Adadi da Kadarorinsu
Integers suna ɗaya daga cikin mahimman ra'ayoyi a fannin lissafi, duk da haka suna taka muhimmiyar rawa a rayuwar yau da kullun. Idan muka ƙirga abubuwa, muka ƙayyade yanayin zafi ƙasa da sifili, muka ƙirga adadin benaye a cikin ginshiki, ko muka rubuta riba da asara a cikin kasuwanci, a zahiri muna mu'amala da integers. Fahimtar integers da kaddarorinsu zai sauƙaƙa koyon ilimin lissafi, algebra, har ma da manyan manufofin lissafi.
Ma'anar Ma'auni
Lambobin lambobi sune saitin lambobi waɗanda suka ƙunshi lambobi marasa kyau, sifili, da lambobi masu kyau. Alamar da ake amfani da ita wajen wakiltar saitin lambobi ita ce Z (wanda aka samo daga kalmar Jamusanci Zahlen, ma'ana "lambobi"). Gabaɗaya, ana iya rubuta lambobi kamar haka:
…, -3, -2, -1, 0, 1, 2, 3, …
Ba kamar cikakkun lambobi waɗanda ke ɗauke da lambobi 0 kawai da lambobi masu kyau ba, lambobi suna haɗa da lambobi marasa kyau don haka sun fi cika wajen bayyana yanayi daban-daban.
Misalan Ma'auni a Rayuwar Yau da Kullum
Integers suna da alaƙa ta kud da kud da ayyukan ɗan adam. Wasu misalan amfani da su sun haɗa da:
1. Zafin Iska: Zafin -5°C yana nuna cewa zafin yana da digiri 5 ƙasa da sifili.
2. Tsawo: Tsawon mita -20 na iya nufin matsayi mita 20 a ƙasa da matakin teku.
3. Kudi: Jimlar -50.000 tana nuna bashi ko ƙarancin 50.000.
4. Bene na gini: Ana amfani da bene -1 ko -2 a ginshiki.
5. Maki na wasa: A wasu wasanni, maki na iya ƙaruwa ko raguwa, don haka ya haɗa da lambobi masu kyau da marasa kyau.
Daga waɗannan misalan, za a iya ganin cewa lambobi suna taimaka mana mu bayyana yanayin "fiye da", "ƙasa da", ko "daidai" tare da sifili a matsayin wurin tunani.
Layin Lamba Gabaɗaya
Domin fahimtar lambobi gaba ɗaya, sau da yawa muna amfani da layin lamba. Layin lamba layi ne madaidaiciya wanda ke sanya lambobi a takamaiman matsayi:
– Lambobi masu kyau suna gefen dama na sifili.
– Lambobin da ba su da kyau suna gefen hagu na sifili.
– Da zarar darajar ta kai ga dama, to, hakan zai fi girma.
– Mafi nisa daga hagu, ƙimar ta fi ƙanƙanta.
Da layin lamba, za mu iya kwatanta lambobi gaba ɗaya cikin sauƙi. Misali, -2 ya fi -5 girma saboda -2 ya fi -5 nesa da dama.
Halayen Ma'auni
Integers suna da wasu muhimman siffofi a cikin ayyukan lissafi. Waɗannan halaye suna da matuƙar amfani lokacin da muke yin lissafi, sauƙaƙa maganganun algebra, ko kuma tabbatar da ra'ayoyi.
1. Yanayin Rufewa (Rufewa)
An rufe saitin lambobi a ƙarƙashin wasu ayyuka, ma'ana idan muka yi waɗannan ayyukan akan lambobi, sakamakon zai kasance lamba.
– Ƙari: Idan a da b lambobi ne, to a + b shima lamba ce.
Misali: -3 + 7 = 4
– Ragewa: Idan a da b lambobi ne, to a − b shima lamba ce.
Misali: 5 − 12 = -7
– Riba: Idan a da b lambobi ne, to a × b shima lamba ce.
Misali: (-4) × 6 = -24
Duk da haka, ba koyaushe ake rufe lambobi ba idan aka kwatanta da rarrabuwa, saboda sakamakon rarrabuwa na iya zama ƙashi.
Misali: 1 ÷ 2 = 1/2 (ba cikakken lamba ba).
2. Kadarar Mota (Musaya)
Haƙƙin sauye-sauye yana nufin cewa ana iya musanya tsarin lambobi ba tare da canza sakamakon ba.
– Ƙari: a + b = b + a
Misali: 2 + (-5) = (-5) + 2 = -3
– Yawan: a × b = b × a
Misali: (-3) × 4 = 4 × (-3) = -12
Yana da muhimmanci a tuna cewa ragi da rabawa ba sa canzawa.
Misali: 7 − 2 ≠ 2 − 7.
3. Halayen Haɗin gwiwa (Rukunin ƙungiya)
Haƙƙin haɗin gwiwa yana nufin cewa yadda aka haɗa lambobi a cikin aiki ba ya canza sakamakon.
– Ƙari: (a + b) + c = a + (b + c)
Misali: (1 + 2) + 3 = 1 + (2 + 3) = 6
– Yawan: (a × b) × c = a × (b × c)
Misali: (2 × 3) × 4 = 2 × (3 × 4) = 24
Kamar yadda ake amfani da kalmar "commutative", kalmar "associative property" ba ta shafi "ragewa" da "rabawa" ba.
4. Kayayyakin Rarrabawa (Yaɗawa)
Kadarar rarrabawa tana da alaƙa da ayyukan ninkawa zuwa ƙari ko ragi:
– a × (b + c) = (a × b) + (a × c)
Misali: 3 × (2 + 5) = (3 × 2) + (3 × 5) = 6 + 15 = 21
– a × (b − c) = (a × b) − (a × c)
Misali: 4 × (10 − 3) = 40 − 12 = 28
Wannan siffa tana da matuƙar muhimmanci wajen sauƙaƙa siffofin algebra da kuma warware daidaito.
5. Abubuwan da ke da Alaƙa (Ƙari da Riba)
Identity lamba ce da ba ta canza darajar wata lamba idan aka yi aiki da ita.
– Asalin ƙari shine 0: a + 0 = a
Misali: -9 + 0 = -9
– Asalin ninkawa shine 1: a × 1 = a
Misali: 8 × 1 = 8
Identities suna taimaka mana mu fahimci ra'ayin tsaka-tsaki a cikin ayyukan lissafi.
6. Abubuwan da suka juye (Lambobin da ba sa juyewa)
Kowace lamba tana da juzu'i na ƙari, wanda shine akasin lamba wanda idan aka haɗa ta tare zai haifar da 0.
– Juyin a shine -a, don haka: a + (-a) = 0
Misali: 6 + (-6) = 0, da kuma -4 + 4 = 0
Duk da haka, ba duk lambobi ne ke da juyi a ninkawa ba. Misali, juyi mai ninkawa na 2 shine 1/2, wanda ba lamba ba ce.
7. Alamomi Masu Kyau da Marasa Kyau a Aiki
A cikin lambobi, alamu masu kyau da marasa kyau suna ba da ƙa'idodi na musamman a cikin ninkawa da rabawa:
– (+) × (+) = (+)
– (+) × (-) = (-)
– (-) × (+) = (-)
– (-) × (-) = (+)
Misali:
– 3 × (-2) = -6
– (-5) × (-4) = 20
Don ƙari da ragi, sau da yawa muna amfani da manufar layin lamba ko ƙa'idar:
– Ƙara lambobi marasa kyau iri ɗaya ne da “ragewa”.
– Rage lamba mara kyau iri ɗaya ne da "ƙara".
Misali:
– 7 + (-3) = 4
– 5 − (-2) = 7
Kammalawa
Lambobin lambobi ne da suka ƙunshi lambobi marasa kyau, sifili, da lambobi masu kyau. Waɗannan lambobi suna da matuƙar muhimmanci domin suna iya bayyana yanayi daban-daban a rayuwa, kamar yanayin zafi ƙasa da sifili, asarar kuɗi, ko wurare a ƙasa da matakin teku. Lambobin lambobi kuma suna da muhimman halaye kamar rufewa (don ƙari, ragi, da ninkawa), juyawa, haɗin kai, rarrabawa, asali, da ƙari juyewa. Ta hanyar fahimtar halayen lambobi, za mu fi sauƙin yin lissafi, sauƙaƙe ayyuka, da kuma koyon batutuwan lissafi na gaba kamar algebra, lissafi, da sauran tsarin lambobi.
Idan kana so, zan iya yin sigar wannan labarin don matakin firamare/tsakiya (mai sauƙi) ko kuma sigar da ta fi dacewa kamar cikakken takarda tare da tambayoyi da tattaunawa na aiki.
