Ngā Tau Whakamārama me ngā Tau Whakamārama: He Māramatanga Taketake mō te Pāngarau
He pūtaiao te pāngarau, e kī ana i ngā ariā me ngā ariā hei pūtake mō tō tātou māramatanga ki te ao. Ko tētahi o ngā ariā matua o te pāngarau ko te tau, e wehea ana ki ngā momo kāwai. I roto i tēnei tuhinga, ka matapakihia e tātou ngā kāwai nui e rua o ngā tau: ngā tau whaitake me ngā tau kore whaitake.
Ngā Tau Whakamārama: Te Whakamārama me ngā Tauira
Ko te tau whaitake he tau ka taea te whakaatu hei ōwehenga o ngā tauoti e rua. Arā, ko te tau whaitake he tau ka taea te tuhi hei hautanga \(\frac{a}{b}\), ko \(a\) me \(b\) he tauoti, ā, kāore a \(b\) i te ōrite ki te kore. Mā tēnei āhua ka taea te whakamahi ngāwari i ngā tau whaitake i roto i ngā mahi pāngarau maha.
Hei tauira, ko \(\frac{1}{2}\), \(\frac{5}{3}\), me \(-\frac{4}{7}\) he tau taurite katoa. Ahakoa ngā tau katoa pēnei i te 3, te -5 rānei, he tauira o ngā tau taurite nā te mea ka taea te tuhi hei \(\frac{3}{1}\) hei \(\frac{-5}{1}\ rānei.
Ngā Āhuatanga o ngā Tau Whakapūmau
1. Kiato: He tino kiato ngā tau taupū, arā, kei waenganui i ngā tau taupū e rua tētahi atu tau taupū. Hei tauira, i waenganui i te 0 me te 1, ka kitea e tātou \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{2}{3}\), me ērā atu. E whakaatu ana tēnei kāore he 'āputa', he tawhiti rānei kāore i tautuhia i waenganui i ngā tau taupū.
2. Ngā Mahi Pāngarau: Ka kati ngā tau taupū i raro i te tāpiri, te tango, te whakarea, me te wehewehe (haunga te wehewehe mā te kore). Hei tauira, mēnā he tau taupū a \(a\) me \(b\), ka noho ko \(a + b\), \(a – b\), \(a \cdot b\), me \(\frac{a}{b}\) hei tau taupū anō hoki (mēnā ko \(b \neq 0\) mō te wehewehe).
3. Te Whakawhanuitanga Ira-ā-ira: Ka taea te whakaatu i ngā tau taurite hei ira-ā-ira e tāruarua ana, e mutu ana rānei. Hei tauira, \(\frac{1}{2} = 0.5\) (te whakawhanuitanga ira-ā-ira e mutu ana) me \(\frac{1}{3} = 0.333…\) (te whakawhanuitanga ira-ā-ira e mutu ana).
Ngā Tau Tauritekore: Whakamārama me ngā Tauira
He rerekē ngā tau koretake i ngā tau e kore e taea te whakaatu hei ōwehenga o ngā tauoti e rua. Kāore e taea te tuhi i ngā tau koretake hei hautanga \(\frac{a}{b}\) ina he tauoti a \(a\) me \(b\) ā, ko \(b \neq 0\). Engari, he whakawhanuitanga ira tā rātou e kore e mutu, e kore hoki e pūrua i ia wā.
Ko ngā tauira rongonui o ngā tau koretake ko \(\pi\) (pi) me \(\sqrt{2}\) (te pūtake tapawhā o te 2). Ko te uara o \(\pi\) he tata ki te 3.14159…, ā, ka haere tonu ōna tau ira-ira me te kore he tauira e whakahokia ana. Waihoki, ko \(\sqrt{2}\) he tata ki te 1.41421…, ā, ka haere tonu ōna tau ira-ira me te kore he tauira e whakahokia ana.
Ngā Āhuatanga o ngā Tau Tauritekore
1. Te Whanuitanga Ira Mutunga Kore: He mutunga kore ngā whanuitanga ira o ngā tau koretake, kāore e mutu, kāore hoki e tāruarua. Hei tauira, he mutunga kore ngā whanuitanga ira o te tau \(\pi\) kāore he tauira tāruarua.
2. Kāore e taea te tuhi hei hautau: Kāore he huarahi hei tuhi i tētahi tau koretake hei hautau \(\frac{a}{b}\), ina ko \(a\) me \(b\) he tauoti, ā, ko \(b \neq 0\). Koinei te āhuatanga e wehewehe ana i ngā tau koretake mai i ngā tau whaitake.
3. Kiato: Pērā i ngā tau taurite, he tino kiato hoki ngā tau koretake. I waenganui i ngā tau koretake e rua, kei reira tētahi atu tau koretake.
Te Hononga i waenganui i ngā Tau Whakamārama me ngā Tau Whakakore
Ahakoa he huinga motuhake ngā tau whaitake me ngā tau koretake, ka hangaia e rāua te rārangi tau tūturu katoa. Ko ia pūwāhi i runga i te rārangi tau tūturu he tau whaitake, he tau koretake rānei. Nā tēnei ka tino kikī, ka haere tonu hoki ngā tau tūturu (te huinga o ngā tau whaitake me ngā tau koretake).
He mea whakamīharo, ahakoa he mutunga kore ngā tau whaitake, he onge ake i ngā tau koretake. I roto i ngā kupu ariā huinga, he nui ake te kātinaratanga (he maha atu ngā tau) o ngā tau koretake i ngā tau whaitake.
Te Hiranga i roto i te Pāngarau me te Pūtaiao
He mea nui te mahi a ngā tau taupū me ngā tau kore taupū i roto i ngā momo mara o te pāngarau me te pūtaiao. He maha ngā whakamahinga o ngā tau taupū i roto i ngā tataunga o ia rā, me ngā tono pēnei i te tatauranga, te ōhanga, me te hangarau. Ka āwhina i a tātou ki te mahi tataunga e pā ana ki ngā hautau me ngā ōrau.
I tētahi atu taha, he mahi motuhake tā ngā tau koretake i roto i te āhuahanga, te ine whārite, me te tātari. Hei tauira, ko te pūmau \(\pi\) he tau koretake e puta ana i roto i ngā tātaitanga āhuahanga maha e pā ana ki ngā porowhita. Waihoki, ko te pūmau \(e\) (tata ki te 2.718), he koretake hoki, he mahi nui tāna i roto i te tātaitanga me te ariā o te tipu taupū.
Whakamutunga
He tūāpapa nui te mārama ki ngā tau whaitake me ngā tau koretake i roto i te pāngarau. Ka taea te whakaatu i ngā tau whaitake hei hautau, ā, he whakawhanuitanga ira mutunga, he whakawhanuitanga ira tāruarua rānei, engari kāore e taea te tuhi i ngā tau koretake hei hautau, ā, he whakawhanuitanga ira kore-mutunga, kore-whakawhanuitanga ira kore-whakawhanuitanga.
Ahakoa he rerekē ēnei momo tau e rua, ka hangaia ngātahitia he rārangi tau tūturu tonu, kiato hoki. He mea nui tā rāua mahi i roto i ngā momo mara o te pāngarau me te pūtaiao, ā, koinei te pūtake o ngā ariā me ngā ariā uaua ake.
Mā te māramatanga pakari ki ngā tau whaitake me ngā tau kore whaitake, ka taea e tātou te tūhura haere i te ao pāngarau me te pārekareka ki te ataahua me te uaua o te ao whānui e tukuna ana e tēnei pūtaiao.
