Radix. QAUV nepozitsionnyh tooj tshuab

tooj system - yog dab tsi? Txawm tsis paub txog cov lus teb rau lo lus nug no, peb txhua tus tas nyob rau hauv koj lub neej nyiam numeration systems thiab tsis paub txog nws. Qhov ntawd yog txoj cai, nyob rau hauv lub plural! Qhov ntawd tsis yog ib tug, tab sis ob peb. Ua ntej muab piv txwv nepozitsionnyh notations, qhia rau peb saib qhov teeb meem no, peb yuav tham txog positional systems, ib yam nkaus thiab.

Qhov yuav tsum tau tus account

Txij ancient sij hawm, cov neeg muaj qhov yuav tsum tau khiav, uas yog intuitively paub hais tias koj yuav tsum cas qhia cov quantitative saib txog tej yam uas thiab cov txheej xwm. Lub hlwb qhia rau koj tias koj yuav tau siv cov khoom rau suav. Cov yooj yim tshaj plaws ib txwm tau nws cov ntiv tes, thiab qhov no yog to taub, vim hais tias lawv yeej ib txwm muaj nyob (nrog ob peb yam xwb).

Tias tau muaj tus hiob tug tswv cuab ntawm cov haiv neeg khoov nws cov ntiv tes nyob rau hauv lub literal siab - txhais lub xov tooj uas twb tuag lawm mammoths, piv txwv li. Lub npe ntawm tej nyiaj hais tsis muaj nyob, tiam sis tsuas yog ib tug yeeb yam duab, ib tug kev sib piv.

Niaj hnub nimno positional tooj system

Tus nab npawj System - ib txoj kev (txoj kev) repose ntau qhov tseem ceeb thiab ntau los ntawm tej yam cim (cov tsiaj ntawv los yog cov cim).

Nws yuav tsum tau to taub hais tias xws li positional nepozitsionnyh thiab cov hlau lead ua ntej yuav muab piv txwv nepozitsionnyh tooj systems. Positional tooj system teem. Tam sim no siv nyob rau hauv ntau yam teb raws li nram no: lub binary (muaj xws li tsuas yog ob tug loj Cheebtsam: 0 thiab 1) Senary (tus naj npawb ntawm cov cim - 6), octal (xaus-npawb tsab - 8) duodecimal (kaum ob cim), hex (muaj xws li kaum cim). Ib kab ib kab cim nyob rau hauv lub tshuab pib thaum pes tsawg. Niaj hnub nimno computer technology raws li nyob rau kev siv ntawm binary code - lub binary positional cim.

Zauv xov tooj system

Positional yog muaj nyob rau hauv cov zaubmov ntawm tseem ceeb txaus qhia, uas yog nyob ib tug xov tooj kos npe rau. Qhov no yog qhov zoo tshaj plaws taw qhia los ntawm tus zauv xov tooj system. Tom qab tag nrho, peb yeej ib txwm mloog nws los ntawm thaum yau. Tej yam tshwm sim nyob rau hauv no system kaum: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Ua tus xov tooj 327. Muaj peb tug lej tom qab 3, 2, 7. Txhua tus ntawm lawv yog nyob rau ntawm nws txoj hauj lwm ( qhov chaw). Xya yuav siv sij hawm rau txoj hauj lwm muab rau ib tug tib tus nqi (units), deuce - dozens, thiab cov triple - pua pua. Txij li thaum lub peb-lej, yog li ntawd, txoj hauj lwm nws cia li peb.

Raws li cov saum toj no, ib tug peb-ntiv decimal tooj yuav piav raws li nram no: peb puas, thiab nees nkaum xya lwm. Thiab qhov tseem ceeb (tseem ceeb) txoj hauj lwm suav los ntawm sab laug mus rau sab xis, los ntawm ib tug tsis muaj zog txoj hauj lwm (chav tsev) los muaj zog (pua pua).

Peb twb nyob heev xav nyob rau hauv tus zauv positional tooj system. Peb nyob rau hauv lub ob txhais tes ntawm kaum tus ntiv tes rau lawv ko taw - zoo li. Tsib ntxiv tsib - yog li ntawd, ua tsaug rau cov ntiv tes, peb tau yooj yim xav txog tej yam lub thaum yau ntawm kaum tawm. Uas yog vim li cas muaj yog ib qho yooj yim rau cov me nyuam mus kawm rau hauv lub npaug rooj ntawm tsib thiab kaum. Thiab yog li ntawd ib qho yooj yim los mus kawm rau suav banknotes, uas yog feem ntau cov zauv (ie faib tsis muaj ib tug seem) ntawm tsib thiab kaum.

Lwm yam positional tooj system

Rau cov surprise ntawm ntau yam, nws yuav tsum tau hais tias tsis tsuas yog peb lub hlwb yog accustomed mus rau ua ib co lus teb nyob rau hauv cov zauv suav system. Kom txog rau thaum tam sim no, noob neej siv Senary thiab Duodecimal. Uas yog, nyob rau hauv no system muaj tsuas yog rau cim (nyob rau hauv Senary): 0, 1, 2, 3, 4, 5. Thaum lawv kaum ob duodecimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 , A, B, qhov twg A - yog tus naj npawb 10, - tus naj npawb 11 (txij li thaum lub kos npe rau yuav tsum tau ib qho).

Txiav txim rau koj tus kheej. Peb ntseeg hais tias lub sij hawm sixes, tsis yog nws? Ib teev - rau caum feeb (rau caum), muaj ib hnub - nws yog nees nkaum plaub teev (ob lub sij hawm kaum ob) xyoo - kaum ob lub hlis, thiab thiaj li nyob rau ... Tag nrho lub sij hawm slots yooj yim haum mus rau hauv six- thiab duodecimal xov tooj. Tab sis peb muaj ces siv mus rau nws, peb tsis txawm xav hais tias nyob rau hauv kev nyeem ntawv lub sij hawm.

Nonpositional tooj system. unary

Koj yuav tsum txiav txim siab yog dab tsi nws yog - nepozitsionnyh tooj system. Qhov no yog xws li ib tug leb system, nyob rau hauv uas muaj yog tsis muaj txoj hauj lwm rau tus xov tooj ntawm cov cim, los yog lub hauv paus ntsiab lus ntawm "nyeem" ntawm txoj hauj lwm yog ywj siab. Nws kuj muaj nws tus kheej nkag cai thiab suav.

Ntawm no yog ib co piv txwv nepozitsionnyh tooj systems. Wb rov qab mus rau lub sij hawm ancient. Cov neeg siv yuav tsum tau tus account thiab tuaj mus nrog rau feem ntau yooj yim invention - nodules. Nonpositional tooj system yog nodular. Ib tug kawm (mov hnab, nyuj, haystack , etc.) suav, piv txwv li, thaum yuav los yog muag thiab khi pob caus nyob rau hauv txoj hlua.

Raws li ib tug tshwm sim, cov hlua tau txais raws li ntau cov pob caus ntawd, yuav ua li cas muaj ntau yam hnab mov muas (raws li ib qho piv txwv). Tab sis nws kuj yuav yog ib tug nqis siab tshaj cov nyob ib tug ntoo stick rau ib lub pob zeb slab, etc. Qhov no numbering system twb muaj npe lumpy. Nws muaj ib tug thib ob lub npe - unary, los yog ib ( "UNO" nyob rau hauv Latin txhais tau tias "ib tug").

Nws yuav tshwm hais tias tus xov tooj system - nepozitsionnyh. Tom qab tag nrho, hais txog dab tsi txaus yog peb tham txog thaum nws (txoj hauj lwm) tsuas yog ib qho! Ironically, nyob rau hauv tej qhov chaw ntawm lub ntiaj teb no tseem nyob rau hauv Vogue nepozitsionnyh unary tooj system.

Tsis tas li ntawd mus nepozitsionnyh tooj system muaj xws li:

• Roman (sau ntawv xov tooj siv cov tsiaj ntawv - Latin cim);
• Ancient Iyiv (ib yam li cov Roman, kuj siv tej cim);
• tsiaj ntawv (siv cov tsiaj ntawv ntawm cov tsiaj ntawv);
• Babylonian (cuneiform - siv ncaj qha thiab prevernuty "npoo");
• Greek (kuj raug xa mus rau raws li cov tsiaj ntawv).

Cov Roman tus nab npawj system

Ancient Roman faj tim teb chaws, raws li tau zoo raws li nws cov kev kawm, yog heev hnyav. Cov Loos muab lub ntiaj teb no muaj ntau pab tau inventions ntawm science thiab art, nrog rau nws cov account system. Ob puas xyoo dhau los, Roman numerals tau siv los txhais tus nqi ntawm ua lag ua luam cov ntaub ntawv (yog li tsis txhob cuav).

Roman numerals - piv txwv li nonpositional tooj system, nws yog hu rau peb tam sim no. Roman system kuj nquag siv, tab sis tsis yog rau zauv, thiab rau narrowly tsom ua. Piv txwv li, siv Roman numerals mus txhais keeb kwm hnub, xyoo pua, ntim cov xov tooj, khej, thiab tshooj nyob rau hauv phau ntawv cov ntaub ntawv. Feem ntau siv rau dai kom zoo ntawm Roman tej yam tshwm sim ntawm dials teev. Thiab ib qho piv txwv ntawm Roman numerals nonpositional Radix.

Cov Loos xws tooj tsiaj ntawv ntawm lub Latin niam ntawv. Thiab tus naj npawb ntawm lawv kaw los ntawm tej yam cov kev cai. Muaj ib daim ntawv teev tseem ceeb cim nyob rau hauv lub Roman tus nab npawj system, los ntawm txoj kev lawv raug kaw tag nrho cov zauv, tsis muaj kev zam.

Cov kev cai ntawm nqus tau cov zauv

Qhov yuav tsum tau tus xov tooj yog tau los ntawm kev ntxiv cim (Latin ntawv) thiab xam xyuas lawv zaum. Xav txog yuav ua li cas symbolically sau tej yam tshwm sim nyob rau hauv lub Roman system, thiab yuav ua li cas lawv yuav tsum tau "nyeem". Peb sau cov yooj yim cov kev cai ntawm tsim ntawm tus xov tooj nyob rau hauv lub Roman tus nab npawj system nonpositional.

1. Tus xov tooj plaub - IV, yog muaj li ntawm ob cim (kuv, V - ib tug thiab tsib). Nws yog tau los ntawm subtracting tus me kos npe rau ntawm ntau yog hais tias nws sawv mus rau sab laug. Thaum tus me cim nyob rau txoj kev, nws yog tsim nyog los ntxiv, ces tau txais xov tooj rau - VI.
2. Nws yog tsim nyog los ntxiv ob tug zoo tib kos npe rau sawv nyob ze. Piv txwv li: SS - yog 200 (C - 100) los yog lub XX - 20.
3. Yog hais tias tus thawj cim tus xov tooj yog tsawg tshaj li lub thib ob, thib peb nyob rau hauv lub series tej zaum yuav ib lub cim uas nws muaj nqis yog tseem me me dua thawj zaug. Kom tsis txhob tsis meej pem, peb yuav muab ib qho piv txwv: CDX - 410 (decimal).
4. Ib txhia ntawm cov loj xov tooj yuav tsum muaj tuaj nyob rau hauv ntau txoj kev, uas yog ib tug ntawm cov downsides ntawm lub Roman suav system. Ntawm no yog ib co piv txwv: MVM (Roman system) = 1000 + (1000 - 5) = 1995 (decimal system) los yog MDVD = 1000 + 500 + (500 - 5) = 1995. Thiab hais tias yog tsis yog txhua txhua txoj kev.

xam tricks

Nepozitsionnyh tooj system - qhov no yog tej zaum ib tug complex set ntawm cov kev cai rau txoj kev ua zauv, lawv ua (ua hauj lwm rau lawv). Xam ua hauj lwm nyob rau hauv nepozitsionnyh tooj tshuab - yog tsis yooj yim rau niaj hnub tus neeg. Peb tsis envy ib tug Roman mathematicians!

QAUV ntxiv. Wb sim ntxiv ob tus xov tooj: XIX + XXVI = XXXV, qhov no ua hauj lwm yog ua nyob rau hauv ob kauj ruam:

1. Tus thawj - thiab coj ib tug me me kev faib ua feem ntawm cov zauv ntxiv mus: IX + VI = XV (kuv V thiab kuv tom qab ua ntej X "tua" ib leeg).
2. Ob - ntxiv txog shares loj ntawm ob tus xov tooj: X + XX = XXX.

Rho yog ua dog dig nyuab dua. Thiaj li muaj cov xov tooj uas yuav tsum tau phua rau hauv nws cov constituent hais, thiab tom qab ntawd txo thiab subtracts los txo duplicate cim. Ntawm 500 Rho 263:

D - CCLXIII = CCCCLXXXXVIIIII - CCLXIII = CCXXXVII.

Npaug Roman numerals. Los ntawm txoj kev, nws yog tsim nyog los hais tias cov neeg Loos tsis muaj tej yam tshwm sim arifmetichekih ua hauj lwm, lawv tsuas lo lus rau lawv.

Multiplicand muab tus xov tooj uas yuav tsum tau rau txhua tus neeg d cim, tau txais ob peb daim uas yuav tsum tau yuav tsum tau daim ntau quav tso. Nyob rau hauv no yav tas ua ib tug npaug ntawm polynomials.

Nrog kev xav txog cov division, tus txheej txheem nyob rau hauv lub Roman tus nab npawj system yog thiab tseem yog cov feem ntau nyuaj. Ces thov lub ancient Roman cov qhab nees - abacus. Mus ua hauj lwm nrog nws qhia tshwj xeeb uas kawm cov neeg (thiab tsis yog txhua tus neeg muaj peev xwm mus kawm tau science).

Nyob rau shortcomings nepozitsionnyh tshuab

Raws li hais saum toj no, muaj drawbacks, inconveniences nyob rau hauv kev siv nepozitsionnyh tooj systems. Unary yog yooj yim txaus rau ib tug yooj yim account, tab sis xam thiab complex suav, nws tsis yog tsim nyog rau tag nrho cov.

Nyob rau hauv lub nroog Loos muaj cov tsis muaj ntau cov kev cai rau cov tsim ntawm loj tus xov tooj thiab muaj ib tug ib niag pawg, thiab nws yog ib qhov nyuaj ua suav. Nyob rau hauv tas li ntawd, feem ntau loj xov tooj, uas yuav tsum tau sau los ntawm cov neeg Loos kev pab los ntawm nws txoj kev, yog 100,000.