) namba ey’omuteeberezo e; e = 2.71828.... (d) omurandira gwa kyebiriga ogwa 2 (e) Omurandir gwa kyebiriga ogwa 3 25. Kiri wa ekikyamu ku bino wansi: (a) Yintegya zonna namba za mugerageranyo kuba ziyinza okuwandiikibwa nga n/1 (b) Emikutule gyonna nagyo namba za mugerageranyo kubanga girina yintegya wansi ne waggulu (ezitali ziro) (c) Emitonnyeze gyonna namba za migerageranyo 26. Lwaki ne namba nga 13.3168980325 nayo ya mugerageranyo? (a) kubanga eyinza okuwandiikibwa nga omukutule: 13.3168980325 = 133,168,980,325 / 10,000,000,000 (b) Kubanga tesobola kuwandiikibwa nga mukutule 27. Lwaki omurandira gwa kyebiriga ogwa 2(=.4142135623730950...) si namba ya mugerageranyo ? (a)kubanga guba mutonnyeze muwanvu nnyo (b) Kuba ansa tesobola kukola mugerageranyo gwa yintegya bbiri kubanga omurandira gwa kyebiriga ogwa 2 ≠ x/y. 28. Lwaki emiramwa “omugerageranyo”ne “namba y’omugerageranyo” gya njawulo? (a) Byonna bye bimu (b) Mu mugerageranyo, kinnawaggulu ne kinnawansi biyinza obutabeerako yintegya wabula ne bibaako mitonnyeze kyokka mu “namba y’omugerageranyo kinnawansi ne kinnawaggulu birina okuba yintegya.

29. Namunigina ya namba y’omuteeberezo (imaginaery number unit) ye (a) √(-5) (b) √(-5) (c) √(-7) (d) √(-1) (omurandira gwa kyebiriga ogwa emu eya negatiivu era akabonero kayo ke i, ekitegeeza nti 30. Namba zino zigwa mu kiti ki? (i) 0.25 (ii) 3.1415926535897932384626433832795028841971693993…. (iii) 3.1415926535 (iv) 30 (v) 7/9 (vi) 4 2/5 (vii) Omurandira gwa kyebiriga ogwa 256 (viii) -81/9 Ekibazo: Ekikulu kwe kumanya namba okusinziira ku bika bya zo nga oziwulidde.Eky’okulabirako oli bw’ayogera ku “yintegya”, olina okumanya nti taminologiya eno ekwata ku namba eza kibazo, kiddannyuma(negatiivu) zazo ne ziro oba ka tugambe namba eza negatiivu n’eza pozitiivu. (i) 0.25 Guno mutonnyeze ogwekomya (terminating decimal), n’olwekyo eyinza okuwandiikibwa nga omukutule: 25/100 = 1/4. Olw’okuba omukutule guno tegukendezebwa kukola namba enzijuvu, namba eno si yintegya yadde okuba namba y’ensibo. Okiraba nti buli kintu ky’ekyo kyennyini. Eno namba ey’omuugerageranyo ate nga namba yennyini. (ii) 3.1415926535897932384626433832795028841971693993…. Eno oteekwa okuba nga omaze okulaba nti paayi(pi), yadde nga eriko ebifo by’emitonnyeze bingi okusinga by’otera okukozesa oba okulaba. Ekikulu kiri nti omutonnyeze guno tegweddingana (it does mugerageranyo. N’olwekyo paayi eba namba etali ya mugerageranyo kyokka nga namba yennyini. (iii) 3.1415926535 Kino kyefaananyirizako ne paayi (pii) kyokka si paayi kubanga guno mutonnyeze ogw’omukunuukirizo omuzingeko(rounded decimal approximation). Olwokouba omukunuukirizo guno gwekomya(terminates), namba eno ya mugerageranyo obutafaanana na pi etali ya mugerageranyo. Namba eno ya mugerageranyo ate nga namba yennyini. (iv) 30 Kyeyorekerawo eno namba ya kibazo. Kino kitegeeza era namba nzijuvu ate era nga yintegya. Okusinziira ku musomesa ky’ayagala okukkaatiriza, era eno eyinza okulagibwa nga ey’omugerageranyo kubanga mu butuufu eno namba ya mugeraageranyo. Kyeyolekerawo nti era namba yennyini. Eno namba ya buttonde (ya nsibo), nzijuvu, yintegya, ya mugerageranyo ate nga namba yennyini. (v)7/9 Guno mukutule (namba enkutulemu) ate era namba yennyini n’olwekyo ya mugerageranyo. (vi) 4 2/5 Eno era eyinza okuwandiikibwa nga 22/5, kino nga kye kimu ne kyetuvaako waggulu.N’olwekyo eno namba ya mugerageranyo ate nga namba yennyini. (vii) Omurandira gwa kyebiriga ogwa 256 Ky’oyinza okusooka okulowooza kiyinza okuba nti namba eno si ya mugerageranyo kubanga murandira gwa kyebiriga kyokka wetegereze olabe nti omurandira gwa kyebiriga guno gusonjoka nga: 256 = 16, eno nga yintegya. N’olwekyo namba eno yintegya, ya mugerageranyo ate nga namba yennyini.

(viii)  – 81/9

Guno mukutule kyokka laba nti gusonjoddwa okutuuka ku –3, n’olwekyo eno era eyinza okutwalibwa nga yintegya. Namba eno yintegya, ya mugerageranyo ate nga namba yennyini.

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