# Inani

### Amanani eematshini

Enye inkqubo yamanani exhaphakae kakhulu yeematshini. Inkqubo yenani lematshini ibizwa ngokuba yi-binary number system. Le binary number system ikwaziwa ngokuba yi-base two number system. Zimbini ke ii-symbols ngokwahlunana kwazo zezi-(0 and 1) ezisetyenziswa kwi-base two number system. ezi ke zimbini zibizwa ngokuba zii-bits.[2]
I-symbol ye-binary number nganye yenziwe ngezi-bit symbols zimbini. indawo ye-bit symbol nganye ibonisa ubukhulu benani. Umzekelo, inani le-10 kwi-binary number. Indawo emi kuyo i-bit symbols ibonisa ukuba ingaba likhulu kangakanani na inani. umzekelo, inani elingu-10 xa likwi-binary number system, enenei lithetha ukuthi 1 times 2 plus 0, aze u-101 athethe ukuthi 1 times four (=4) plus 0 times two (=0) plus 1 times 1 (=1). I-binary number 10 yinto enye ne-decimal number 2. Ize yona i-binary number 101 ibe yinto enye ne-decimal number 5.

### Amanani a-Irrational

Amanani angenangqiqo ngalawo manani aqhubeka ngokungenasiphelo, kodwa awayiyo amaxesha ngamaxesha kwaye akanalo inani elifanayo ngelo xesha lide. oludidi lwenani lunokufumaneka kwiingcambu ezikwere

**Amanani a-Rational** ngamanani anokubhalwa njengee-fractions. Oku kuthetha okokuba asenokubhalwa njengo-*a* owahlulwa-hlulwa ngo-*b*, apho ke amanani amelwe ngu-*a* no *b* azii-integers, aze ke yena u-*b* angalingani no-0.

Amanye amanani a-rational, afana no-1/10, adinga inani lee-digits eli-finite, emva kwe-decimal point ukuze zibhalwe ngendlela e-decimal. The number one tenth is written in decimal form as 0.1. Numbers written with a finite decimal form are rational. Some rational numbers, such as 1/11, need an infinite number of digits after the decimal point to write them in decimal form. There is a repeating pattern to the digits following the decimal point. The number one eleventh is written in decimal form as 0.0909090909... .

### Amanani a-Natural

*For the book in the Bible, see Numbers (Bible)*.
### Ii-integers

I-puzzle ye-Sudoku
**Ii-integers** ngawo onke amanani a-natural, zonke ii-opposites zawo, kunye nenani eling-u-zero. Amanani azii-Decimal nalawo azii-fractions akazizo ii-integers.

## Iindlela zokusetyenziswa kwamanani

Inani ngumbhalo ophuhlisa umbono wezibalo, osetyenziswa xa kubalwa okanye xa kusenziwa umlinganiselo. kuxhomekeke kwelo candelo lezibalo kuthethwa ngalo, apho kusetyenziswa khona amanani, kubakho iinkcazo ezahlukeneyo:

* 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).

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.

## IIndlela zokubala

### Inani labantu

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.

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.

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- WIKINIUM\_MATH\_PH\_0 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 azii-Irrational

**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

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

**Amanani azii-imaginary** enziwa ngamanani a-real aphindwe-phindwe by the number **i**. Eli nani lisi-square root sika minus one (−1).

Akukho nani eliyi-real number elinokuthi xa iphinda-phindiwe elingu-1. Akukho nani kumanani a-real xa 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, 2**i** + 3**i** = (2 + 3)**i** = 5**i**.
* Inani elishiyekayo okanye elenza umahluko kulamanani a-marginal or kwezi imaginal numbers zimbini lifumaneka kwangolu hlobo lunye. Umzekelo, 5**i** − 3**i** = (5 − 3)**i** = 2**i**.
* Xa ezi imaginary numbers zimbini ziphindwa-phindwa, khumbula okokuba u-**i** × **i** (i2) is −1. umzekelo, 5**i** × 3**i** = ( 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

**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 ngomgca-manani we-real number, kunye nomgca-manani we-imaginary number.

```
           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 + 3**i**) + (3 + 2**i**) = (2 + 3) + (3 + 2)**i**= 5 + 5**i**.
* Xa uphungula okanye uthatha i-complex number enye kwenye, phungula okanye thatha amacala e-real nele- imaginary parts ecaleni. Nanku umzekelo, (7 + 5**i**) − (3 + 3**i**) = (7 − 3) + (5 − 3)**i** = 4 + 2**i**.

Kunzima ukuphinda-phinda ii-complex numbers ezimbini. Kulula kakhulu ukuchaza nje ngokubanzi, ngee-complex numbers ezimbini  + b**i** and c + d**i**.

 WIKINIUM\_MATH\_PH\_2

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

### Ii-Transcendental numbers

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

 WIKINIUM\_MATH\_PH\_4

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.

Amanani a-transcendental awaziwa kakhulu:

* e
* π
* ea for algebraic a ≠ 0
* WIKINIUM\_MATH\_PH\_6

## Ii-Notes

## Amagama amanani

Jonga nakulo mbhalo uthi: amagama amanani abekwe ngesiNgesi
IsiNgesi sinamagama akhethekileyo samagama athile kwi-decimal number system, laa manani azii-"powers of ten". Onke ke la mamnani angamanani abizwa ngokuba zii-*power of ten numbers* kwi-decimal number system asebenzisa nje i-symbol engu-"1" kunye ne-symbol engu-"0". Umzekelo, ishumi lamashumi ekuthiwa ngesiNgesi u-ten tens, yinto enye no-ten times ten, okanye ikhulu (one hundred). Ngokwee-symbols singathi "10 × 10 = 100". Kananjalo, amashumi amakhulu (ten hundreds) yinto enye neshumi eliphinda-phindwe ngekhulu elinye (ten times one hundred),okanye sithi liwaka (or one thousand). Kwii-symbols, oku singathi ngu-"10 × 100 = 10 × 10 × 10 = 1000". Kukwakho nezinye ii-power of ten numbers ezikwanamagama awodwa.

* 1 – ngu-one
* 10 – lishumi
* ikhulu – one ikhulu elinye
* 1000 – one [[iwaka ekinye}}
* 000,000 – isigidi esinye

Xa kusetyenziswa amanani amakhulu kunala, zimbini iindlela zokuthiya amanani ngesiNgesi. phantsi kwe sikali eside ("the long scale") kuqala kuvele igama elitsha qho xa inani liye lalikhulu ngesigidi kunenani lokugqibela ukuthiwa. esi sikali sikwabizwa ngokuba siyi-"British Standard". esi sikali ke sasisaya kuxhaphaka eBritani, kodwa ke kungoku nje asisasetyenziswa kangako kumazwe antetho yazo isisiNgesi. Nangona kunjalo sisasetyenziswa kwezinye izizwe zaseYurophu. Esinye isikali yi-"short scale" kulapho ke igama elitsha elithi linikwe qho xa inani liye langaphezulu ngokuphindwa-phindwe kaliwaka ngobukhulu kunelo nani lokugqibela ebelivele lilkho. fikelela kwisigidi xa kuthelekiswa nenani lokugqibela elikhoyo. lafikelela kwisigidi kwinani lokugqibela elinikiweyo. Kungoku nje esi sikali ke sesona sixhaphakileyo kwizizwe apho abantu balhona bantetho yabo isisiNgesi.

* 1,000,000,000 – isigidi-gidi esinye (isikali esifutshane), i-milliard enye (isikali eside)
* 1,000,000,000,000 – i-trillion enye (isikali esifutshane), isigidi-gidi esinye (isikali eside)
* 1,000,000,000,000,000 – i-quadrillion enye (isikali esifutshane), {[amagama amanani amakhulu}} (isiklai eside)

### Amanani a-rational (Ii-rational numbers)

### Amanani a-negative

## Iindidi zamanani