Yuav ua li cas to taub yog vim li cas lub "plus" mus rau "tsis zoo" muab lub "rho tawm"?

Mloog tus kws qhia ntawv ntawm kev kawm txog zauv, feem ntau ntawm cov menyuam kawm ntawv paub cov ntaub ntawv uas raws li ib tug axiom. Tab sis ob peb cov neeg ua tau mus rau hauv qab thiab paub yog vim li cas lub "rho tawm" mus rau "plus" muab ib tug "rho tawm" kos npe rau, thiab thaum twg txawj ob cov zauv negative los positive.

cov kev cai ntawm kev kawm txog zauv

Feem ntau cov neeg laus yuav tsis tau qhia rau lawv tus kheej los yog rau lawv cov me nyuam vim li cas qhov no yog li ntawd. Lawv nrees tsuab cov khoom nyob rau hauv lub tsev kawm ntawv, tab sis nws tsis txawm sim mus nrhiav qhov twg puas tau cov kev cai no. Thiab rau zoo yog vim li cas. Feem ntau, niaj hnub no cov me nyuam yog tsis yog li ntawd gullible, lawv yuav tsum tau mus rau hauv qab thiab to taub, rau Piv txwv li, yog vim li cas lub "plus" mus rau "tsis zoo" muab "rho tawm". Thiab tej zaum urchins yeej nug lo qhia lus nug, nyob rau hauv thiaj li yuav txaus siab rau lub sij hawm thaum cov neeg laus yuav tsis muab ib tug ntshiab lus teb. Thiab nws yeej yog hais tias ib tug hluas kws qhia ntawv twg thiaj li raug daig ...

Incidentally, nws yuav tsum tau muab sau tseg tias saum toj no-hais txoj cai no yuav siv tau rau lub npaug thiab rau fission. Cov khoom ntawm qhov tsis zoo thiab zoo xov tooj xwb "muab ib tug rho tawm. Yog hais tias muaj yog ob tug zauv nrog cov kos npe rau "-", qhov tshwm sim yog ib tug zoo xov tooj. Tib yam siv rau lub division. Yog hais tias ib tug ntawm cov zauv yuav tsum tsis zoo, ces tus quotient kuj yuav nrog tus kos npe rau "-".

Yuav kom piav qhia txog lub correctness ntawm cov kev cai lij choj ntawm kev kawm txog zauv, nws yog tsim nyog los npaj lub axiom rings. Tab sis yuav tsum xub to taub dab tsi nws yog. Nyob rau hauv kev kawm txog zauv hu ua nplhaib set nyob rau hauv uas ob haujlwm muab kev koom tes nrog ob ntsiab. Tab sis kom to taub nws zoo dua nrog ib qho piv txwv.

axiom nplhaib

Muaj ntau ntau zauv cai.

• Tus thawj ntawm cov commutative, raws li nws, C + V = V + C.
• Qhov thib ob yog hu ua associative (V + C) + D = V + (C + D).

Lawv kuj ua raws li thiab npaug (V x C) x D = V x (C x D).

Tsis muaj leej twg muab tso tseg thiab kev cai los ntawm cov uas qhib bracket (V + C) x D = V x D + C x D, nws kuj muaj tseeb hais tias C x (V + D) = C x V + C x D.

Tsis tas li ntawd, nws twb pom hais tias lub nplhaib yuav nkag mus rau ib tug tshwj xeeb nruab nrab los ntawm tas li ntawd ntawm ib tug keeb, kev siv cov uas cov nram qab no muaj tseeb: C + 0 = C. Nyob rau hauv tas li ntawd, rau txhua opposite C yog ib qho lub caij uas yuav tau xaiv raws li (C). Yog li C + (C) = 0.

Deducing axioms rau cov zauv negative

Los ntawm adopting lub saum toj no cov nqe lus, nws yog ua tau rau teb rau lo lus nug: "" plus "mus rau" tsis zoo "muab kev kos npe rau?" Thaum koj paub qhov axiom txog lub npaug ntawm cov zauv negative, koj yuav tsum kom paub meej tias uas tseeb (C) x V = - (C x V). Thiab kuj, dab tsi yog qhov tseeb yog sib npaug zos: (- (- C)) = C.

Ua li no, ua ntej peb muaj los ua pov thawj hais tias txhua tus ntawm lub ntsiab muaj no tsuas muaj ib yam txawv nws "tus kwv tij." Xav txog cov nram qab no ua pov thawj. Wb sim xav txog tej yam dab tsi rau hauv lub C opposite yog ob tug zauv - V thiab D. Los ntawm no nws hais ntxiv hais tias C + V = 0 thiab C + D = 0, piv txwv li C + V = 0 = C + D. Recalling lub commutative txoj cai thiab nyob rau thaj chaw ntawm cov zauv 0, peb yuav xav txog lub sum ntawm tag nrho peb tus xov tooj: C, V, thiab ua kom nrhiav tau tawm tus nqi ntawm D. V. Txawj, V = V + 0 = V + (C + D) = V + C + D, txij li thaum tus nqi ntawm C + D, tau txais raws li cov saum toj no, nws sib npaug 0. yog li ntawd, V = V + C + D.

Ib yam li ntawd, cov qhov tso zis nqi thiab rau D: D = V + C + D = (V + C) + D = 0 + D = D. Los ntawm no, nws yuav paub tseeb hais tias V = D.

Nyob rau hauv thiaj li yuav to taub vim li cas tag nrho cov "plus" mus rau "tsis zoo" muab ib tug "rho tawm", nws yog tsim nyog los nkag siab txog cov nram qab no. Yog li, rau lub caij (C) yog tau tawm tsam txoj thiab C (- (- C)), i.e. lawv yog muaj sib npaug rau txhua tus lwm yam.

Ces nws yog cuab kev tias 0 x V = (C + (C)) = C x V x V + (C) x V. Los ntawm no nws hais ntxiv hais tias C x V oppositely (-) C x V, yog li ntawd, (- C) x V = - (C x V).

Rau ib tug tag nrho zauv rigor yuav tsum tau kom paub tseeb tias 0 x V = 0 rau tej lub caij. Yog hais tias koj ua raws li cov logic, ces 0 x V = (0 + 0) x 0 x V = V + 0 x V. Qhov no txhais tau hais tias qhov sib ntxiv ntawm cov khoom 0 x V tsis hloov lub tshuaj npaum li cas. Tom qab tag nrho cov no ua hauj lwm yog pes tsawg.

Paub tag nrho cov ntawm cov axioms yuav muab tsis tau tsuas yog raws li lub "plus" mus rau "tsis zoo" muab, tab sis hais tias yog tau ua los ntawm txawj zauv negative.

Npaug thiab sib faib ntawm ob tug zauv uas muaj tus kos npe rau "-"

Tsis tas yuav mus rau hauv cov zauv nuances, koj yuav tau sim ib tug yooj yim txoj kev uas yuav piav txog cov kev cai ntawm kev txiav txim nrog cov zauv negative.

Xav hais tias hais tias C - (-V) = D, nyob rau hauv no lub hauv paus, C = D + (-V), i.e. C = D - V. Peb hloov thiab V peb pom tias C + V = D. Yog li ntawd, lub C + V = C - (-V). Qhov kev piv txwv qhia txog yog vim li cas cov kev qhia, qhov twg muaj ob lub "rho tawm" nyob rau hauv ib tug kab, hais tias cov cim yuav tsum tau hloov rau "plus". Tam sim no cia tus nrog sib npaug.

(C) x (-V) = D, nyob rau hauv cov kev qhia ntxiv thiab rho ob tug zoo tib daim uas yuav hloov tsis tau nws cov nuj nqis: (C) x (-V) + (C x V) - (C x V) = D.

Cia peb nco ntsoov cov kev cai ntawm lub staple lag luam, peb tau txais:

1) (C) x (-V) + (C x V) + (C) x V = D;

2) (C) x ((-V) + V) + C x V = D;

3) (C) + C x 0 x V = D;

4) C x V = D.

Los ntawm no nws hais ntxiv hais tias C x V = (C) x (-V).

Ib yam li ntawd, ib tug yuav ua pov thawj hais tias ib tug tshwm sim ntawm lub division ntawm ob cov zauv negative yuav zoo.

General zauv cai

Ntawm cov hoob kawm, qhov no piav yog tsis haum rau lub tsev kawm ntawv cov me nyuam uas yog cia li pib mus kawm paub daws teeb cov zauv negative. Lawv xav zoo piav rau tus pom yam khoom, manipulating lub sij hawm paub lawv los ntawm daim iav. Piv txwv li, yees, tab sis tsis muaj uas twb muaj lawm cov khoom ua si yog muaj. Lawv thiab yuav muab tso tawm kom nrog tus kos npe rau "-". Npaug ntawm ob yam khoom transmirror thauj lawv mus rau lwm lub ntiaj teb no, uas yog ncaj rau tus tam sim no, uas yog, raws li ib tug tshwm sim, peb muaj zoo xov tooj. Tab sis lub sib npaug ntawm paub daws teeb tsis zoo pes tsawg rau ib tug zoo muab tsuas tau lub npe hu mus rau tag nrho. Tom qab tag nrho, lub "plus" khoo los ntawm "rho tawm" muab lub "rho tawm". Txawm li cas los, nyob rau hauv lub tsev kawm ntawv uas muaj hnub nyoog cov me nyuam tsis heev ua tau mus rau hauv tag nrho cov zauv nuances.

Txawm tias, yog tias koj lub ntsej muag rau qhov tseeb, rau ntau tus neeg, txawm nrog kawm ntawv qib siab tseem yog ib tug paub tsis meej ntau cov kev cai. Tag nrho nws yuav siv sij hawm rau kev tso cov xib fwb qhia lawv, tsis yog teeb meem ntau heev rau delve rau hauv tag nrho cov teeb meem xam qhovkev nyob rau hauv lub lej. "Tsis zoo" mus rau "tsis zoo" muab "plus" - txhua leej txhua tus paub txog nws, tsis muaj kev zam. Qhov no yog raws li muaj tseeb rau tus kheej, thiab rau fractional tus xov tooj.