INANI

Ngokwencwadi yebhayibhile, jonga kwincwadi ye-Numbers (eBhayibhileni).

Inani ligama elisisigama esisuka kwa-mathematika, elisetyenziswa xa kubalwa okanye kusenziwa umlinganiselo. 

I-puzzle ye-Sudoku 

A Sudoku puzzle

Igama elithi nani lingachazwa ngeendlela ngeendlela,  elisetyenziswe ngalo nakwiliphi na isebe lakwamathe lakwamathe. 

•  Abantu basebenzisa uphawu oluthile olumele amanani; bathi xa belubiza ludweliso lwamanani (zii-numerals) zii-numerals. Apho zixhaphake khona ezi numerals kukwimibhalo echaza izinto ukuba ziyintonina, njengakwiminxeba,ukuze kubekwe izinto ngokulandelelana kwazo, njengakumanani alandelelananyo, okanye ukuze kubekwe isiphawuli esinomahluko, njengakwi-ISBN, inani elinomahluko nenokubonwa ngalo incwadi.
  
•  Amanani engqokolela yezinto asetyenziswa kumlinganiselo wokubona okokuba izinto ezo zibonakala kangaphi na kuloo ngqokolela yezi zinto]]. U-{A,B,C} unomlinganiselo wesi-"3".
  
•  Amanani e-Ordinal asetyenziswa xa kubhekiswa luloo nto ithile kwingqokolela okanye ngokulandelelana kwazo (Eyokuqala, eyesibini, eyesithathu).
  

• Abantu basebenzisa ii-symbols ukuze ibe zizo ezima kwindawo yamanani; batsho bazibize ezi-symbols ngokuba ngokuba zii- numerals. Iindawo ezithanda ukusetyenziswa kuzo ezi numerals kuxa ku-label-ishwa, njengoba sibona kumanani efowini, naxa si-oda, njengakumanani alandelelanayo kwii-oda zethu, okanye sibeke inani eliyakuthi yahluke ngalo i-oda leyo phakathi kwezinye ii-oda, njengakwi-ISBN, inani elisetyenziswa ukwahlula loo ncwadi kwezinye iincwadi, sitsho ke thina siyazi ngelo nani loo ncwadi xa siyifuna. 
  
• Amanani a-Cardinal asetyenziswa xa kujongwa   okokuba ingaba zingaphina izinto ezikwingqokolela ethile. Umzekelo nanku ingqokolela yaba nobumba  {A,B,C} ilingana nenani elingu-"3".
• Amanani a-ordinal asetyenziswa xa kugqalwa okanye kuboniswa ngokukodwa loo nto kthethwa ngayo ukuze yahluke kwezinye izinto eziqokolelwe ndawonye, into nganye ke kwezi iyakuthiwa ngokulandelelana kwazo kuthiwe yeyo-(kuqala, yeyesibini, yeyesithathu). Umzekelo, xa kubuzwa okokuba yeyiphi na le ntombi kwezi ntombi zikaKwayi ziluthoba, ebonelwe unyana kamfundisi uYekani? Impendulo isenokuthi yintombi yesithathu.
  

amanani asetyenziswa nakwezinye izinto ezingengomanani. amanani asetyenziswa xa kusenziwa umlinganiselo wezinto. Amanani asetyenziswa xa kufundwa okokuba lisebenza kanjani na ihlabathi. Izibalo yindlela yokusetyenziswa kwamanani ukuze kufundwe ngehlabathi kwenziwe nezinto. Isifundo ngemithetho yehlabathi kuthiwa zizifundo zenzululwazi. umsebenzi owenziwa ngokuthi kusetyenziswe amanani ukuze kwenziwe izinto kuthiwa zizifundo zobunjineli.

Amanani akwasetyenziswa nakwezinye izinto ngaphandle kokubala.  kusetyenziswa amanani xa kusenziwa umlinganiselo wezinto okanye xa kumentwa izinto. Amanani asetyenziswa xa kuphandwa ngendlela elisebenza ngayo ihlabathi.  I-Mathematics yindlela ekusetyenziswa ngayo amanani xa kufundwa ngehlabathi naxa kusenziwa izinto. Isifundo okanye uphando ngemithetho nemigaqo yehlabathi lemveli lubizwa ngokuba yi-science. Umsebenzi osebenzisa amanani xa kusenziwa into ubizwa ngokuba yi-  engineering.

Ukunikwa kwamanani iimpawu ezithile kwenziwa ngeendlela-ngeendlela.  Ezi ndlela ke zibizwa ngokuba zii-number systems. Eyona-number system ethanda ukusetyenziswa ngabantu yi-base ten number system. I-base ten number system ikwaziwa ngokuba yi-decimal number system. I-base ten number system ixhaphakile kuba abantu baneminwe elishumi neenzwane ezilishumi. Zili-10 iimpawu  zamanani {0, 1, 2, 3, 4, 5, 6, 7, 8, kunye no-9} ezisetyenziswa kwi-base ten number system. Ezimpawu zilishumi zibizwa ngokuba zii-digits.

Iindlela zokubala

Inani labantu 

Kukho iindidi ngeendidi zeendlela ekunikwa ngazo iimpawu zamanani. Ezi ndlela zibizwa ngokuba zii-number systems.  Olona hlobo luxhaphake kakhulu nelusetyenziswa kakhulu ngabantu lolu kuthiwa yi-base ten number system. I-base ten number system ikwabizwa ngokuba yi-decimal number system. I-base ten number system ixhaphakile kuba abantu baneminwe elishumi, kwaneenzwane ezikwalishumi. Ngoko ke kukho iimpawu ezili-10 ngokwahlukana kwazo, nazi: {0, 1, 2, 3, 4, 5, 6, 7, 8, and 9} zisetyenziswa kwi-base ten number system. Ezi mpawu zilishumi zibizwa ngokuba zii-digits.

Uphawu lwenani lwenziwe ngezi-digits zilishumi. Indawo ekuyo idigit nganye ibonisa ubukhulu nobuncinane benani. Umzekelo, inani elingama-23 kwi-decimal number system, eyonanto liyithethayo yile (u-2 xa ephindwa-phindwe kali-10) ize kuthi kwelo nani liphumileyo songeze inani elingu-3, kananjalo inani elingu-101 lithetha ukuthi inani elingu-1 xa liphinda-phindwe kalikhulu kuzakuphuma u- (=100) songeze u-0 esiyakumphindaphinda nge-10 kutsho kuphume u (=0) kongezwe u-1 ozakuphindwaphindwa ka-1 kutsho kuphume u-(=1).

Enye i-number system ixhaphake kakhulu kwiimatshini.  I-machine number system ibizwa ngokuba yi-binary number system. I-binary number system isoloko ibizwa ngokuba yi-base two number system. Zimbini ke iimpawu, (nazi: ngu-0 no-1) ezisetyenziswa kwi-base two number system. Ezi mpawu zimbini zibizwa ngokuba zii-bits.

I-simboli okanye uphawu lwe-binary number lwenziwe ngezi-bit symbols zimbini. indawo ezikuyo ezi-bit symbols zibonisa ubukhulu benani. Umzekelo, eyona nto ithethwa linani elingu-10 kwi-binary number system kukuba u-1 xa ephindwa-phindwa ka-2 kongezwe u-0, kunye no-101 uthetha ukuba u-1 xa ephindwa-phindwe kane (=4) kongezwa u-0 kuphinda-phindwe kabini (=0) kongezwe u-1 kuphinda-phindwe ka-1 (=1). I-binary number 10 iyinto enye ne-decimal number 2. I-binary number 101 iyinto enye ne-decimal number 5.

Iindlela zokunika amanani

Amagama amanani

Numbers for machines

Another number system is more common for machines. The machine number system is called the binary number system. The binary number system is also called the base two number system. There are two different symbols (0 and 1) used in the base two number system. These two symbols are called bits.

IsiNgesi sinamagama akhethekileyo abekelwe bucala ukuze asetyenziswe kumanani athile akwi-decimal number system nabizwa ngokuba ngamanani e--"powers of ten". onke ke laa manani e-power of ten kwi-decimal number system asebenzisa isimboli engu-"1" kunye nesimboli engu-"0" kuphela. Umzekelo, ishumi lamashumi yinto enye neshumi eliphinda-phindwe kalishumi, okanye lilkhulu elinye. Ngokweesimboli, iba ngu-"10x10=100". Kananjalo, ishumi lamakhulu liyinto enye neshumi eliphinda-phindwe ngekhulu, okanye singathi liwaka elinye. Ngokweesimboli ke sithi ngu- "10 × 100 = 10 × 10 × 10 = 1000". Namanye amanani e-power of ten numbers anamagama awo akhethekileyo. 

• 1 – inye
• 10 – zilishumi
• 100 – likhulu elinye
  
• 1000 – liwaka elinye
• 1,000,000 – sisigidi esinye
  

Xa usebenza ngamanani amakhulu kangaka, esiNgesini zimbini iindlela ezahlukeneyo zokubiza laa manani. Phantsi kwe-"long scale" inani linikwa igama elitsha ngalo lonke ixesha lilikhulu ngesigidi kune nani lokugqibela esele linikwe igama. lo ke ubizwa ngokuba ngumgangatho waseBritani. Esi sikali sasidla ngokuxhaphaka eBritani, kodwa asisetyenziswa njalo kumazwe antetho yawo isisingesi kule mihla. Nangona kunjalo sisasetyenziswa kwezinye izizwe zaseYurophu.    Esinye isikali sisi-"short scale" apho ke igama elitsha linikwa qho xa inani lilikhulu ngokuphindaphindwe kaliwaka ngaphezulu kwenani lokugqibela esele linegama. Esi sikali sixhaphake kakhulu kumazwe amaninzi antetho yawo isisiNgesi namhlanje. 

• 1,000,000,000 – i-billion enye (isikali esifutshane), i-milliard enye (isikali eside)
• 1,000,000,000,000 – i-trillion enye  (isikali esifutshane), i-billion enye (isikali eside)
• 1,000,000,000,000,000 –i-quadrillion (isikali esifutshane), i-billiard enye (isikali eside)

Amanani a-natural angamanani esihlala siwasebenzisa xa sibala, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 etc. Abanye abantu bathi u-0 naye ukwalinani eli-natural. 

Elinye igama lala manani kuthiwa angamanani a-positive. ngamanye amaxesha laa manani abhalwa njengo-+1 ukubonisa okokuba ahlukile kwamanye amanani waziwa ngokuba ngamanani a-negative numbers. Nangona kunjalo qaphela okokuba akusingawo onke amanani a-positive abanatural. Inani lisenokuba positive kodwa lingabikho natural (Umzekelo, i- ${\frac {1}{2}}$ linani eli-positive, kodwa alikho natural).

Ukuba u-0 ubizwa ngokuba linani eli-natural, ngoko ke amanani a-natural ayafana namanani azii-whole numbers. Ukuba u-0 akabizwa ngokuba linani eli-natural,  oko kuyakuthetha okokuba amanani a-natural ayinto enye namanani okubala (counting numbers ngesiNgesi).  Ukuba amagama athi "amanani a-natural" "natural numbers" ngesiNgesi, akasetyenziswa, kuyakweza ukubhideka ekubeni ingaba u-0 uyabandakanyeka na okanye hai. Kodwa ngelishwa, abanye abantu bathi u-O akalonani lipheleleyo (akayo-whole number, ngesiNgesi),   bakwathi abanye amanani apheleleyo (iiwhole numbers) zisenokuba negative. "ii-Positive integers" nee-"non-negative integers" ziyenye indlela yokubandakanya okanye yokukhuphela ngaphandle u-zero, kodwa oko kungenzeka kuphela xa abantu bewazi loo magama. 

Amanani a-negative ngamanani angaphantsi kuka-zero. 

Enye indlela yokucinga ngamanani a-negative, kukusebenzisa umgca-manani. kulo mgca-manani ke kukho indawo esiyibiza u-zero. Size ke emva koko sibhale ukubonisa (senze i-label)  (sibhale igama) yonke i-point esemgceni sijonge okokuba ingaba ikude kangakanani na kwicala langasekunene ukusuka kwi-point akuyo u--0, umzekelo, i-point engu-inye (u-point one)  uyi-centimeter enye ngokomgama ukuya ngasekunene, i-point engu-zimbini (u-point two) uzisentimitha ezimbini ukuya ngasekunene. 

Ngoku cinga nge-point eyi-centimeter enye ukuya kwicala langasekhohlo le-point engu-zero. Asinakuthi le yi-point u-inye (ngu-point one), njengoko kukho u-point osele ebizwa ngokuba ngu-point inye (ngu-point one).  Lo ke siyakumbiza njengo-point minus 1 (−1) (njengoko ikude nge-centimeter enye ku-0, kodwa lo mgama ukwelinye icala eli lingasekhohlo, icala elichaseneyo ke neliya lasekunene sele silibonile ngasentla apha). 

Umzobo weomgca-manani nanku ngezantsi. 

Yonke imisebenzi yokubala eyenzeka ngokufanelekileyo kwamathematika ingenzeka ngamanani a-negative:

Ukuba abantu bongeza inani eli-negative kwelinye inani, oko kuyinto enye nokususa okanye ukuthabatha kweli nani li-positive ngokwale ndlela alandelelana ngayo amanani (ukulandelelana kwamanani kubizwa ngokuba zii-numerals). Umzekelo, 5 + (−3) is the same as 5− 3, and equals 2. Khumbula ke, xa inani lingenaphawu lungu +, loo nto ithetha ukuba inani li-positive. Kodwa lona eli-negative liyakuhlala linophawu elingu -

Ukuba bathabatha inani eli-negative kwelinye inani eli-negative, oko kufana nqwa nokudibanisa okanye ukongeza inani eli-positive ngokokulandelelana kwawo laa manani (Khumbula, amanani alandelelanayo azii-numerals). Umzekelo, 5 − (−3) ufana nqwa no-5 + 3, abenza u-8. Xa amanani engoo-negative omabini, xa edityaniswa enza u-positive omnye.

Xa bephinda-phinda amanani amabini a-negative, sisiphumo iyakuba linani eli-positive.  Umzekelo, −5 xa simphinda-phinda ngo −3 kuyakwenzeka inani elingu- 15.

Ukuba baphinda-phinda inani eli-negative ngeli-positive, okanye baphinda-phinde inani eli-positive ngeli-negative. Bayakufumana iziphumo ezi-negative  umzekelo, u-5 xa ephindwa-phindwa ngo- −3 kuphuma u- −15.

Njengokuba ukufumana i-square root senani eli-negative kuyinto engenakwenzeka, kuba kaloku u-negative xa ephinda-phindwe ngomnye u-negative wenza okanye ulingana no-positive. siye ke sisinike uphawu olungu-i  i-square root senani eli-negative. Ngoko ke u-i luphawu lwe-square root senani eli-negative.

Ii-integers ngawo onke amanani a-natural. Khumbula ke okokuba amanani a-natural ngala sihlala sibala ngawo imihla nezolo, sisithi 1,2,3, njalo njalo. Onke ke amanye amanani anje kwelinye icala lomgca manani anje ngala -1, -2, -3, njalo njalo; kwanenani elingu-zero. Akazizo ii-intergers. Kanti nawo amanani a-decimal, oko kukuthi ii-decimal numbers kunye neefractions akazo-intergers.

Amanani a-rational, ii-Rational numbers ke ngesiNgesi ngamanani anokubhalwa njenee-fractions. Oku kuthetha ukuthi asenokubhalwa njengamanani anokohlulwa-hlulwa ngo-b, kodwa oko kunokwenzeka kuphela xa amanani angu- a no- b ezii-integers, kuyo yonke loo nto siqiniseke okokuba u-b akalingani no-0. 

Amanye amanani a-rational (ii-rational numbers), afana no-1/10, zidinga inani eli-finite (i-finite number) lee-digits emva kwe-decimal point ukuze abhalwe ngendlela e-decimal. Ishumi lokuqala (ebizwa ngokuba yi-number one tenth, ngesiNgesi) ibhalwa ngokwe-decimal ibe ngu-0.1. Amanani abhalwe ngokwendlela ye-finite decimal ngala aba-rational. Amanye amanani a-ratiinal  (azi-rational numbers), afana no-1/11, adinga inani eli-finate (i-infinite number)  lee-digits emva kwe-decimal point ukuze zibhalwe ngokwendlela ye-decimal. kukho ukuphindaphindwa kwe-pattern yee-digits okulandela emva kwe-decimal point. Inani leshumi elinanye lokuqala (i-number one eleventh) ibhalwa ngendlela ye-decimal eyile 0.0909090909 ... .

I-percentage isenokubizwa ngokuba ilinani eli-rational libe ke yi-rational number, kuba i-percentage efana no-7% isenokubhalwa njenge-fraction 7/100. Kananjalo isenokubhalwa njenge-decimal engu-0.07. Maxa wambi, i-ratio thathwa njengenani eli-rational i-rational number ke ngesiNgesi.

Amanani a-rational  ngamanani angenakubhalwa njenee-fractions, kodwa abe engenazo iindawo ezi- imaginary (kuyakucaciswa kamva).

√2 is irrational.

Amanani a-Irrational asoloko esenzeka kwi-geometry. Umzekelo, ukuba sine-square esinecala eliyimitha enye ngobude,  umgama okwiikona engaphesheya kwenye okanye ekwelinye icala, kwi-square-root sezi kona zimbini, ezilingana no-1.414213 ... . Le ke linani eli-irrational. I-Mathematika iye yabonakalisa okokuba i-square root salo naliphi na inani eli-natural lisenokuba yi-integer okanye sibe linani eli-irrational.

Inani elaziwa kakhulu ngokuba irrational ngu-pi. Le y-i-circumference (umgama ojikeleze) yesangqa yahlulwahlulwe nge-diameter yaso (umgama onqumla kwisangqa). Inani linye kuzo zonke izangqa. Inani eliyi-pi lingaphantse lithi 3.1415926535 ... .

inani eli-irrational alinakubhalwa phantsi ngokuphelelyo ngendlela ekubhalwa ngayo i-decimal . Liyakuba nenani elinee-digits ezi-infinite emva kwe-decimal point. Umahluko ophakathi kuka-0.333333 ..., ezi digits azinakuphinda-phina into engenasiphelo.

Amanani a-real abhekisa kuzo zonke ingqokolela zamanani ezidweliswe apha ngasentla:

• Amanani a-rational, aquka ii-integers
• Amanani a-irrational
  

Laa ke ngawo onke amanani angabandakanyi amanani azii-imaginary.

Amanani azii-imaginary are formed by real numbers multiplied by the number i. Eli nani lisi-square root sika minus one (−1).

Akukho nani eliyi-real number elinokuthi xa iphindaphindiwe lenze inani elingu--1. There is no number in the real numbers which when squared makes the number −1. Therefore, mathematicians invented a number. They called this number i, or the imaginary unit.

Amanani a-imaginary asebenza phantsi kwemiqathango nemithetho efanayo nale yee-real numbers:

• Isiphumo samanani amabini a-imaginary sifumaneka ngokuchwethela ngaphandle (ukukhuphela ngaphandle, ekuthiwa ngesiNgesi yi-factorising out) lo-i. Umzekelo, 2i + 3i = (2 + 3)i = 5i.
• Inani elishiyekayo okanye elenza umahluko kulamanani a-marginal or kwezi imaginal numbers zimbini lifumaneka kwangolu hlobo lunye. Umzekelo, 5i − 3i = (5 − 3)i = 2i.
• Xa ezi imaginary numbers zimbini ziphindwa-phindwa, khumbula okokuba u-i × i (i²) is −1. umzekelo, 5i × 3i = ( 5 × 3 ) × ( i × i ) = 15 × (−1) = −15.

Ii-imaginary numbers zabizwa ngokuba zii-imaginary kuba ukufunyanwa kwazo okokuqala, iingcali zemathematika ezininzi zange zicinge okokuba ngamanani akhoyo. iThemplethi:Fact umntu owafumanisa ezi-imaginary numbers yaba ngu-Gerolamo Cardano ngoo-1500s. Umntu wokuqala ukusebenzisa amagama athi 'imaginarly numbers yaba ngu-René Descartes. Abantu bokuqala ukusebenzisa laa manani kwaba ngoo-e Leonard Euler no-Carl Friedrich Gauss. Bobabini babephila ngomnyaka wenkulungwana ye-18th century.

Ii-Complex numbers ngamanani anamacala amabini; icala eliyi- real part kunye ne-imaginary part. Zonke ke ezi ntlobo zamanani achazwe apha ngasentla akwazii-complex numbers.

Ii-Complex numbers ziluhlobo oluquka onke amanani.  Ii-complex numbers zingabhalwa kwi-number plane. Le ke yenziwe ngee-real number line, kunye ne-imaginary number line.

           3i|_
             |
             |
           2i|_          . 2+2i
             |
             |
            i|_
             |
             |
 |_____|_____|_____|_____|_____|_____|_____|_____|
−2    −1     0     1     2     3     4     5     6
             |
           −i|_                .3−i
             |
             |
 .−2−2i   −2i|_
             |
             |
          −3i|_
             |

Yonke imathematika yesiqhelo ingenziwa ngee-complex numbers:

• Xa kusongezwa i-complex number kwenye, ngamanye amazwi xa kudityaniswa ii-complex numbers ezimbini, dibanisa amacala e-real nawe-imaginary ecaleni. Umzekelo, (2 + 3i) + (3 + 2i) = (2 + 3) + (3 + 2)i= 5 + 5i.
• Xa uphungula okanye uthatha i-complex number enye kwenye, phungula okanye thatha amacala e-real nele- imaginary parts ecaleni. Nanku umzekelo, (7 + 5i) − (3 + 3i) = (7 − 3) + (5 − 3)i = 4 + 2i.

IIndlela zokubala

Ii-Notes

Kunzima ukuphinda-phinda ii-complex numbers ezimbini. Kulula kakhulu ukuchaza nje ngokubanzi, ngee-complex numbers ezimbini  + bi and c + di.

           3i|_
             |
             |
           2i|_          . 2+2i
             |
             |
            i|_
             |
             |
 |_____|_____|_____|_____|_____|_____|_____|_____|
−2    −1     0     1     2     3     4     5     6
             |
           −i|_                .3−i
             |
             |
 .−2−2i   −2i|_
             |
             |
          −3i|_
             |

$(a+b\mathrm {i} )\times (c+d\mathrm {i} )=a\times c+a\times d\mathrm {i} +b\mathrm {i} \times c+b\mathrm {i} \times d\mathrm {i} =ac+ad\mathrm {i} +bc\mathrm {i} -bd=(ac-bd)+(ad+bc)\mathrm {i}$

For example, (4 + 5i) × (3 + 2i) = (4 × 3 − 5 × 2) + (4 × 2 + 5 × 3)i = (12 − 10) + (8 + 15)i = 2 + 23i.

I-real okanye i-complex number ibizwa ngokuba yi-transcendental number xa ingenakufunyanwa ngenxa yesiphumo se-algebraic equation inee-with integer coefficients.

$a_{n}x^{n}+\dots +a_{2}x^{2}+a_{1}x+a_{0}=0$

Kunganzima kakhulu ukufumanisa okokuba ingaba inani elithile li-transcendental na okanye hayi. I-transcendental number nganye ikwayi-irrational number. Abantu bokuqala abaqaphela okokuba kukho ii-transcendental numbers yaba ngu-Gottfried Wilhelm Leibniz no-Leonhard Euler. Umntu wokuqala owa-prova okokuba kukho ii-transcendental numbers yaba ngu-Joseph Liouville. Waakwenza oku ngo-1844.

Ii-transcendental numbers ezaziwa kakhulu:

• e
• π
• e^a for algebraic a ≠ 0
• $2^{\sqrt {2}}$

For the book in the Bible, see Numbers (Bible).

Ii- Transcendental numbers

I-real ookanye i-complex number ibizwa ngokuba yi-transcendental number xa ingenakufunyanwa ngenxa ye-algebraic equation ene-integer coefficients.

kunganzima kakhulu ukubona okokuba inani li- transcendental na okanye hayi. I-transcendental nganye ikwayi-rrational number.  Abantu bokuqala ukuphawula okokuba kukho ii-transcendental yayi ngoo- Gottfried Wilhelm Leibniz no- Leonhard Euler. Umntu wokuqala ukuyifumanisa ngokuza nobungqina bokokuba ngenene zikho ii-transcendental numbers yaaba ngu-Joseph Liouville. Waakwenza oku ngo1844 1844.

Ii-transcendental numbers ezaziwa kakhulu:

• e
• π
• e^a for algebraic a ≠ 0
• $2^{\sqrt {2}}$