Nzira dzeUchapupu hweMasvomhu
Humbowo mumasvomhu ndihwo hwaro hwechidzidzo ichi. Nzira dzekusimbisa ndidzo hwaro hwekuona kuti chirevo chemasvomhu ndechechokwadi. Kubva pakufungidzira kwekutanga kusvika pakuguma, danho rega rega rinofanira kuvimbiswa kuti rinoshanda. Kunzwisisa nzira dzakasiyana-siyana dzekusimbisa hakungosimbisi hunyanzvi hwekuongorora chete asiwo kunovandudza ruzivo rwekudzidza nekushandiswa kwemasvomhu muzvikamu zvakasiyana-siyana.
Chinyorwa chino chichakurukura dzimwe nzira huru dzekuongorora masvomhu, kusanganisira humbowo hwakananga, humbowo husina kunanga (kupesana uye kupokana), kupinza masvomhu, uye humbowo nemuenzaniso chaiwo. Nzira imwe neimwe ine mashandisirwo akasiyana, simba, uye kusasimba. Ngatizviongorore zvakadzama.
1. Humbowo Hwakananga
Tsanangudzo uye Mienzaniso
Humbowo hwakananga inzira yatinoratidza nayo chirevo nekuratidza kuti kana pfungwa (fungidziro) dziri dzechokwadi, saka mhedziso yacho ndeyechokwadiwo. Muhumbowo hwakananga, tinowanzotanga nezvinozivikanwa toshandisa matanho ane musoro kuti tisvike pamhedziso.
Muenzaniso:
Ratidza kuti kana \(n\) iri nhamba yakaenzana, saka \(n^2\) ikaenzanawo.
Humbowo:
Ngatitii \(n\) inhamba yakaenzana. Zvadaro, zvichienderana netsananguro yenhamba yakaenzana, zvinogona kunyorwa kuti \(n = 2k\) kune imwe nhamba yakakwana \(k\). Saka,
\[ n^2 = (2k)^2 = 4k^2 = 2(2k^2) \]
Zviri pachena kuti \(n^2\) inogona kuratidzwa kaviri panhamba yese (kureva \(2k^2\)). Sezvo chinodiwa chikuru chenhamba yese ndechekuti inogona kuratidzwa kaviri panhamba yese, saka \(n^2\) inhamba yese.
2. Humbowo Husina Kunanga
Humbowo husina kunanga hunosanganisira nzira mbiri huru: humbowo hunobva mukupesana uye humbowo hunobva mukupesana.
a. Humbowo hweKusawirirana
Tsanangudzo uye Mienzaniso
Nzira iyi inosanganisira kuratidza chirevo chinoreva kuti "kana \(P\), ipapo \(Q\)" nekuratidza zvinopesana nechirevo ichi: "kana zvisiri \(Q\), saka kwete \(P\)".
Muenzaniso:
Ratidza kuti kana \(n^2\) isingawanzoitiki, saka \(n\) haiwanzoitikiwo.
Humbowo:
Zvinopesana nemashoko aya ndeizvi: Kana \(n\) isiri odd (kana even), saka \(n^2\) haisi odd (kana even).
Ngatitii \(n\) yakaenzana, saka \(n = 2k\) yenhamba yese \(k\). Saka,
\[ n^2 = (2k)^2 = 4k^2 = 2(2k^2) \]
Izvi zvinoreva kuti \(n^2\) inhamba yakaenzana. Saka, chiratidzo chekuti contrapositive chinoratidzwa, uye chirevo chekutanga chinovimbiswawo kuti ndechechokwadi.
b. Humbowo hunobva mukupokana
Tsanangudzo uye Mienzaniso
Humbowo hwekupokana hunosanganisira kufunga kuti chirevo chinofanira kusimbiswa ndechenhema uye kuratidza kuti fungidziro iyi inotungamira kupokana kunonzwisisika.
Muenzaniso:
Ratidza kuti \(\sqrt{2}\) inhamba isina musoro.
Humbowo:
Ngatitii, pachinzvimbo, \(\sqrt{2}\) inhamba ine musoro. Zvadaro, \(\sqrt{2} = \frac{a}{b}\), apo \(a\) uye \(b\) ari nhamba dzinonzi prime integers (kubvisa i1), uye \(b \ne 0\). Saka, tinogona kunyora:
\[ \sqrt{2} = \frac{a}{b} \]
\[ 2 = \frac{a^2}{b^2} \]
\[ 2b^2 = a^2 \]
Kubva muequation iyi, tinoona kuti \(a^2\) inhamba yakaenzana, zvinoreva kuti \(a\) inofanirawo kunge yakaenzana. Ngatitii \(a = 2k\), tine:
\[ 2b^2 = (2k)^2 \]
\[ 2b^2 = 4k^2 \]
\[ b^2 = 2k^2 \]
Sezvo \(b^2\) iri nhamba yakaenzana, saka \(b\) inofanirawo kunge iri nhamba yakaenzana. Izvi zvinoreva kuti \(a\) na \(b\) ese ari manhamba akaenzana, zvichipesana nepfungwa yekutanga yekuti \(\frac{a}{b}\) iri muchimiro chayo chiri nyore. Saka, \(\sqrt{2}\) haigone kuva nhamba ine musoro, saka haina musoro.
3. Kudzidziswa kwemasvomhu
Tsanangudzo uye Mienzaniso
Kupinza masvomhu inzira yekuratidza humbowo inoshandiswa kuratidza zvirevo zvine nhamba dzese. Maitiro acho ane matanho maviri: hwaro hwekutanga uye danho rekutanga.
Muenzaniso:
Ratidza kuti huwandu hwenhamba yekutanga yenhamba \(1 + 2 + 3 + … + n = \frac{n(n+1)}{2}\).
Humbowo:
- Hwaro hweInduction:
Kune \(n = 1\),
\[ 1 = \frac{1(1+1)}{2} \]
correct.
- Matanho Ekutanga:
Ngatitii chirevo ichi ndechechokwadi kune nhamba \(k\). Kureva kuti,
\[ 1 + 2 + 3 + … + k = \frac{k(k+1)}{2} \]
Tinofanira kuratidza kuti ichokwadiwo kuna \(k + 1\). Tinowedzera \((k + 1)\) kumativi ese e equation:
\[ 1 + 2 + 3 + … + k + (k + 1) = \frac{k(k+1)}{2} + (k + 1) \]
\[ = \frac{k(k+1) + 2(k+1)}{2} \]
\[ = \frac{(k + 1)(k + 2)}{2} \]
Saka, chirevo ichi ndechechokwadi pa \(k + 1\). Saka, nepfungwa yekudzidziswa kwemasvomhu, chirevo ichi ndechechokwadi panhamba dzese dzinokwana \(n\).
4. Humbowo Hune Mienzaniso Yakananga
Tsanangudzo uye Mienzaniso
Nzira iyi inosanganisira kuratidza nekusarudza mienzaniso chaiyo inoenderana nemamiriro ese akapihwa muchirevo uye inoratidza kuti chirevo chacho ndechechokwadi. Zvisinei, nzira iyi inowanzo shandiswa kuratidza chirevo chenhema.
Muenzaniso:
Ratidza kuti kune nhamba dzisingagone kutsanangurwa sehuwandu hwemasikweya maviri akakwana.
Humbowo:
Edza kushandisa muenzaniso \(3\):
Ngatitii \(3\) inogona kuratidzwa sehuwandu hwemasikweya maviri akakwana, anoti \(a^2 + b^2 = 3\). Mushure mekuedza kusanganisa kwese kwenhamba dzese \(a\) uye \(b\),
1. \(a = 0\), \(b^2 = 3\) (hazvigoneki).
2. \(a = 1\), \(b^2 = 2\) (hazvigoneki).
3. \(a = 2\), \(b^2 = -1\) (hazvigoneki).
4. Nhamba dzisina kunaka kana nhamba dzinopfuura 2 hazvigonekiwo.
Izvi zvinoratidza kuti \(3\) haigone kuratidzwa sehuwandu hwenhamba mbiri dzesikweya. Saka, kune nhamba dzisingagone kuratidzwa sehuwandu hwenhamba mbiri dzesikweya dzakakwana.
Mhedziso
Humbowo mumasvomhu hunoda nzira dzakasiyana uye matanho akarongeka zvichienderana nerudzi rwemashoko ari kuratidzwa. Humbowo hwakananga, humbowo husina kunanga (hunopesana uye hunopesana), kupinza masvomhu, uye mienzaniso yakakosha ndedzimwe dzenzira huru dzehumbowo dzinoshandiswa mumamiriro akasiyana-siyana. Kunzwisisa nzira idzi kuchasimbisa hwaro hwemasvomhu uye kukubatsira kuongorora zvakadzama mapazi akasiyana-siyana emasvomhu.
Nekudzidzira uye kunzwisisa kwakadzama, nzira dzekuongorora masvomhu dzichava chishandiso chichagara chakagadzirira kushandiswa mukugadzirisa matambudziko akaomarara emasvomhu.
