Emiramwa gy’Ekibalangulo emigandawaze

   (Formed Luganda mathematical concepts)

Gino gy'emiramwa egy'enjawulo Mwami Muwanga Charles gy'atukkirizza ffe aba IALI NGO okuweereza abasomi ba Luganda wikipedia okuva mu kitabo kye "Luganda Mathwords: An etymological handbook for Luganda mathmatical terms"

1. Ekibalangulo

           (Mathematics).

"Essomakubala" oba "ekibalangulo" (mathematics) “sessomo” eriwagala obwongo nga lisengeka ebirowoozo ne namba. Essomo lino n’olwekyo liyamba okukulaakulanga “ EkkyO”(kisomwa “Ekkyu”) y’omuntu.

Ekibalangulo mu bumpi kiwandiikibwa “ekibalo”(maths). Gino gye gimu ku mibalanguzo( operational concepts) egyetaagisa mu sessomo lino:

a) Namba (number) b) Nambiso (numeral)

c) Ekikunizo (mathematical problem(question) d) Omweyoreko (mathematical expression) e) Okusonjola (to simplify) f) ) Okubalangula (to calculate) g) okubalanguuza (to calculate) h) Okubaza to compute) i) Ekibazo (solution) j) Ekibalanguzo (formula) k) Omubalanguzo /akabalanguzo(mathematical operation) (

Amasomo g'ekibalangulo

   (the sub-disciplines or sub \fields of study)

i) Okubalirira (arithmetic , mental calculations using numbers only ) ii) Essomabigoberero/Aligebbula ( algebra calculations using both numbers and letters) iii) Omugereeso gwa namba (number theory) iv) Emisingi gya namba(Emisimba) (number bases) v) Ossomankula oba essomampimo (geometry) vi) Emikutule (fractions) vii) Emitonnyeze (decimals) viii) Okukunuukiriza n’okuzingako namba (estimation and rounding off numbers) ix) Nambuluzo n’Okulambulula namba (Factors and factoring numbers) x) Namba ennambulukufu (composite numbers) xi) Namba ezitali nambulukufu (prime numbers) xii) Ennyingo n'Ekibalo ky'Ennyingo (Terms and Polynomials) xiii) Emigereko (Sets) xiv) Emikwanaganyo n’emikwataganyo (Relations and functions) xv) Ekikwataganyo (coordinate) xvi) Emigerageranyo (Ratios) xvii) Ebigerageranyo (Rates) xviii) Emigendaganyo (proportions) xix) Kalonda (Data) xx) Essomakalonda/ Kalonomusengeke (statistics) xxi) Ekibalo ky’Obusuubuzi business maths

2. Omugereeso gwa Namba

                (Number theory)

Mu mugereeso gwa namba mulimu ebika bya namba eby’enjawulo omuntu anabeera omubalanguzi bye yetaaga okunnyonnyoka.Bye bino:

a) Namba eza kibazo(cardinal numbers)

b) Namba ezibala(counting numbers)

c) Namba eza ndagakifo (ordinal numbers).

d) Namba eza ndagalinnya(nominal numbers). Nga:  Omujoozi namba 7  Muteesa II  Paapa Bendikito XVI  Ssimu namba 071000000  Ekifo namba 23  Mmeeza namba 9

e) Namba enzijuvu(Whole numbers).

0,1,2,3,4,5,6,7,8,9, ......Okiraba nti ne ziro ebalibwa mu namba enzijuvu. . Eno wanso ye “layini ya namba” enzijuvu :

f) Yintegya (namba za kirumirampuyibbiri). Omugereko gwa yintegya(set of integers) gufaanana bwe guti :

{...,-5,-4 -3, -2, -1, 0, 1, 2, 3,4,5,….} Gulagibwa ku layini ya namba nga eno wansi:

g) Namba ez’ensibo (natural numbers):

h) Namba za kyegabanya (Even numbers).

i) Namba eza kigaanira (odd numbers).

j) Emikutule /Emwenkutulemu (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) Namba za kyebiriga (square numbers ).

p) Namba za kyesatuza (cube numbers)

3.Emisingi gya namba

          (Number bases)

• 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.

Mui bufunze “omusungi gwa namba” guyitibwa “musiimba”,Buli musinga gwa namba kw’egyo gyendaze waggulu era guyinza okuwandiikibwa mu bufunze nga wano wansi:

Emisingi gya Namba =Emisimba)

               (Number bases)

2 Omunebbirior Omusimba ogwa bbiri(binary)

3 Omunessatu or Omusimba ogwa satu(ternary)

4 Omunannya or Omusimba ogwa nnya(Quaternary)

5 Omunattaano or Omusimba ogwa taano (Quinary)

6 Omunekaaga or Omusimba ogwa mukaaga(Senary)

7 Omusenvu or Omusimba ogwa musanvu(Septenary)

8 Omuneenaor Omusimba ogwa munana(octal)

9 Omunenda or Omusimba ogwa mwenda (nonary)

10 Omunekkumi or omusimba ogwa kkumi(decimal)

11 Omusingi gwa namba ogwa kumi n’emu(undenary) 12 Omusingi gwa namba ogwa kumi na bbiri(duodecimal)

13 Omusingi gwa namba igwa kumi na satu(hexadecimal)

14 Omusingi gwa namba ogwa kumi na nnya (vigesimal)

15 Omusingi gwa namba ogwa kumi na ttaano(sexagesimal)

16 Omusingi gwa namba ogwa kumi na mukaaga

Nga omubalanguzi atandika , manya bino:

• Mu nsengekera y’omuwendo gw’ekifo, buli musimbalaala (column) gubaamu endagamuwendo(digits) ezikiikirira obufunza bwa namba ey’entob 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.

• Endagamuwendo(digits) 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 nmab gino esatu gye :

(i) Omunakkumi(=Omusingi gwa namba ogwa kkumi)

(ii) Omunabbiri (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”

4. Okulambulula Namba

                (Factoring numbers)

Okusobola okunnyonnyoka essomo ly’ekibalangulo eky’okulambulula namba obulungi, wetaagisa okumanya egimu ku miramwa egikozesebwa naddala emiramwa nga :

• Enkubisa(coefficients)

• Enkubisa (multiples)

• entakyuka(constants),

• omugabo, • enkyuso(variable),

• ennyingo(terms),

• mufunza(exponentiation),ne

• kyekubisa(logarithms).

• Namba ennambulukufu(composite numbers)

• Namba ezitali nambulukufu(prime numbers)

• Enfikkizo(remainder)

• Namuluzo (factors)

• Nambuluzo Eyawamu (Common factor)

• Enkubise Eyawamu (Common Multiple)

• Enkubise Eyamu Esembayo (Lowest Common Multiple)

• Nambuluzo Eyawamu Esingayo( Greatest Common )

• Okulambuluza namba ezitali nambulukufu (prime factorisation)

5. Emikutule

     (Fractions)

Emiramwa egyetaagisa okusobola okubaza emikutule gye gino:

(i)Ebitundu by’Omukutule(Parts of a Fraction)

(a) Kinnawansi ne  Kinnawaggulu. Mu mukutule guno  namba eri waggulu eyitibwa kinnawaggulu ate eri wansi kinnawansi:

11 13

Kinnawaggulu (Numerator)

Kinnawansi (Denominator)

(b) Kinnawansi Ekyawamu(Common Denominator)

(ii) Nambuluzo. Muno mulimu :

(a) Nambuluzo ey’awamu(Common Factor)

(b) Nambuluzo Eyawamu Esingayo (Greatest Common Factor)

(c) Enkubise Eyawamu (Common Multiple)

(d) Enkubise Eyawamu Esembayo(Lowest Common Mutlple)

Okugatta emikutule egirina kinnawansi kye kimu kuba kulya mungu buteesokoola naye bwe giba ne kinnawansi ez’enjawulo oba olona okusooka okuzuula “kinnawansi ekya’wamu”(common denominator). Okuzuula kinnawansi ekya’wamu weyambisa enkubise ey’awamu era eno gy’osooka okunonnya.

6. Emigereko

         (Sets)

“Omugereko” (set) liba kungaanyo lya bintu oba bamemba abayitibwa erementi nga bili mu busengeke. Omugereko gutera okulagibwa n’ennyukuta nga A,B,C, ------ U,V,W, n’okweyongerayo. Erementi (bamemba) z’Omugereko zikugirwa mu bukomera bwe buti:

Eky’okulabirako;

Omugereko guyinza okubaamu erementi (bamemba) ez’entakoma (infinite) nga Omugereko gwa namba ez’ensibo (namba z’obutonde) oba ne gubaamu bamemba ab’ekkomo (finite ) nga Omugereko gwa waliifu z’oluganda.

Omugereko guyinza obutabaamu bamemba, ntegeeza obutabaamu kintu kyonna era Omugereko nga guno guyitibwa Omugereko omwangaala (empty set). Kino kiragibwa n’akabonero , ekisomwa (fi). Ebintu oba bamemba ababeera mu musengeko A balagibwa nga n (A), ekitegeeza nti n ( ) = 0, ekitegeeza nti Omugereko A teguliimu erementi yonna. Omugereko gulimu emiramwa gino :

• Omuteeko=emigereko emiteke awamu (Union of sets)

• Endaga y’Omugereko (Set notation)

• Amasang’azzira g’Emigereko /Entabiro y'emigereko (Intersection of sets)

• Enjawuzo y’Emigereko ebiri (The difference of two Sets) • Ekitundu ky’Omugereko (Sub set)

• Veniggulaamu(Venn Diagram)

• Embaranguza z’Emigereko (Set Operations)

• Omuteeko ( union of sets ) gw’emigereko ebiri guba mugereko gwa erementi ezisangibwa mu mugereko ogumu oba gyombi . N’olwekyo , omugereko C guba muteeko gwa migereko A ne B .

7. Emikwanaganyo n’Emikwataganyo

        (Relations and Functions)

Emiramwa egyetaagisa

Omukwanaganyo-relation

Omukwanaganyo – Musengeko gwa mifulumyo (outputs) n’emiyingizo (inputs), ekitera okuwandiikibwa nga emigago emisengeke (ordered pairs), ekiragibwa nga: (muyingizo, mufulumyo). Ebyokulabirako (x, y) oba (5,12).

Omusooka – musengeko gwa miyingizo (inputs) egy’omukwanaganyo(relation) oba

Omukwataganyo -function

Omukwataganyo – mukwanaganyo nga buli muyingizo (input) gulina omufulumyo (output) gumu gwokka. Omukwataganyo gutera okulagibwa nga m(x) oba f(x) mu lungereza.

Omufulumyo – Musengeko gwa myawuzo (ranges) egy’emikwanaganyo oba omikwataganyo.

Engezeso eya layini engalamivu – Singa buli layini engalamivu ekubibwa muggulaafu eyita mu punkuti emu yokka, x guba mukwataganyo gwa y. Kyokka singa okuba layini engalamivu n’eyita mu punkuti 2, x teba mukwataganyo gwa y.

Engezeso eya layini ennesimbu – Singa buli layini ennesimbu ekubibwa eyita mu punkuti emu yokka, y eba mukwataganyo gwa x. Singa okuba layini ennesimbu eyita mu punkuti 2, y teba mukwataganyo gwa x.

“Kifuulannenge w’omukwataganyo”(the inverse of a function) ajjawo nga omusooka(domain) n’omufulunyo (range) bikyusaganyizza ebifo, ermenti zonna ez’omusooka ne zifuuka mufulumyo ate ez’omufulumyo ne zifuuka musooka.

Eky’okulabirako ekya kifuulannenge w’omukwataganya oguli mu ngeri eno:

Omukwatagenyo ogwa Nansangwawo Kifuulanenge wagwo

Singa omukwataganyo “m” gukwataganya “k” ku “g”, kino kitegeeza nti kifuulannenge w’omukwatagenyo, ekiwandiiki bwa nga M-1, akwataganya k ku g.

8. Ennyingo n'Ekibalo ky'ennyingo(Ekiyingo)

       (Terms and Polynomials)

Emiramwa egyetaagisa:

(i) Ennyingo (term , nomial)

(ii) Ennyingo ezifaanagana (Like Terms)

(iii) Nnyingo ezikwatagana (Like Terms)

(iv) Ennyingo ezitafaanagana (Unlike Terms)

(v) Nnyingo ezitakwatagana (Unlike Terms)

(vi) Ekiyingo=Ekibalo ky'ennyingo) (Mathematics of nomials,Polynomials)

(vii) Nnyingo emu (Monomial)

(viii) Nnyingo bbiri (Binomial)

(ix) Nnyingo satu (Trinomial)

(x) Enjawuzo eya kyebiriga ebbiri (The difference of two squares)

9. Ekibalangulo ky’Omukisa /EKM

      (Probabilty)

Mu kibalo ky’Omukisa(Mathematics of chance, prpbability), emiramwa egyetaagisa gye gino:

• EKM (Ekibalo Ky’Omukisa)

• Ekituukiriro oba ekituuko (event)

• Ebisoboko (possibles,outcomes)

• Omugereko gw’ebisoboko (probability space, probability set)

• Ebituuko eby’ejjuuliriza (complementary events)

• Ebituuko ebyetengerevu (independent events)

• Ebituukiriro ebitali byetengerevu (dependent events)

• Ensatuze (dice)

• Ekinusu (coin)

• Omutwe(m) n’ekikira(k) (head and tail)

• Ebituukiriro ebitasobola kubaawo mu kiseera kye kimu (mutually excussive events)

• Ebituukiriro ebisobola okubaawo mu kiseeta kye kimu (mutually inclusive) events)

10. Emigerageranyo ,Ebigerageranyo, n’emigendaganyo

                       (Ratis,Rates, and Proportions)

• Omugerageranyo (ratio)

• Emigerageranyo (ratios)

• Ekigerageranyo (rate)

• Ebigerageranyo (rates)

• Omugendaganyo (proportion)

• Emigendaganyo (proportions)

11. Ebikyusaganyo

    (Transformations)

• Ekifaananyi (Picture)

• Ekifaananyo (Image)

• Ekifaanaganyo (reflection)

• Ekiseetuko (displacement)

• Ekyetoloozo (rotation)

12.Kalonda n'Ekisngekakaloda

         (Data ad statistics)

• Kalonda (data)

• Okusengeka kalonda (to arrange data)

• Ekisengekakalonda/Kalonda omusengeke (statistics)

• Emisengeko gya kalonda (statistical patterns)

• Okukung’anya kalonda (data collection)

• Okutaputa kalonda (to interpret data)

• Kalonda ow’endagabungi (quantitative data)

• Kalonda ow’endagamutindo (qualitative data)

12. Ekibalo eky’obusuubuzi

          (Busines Math)

• Ekyewolo (loan)

• Ebbanjo (loan)

• Omuganyulwo (interest)

• Omuganyulo (interest)

• Omukendeezo (discount)

• Commission (bwasiisi)

• Omugeranyakikumi (percentage)

• Ekigambululo kya Bbanka ( bank statement)

• Omujjuulirizo (subsidy)

• Enzijuulirirzo (subsidy)

13.Nakyenkanyanjuyi ne Nakyekubira

           (Equations and Iqualities)

• Nakyenkanyanjuyi (equation)

• Nakyekubira (Inequality)

• Ekibazamusikiza or Musikiza (substitution)

• Okusikiza namba (to substitute a variable with a number)

• Akenkanyanjuyi (the equals sign)

14. Emifunza ne Kyekubisa

      (Exponents and Logarithms)

• Ekifunza=ekibalo y'emirundi emifunze (exponentiation)

• Kyekubisa (Logarithms)

• Omufunzo (exponent)

• Akafunza (the power of a number)

• Ekikolo/Enfunzo (the base or the number at the base of an exponent)