Chū-jiân-sò͘

Kok-chióng ê sò͘

Ki-pún

$\mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \subseteq \mathbb {C}$ Chèng-sò͘ ${\begin{smallmatrix}\mathbb {R} ^{+}\end{smallmatrix}}$ Chū-jiân-sò͘ ${\begin{smallmatrix}\mathbb {N} \end{smallmatrix}}$ Chèng-chéng-sò͘ ${\begin{smallmatrix}\mathbb {Z} ^{+}\end{smallmatrix}}$ Siáu-sò͘ Iú-hān siáu-sò͘ Bû-hān siáu-sò͘ Sûn-khoân siáu-sò͘ Iú-lí-sò͘ ${\begin{smallmatrix}\mathbb {Q} \end{smallmatrix}}$ Tāi-sò͘-sò͘ ${\begin{smallmatrix}\mathbb {A} \end{smallmatrix}}$ Si̍t-sò͘ ${\begin{smallmatrix}\mathbb {R} \end{smallmatrix}}$ Ho̍k-cha̍p-sò͘ ${\begin{smallmatrix}\mathbb {C} \end{smallmatrix}}$ Gauss chéng-sò͘ ${\begin{smallmatrix}\mathbb {Z} [i]\end{smallmatrix}}$ Hū-sò͘ ${\begin{smallmatrix}\mathbb {R} ^{-}\end{smallmatrix}}$ Chéng-sò͘ ${\begin{smallmatrix}\mathbb {Z} \end{smallmatrix}}$ Hū-chéng-sò͘ ${\begin{smallmatrix}\mathbb {Z} ^{-}\end{smallmatrix}}$ Hun-sò͘ Tan-ūi hun-sò͘ Jī-chìn hun-sò͘ Kui-kí-sò͘ Bû-lí-sò͘ Chhiau-oa̍t-sò͘ Hu-sò͘ Jī-chhù bû-lí-sò͘ Eisenstein chéng-sò͘ ${\begin{smallmatrix}\mathbb {Z} [\omega ]\end{smallmatrix}}$

Iân-sin

Siang-ho̍k-sò͘ Sù-goân-sò͘ ${\begin{smallmatrix}\mathbb {H} \end{smallmatrix}}$ Kiōng-sù-goân-sò͘ Pat-goân-sò͘ ${\begin{smallmatrix}\mathbb {O} \end{smallmatrix}}$ Chhiau-sò͘ Siōng-chhiau si̍t-sò͘ Chhiau-ho̍k-sò͘ Si̍p-lio̍k-goân-sò͘ ${\begin{smallmatrix}\mathbb {S} \end{smallmatrix}}$ Ho̍k-sù-goân-sò͘ Tāi-si̍t-sò͘ Chhiau-si̍t-sò͘ ${\begin{smallmatrix}{}^{\star }\mathbb {R} \end{smallmatrix}}$ Chhiau-hiān-si̍t-sò͘

Kî-thaⁿ

Tùi-ngó͘-sò͘ Siang-khiok ho̍k-sò͘ Sū-sò͘ Sò͘-sò͘ Tông-û Khó-kè-soàn-só͘ Aleph sò͘ Kong-chheng-ti̍t Chhiau-hān-sò͘ Ki-sò͘ P chìn-sò͘ kui-kí-sò͘ Chéng-sò͘ sò͘-lia̍t Sò͘-ha̍k siông-sò͘ ${\begin{smallmatrix}\pi \end{smallmatrix}}$ = 3.141592653… ${\begin{smallmatrix}e\end{smallmatrix}}$ = 2.718281828… ${\begin{smallmatrix}i\end{smallmatrix}}$ = ${\begin{smallmatrix}+{\sqrt {-1}}\end{smallmatrix}}$ ${\begin{smallmatrix}\infty \end{smallmatrix}}$

Chū-jiân-sò͘ 自然數 sī siōng ki-pún ê sò͘-ba̍k, ū lâng kóng sī chiàⁿ-ê chéng-sò͘ hō-chò chū-jiân-sò͘, mā ū lâng kā lêng mā sǹg-chāi-lāi. It-poaⁿ lâi kóng gián-kiù sò͘-lūn ê lâng bô kā 0 sǹg-chāi-lāi, gián-kiù chi̍p-ha̍p-lūn iah-sī tiān-náu kho-ha̍k ê lâng ōe kā 0 sǹg-chāi-lāi.

Chū-jiân-sò͘ ê lō͘-iōng chú-iàu sī the̍h-lâi sǹg mi̍h-kiāⁿ ê sò͘-liōng, he̍k-chiá-sī the̍h-lâi pâi sūn-sū. It-poaⁿ ēng N iah-sī $\mathbb {N}$ chò chū-jiân-sò͘ ê phiau-kì.

Chèng-sek ê tēng-gī

Chū-jiân-sò͘ ōe-tàng ēng Peano kong-siat lâi tēng-gī. Chū-jiân-sò͘ sī 1-ê bô chīn-pōng ê chi̍p-ha̍p

• Ū 1-ê sêng-oân hō-chò 0
• Jīm-hô 1-ê sêng-oân a lóng ōe-tàng tī chū-jiân-sò͘ lāi-tóe chhōe-tio̍h i-ê siong-sio̍k-chiá, kì-chò S(a).
• Chhōe-bô 1-ê sêng-oân ê siong-sio̍k-chiá sī 0.
• Bô-kâng ê sêng-oân, in-ê siong-sio̍k-chiá tō bô-kâng: nā a ≠ b tiāⁿ-tio̍h S(a) ≠ S(b).
• Jīm-hô 1-ê sêng-oân a ū-·ê sèng-chit, nā-sī-kóng tiāⁿ-tio̍h i-ê siong-sio̍k-chiá S(a) mā ài ū chit-ê sèng-chit, jî-chhiáⁿ 0 koh ū chit-ê sèng-chit, nā án-ne só͘-ū ê chū-jiân-sò͘ tō lóng ū chit-ê sèng-chit. Chit-tiâu kong-siat sī kóng sò͘-ha̍k kui-la̍p-hoat ōe-tàng ēng tī chū-jiân-sò͘.

Sèng-chit

Ka-hoat

Chū-jiân-sò͘ ê ka-hoat ēng (recursion) lâi tēng-gī:

1. a + 0 = a
2. a + S(b) = S(a) + b

Koh lâi tō khì chhōe siáng-ê siong-sio̍k-chiá sī b, ēng tē-2-tiâu it-tı̍t sǹg-·lo̍h-khì, kàu-bóe-·á tē-1-tiâu tō ōe-tàng ēng.

Kú 1-ê te̍k-pia̍t ê lē. Nā-sī kā S(0) tēng-gī chò 1, án-ne S(a) = S(a) + 0 = a + S(0) = a + 1.

Chiàu chit-ê tēng-gī, (N, +) chiâⁿ-sêng 1-ê ū tan-ūi-goân 0 ê poàⁿ-kûn.

Sêng-hoat

Chū-jiân-sò͘ ê sêng-hoat mā-sī ēng (recursion) lâi tēng-gī:

1. a × 0 = 0
2. a × S(b) = (a × b) + a

Kú 1-ê te̍k-pia̍t ê lē. a × 1 = a ×S(0) = a. Só͘-í 1 sī chū-jiân-sò͘ ê seng-hoat tan-ūi-goân. (N, ×) mā chiâⁿ-sêng 1-ê ū tan-ūi-goân ê poàⁿ-kûn.