Ua kapa ʻia nā helu decimal palena he mau hakina decimal, no ka mea, he hōʻike ʻokoʻa lākou no nā hakina me nā mana he ʻumi ma ka denominator. Pela no:
$$\frac{z}{n} = \frac{q_1}{1} + \frac{q_2}{10} + \dots + \frac{q_k}{10^k}$$
me \(k \in \mathbb{N}\) a me \(q_k\) ka \(k-1\) -th wahi ma ka akau ma hope o ke koma.
I kēia manawa:
$$\frac{z}{n} = \frac{10^k \cdot q_1 + 10^{k-1} q_2 + \dots + q_k}{10^k} = \frac{10^k \cdot q_1 + 10^{k-1} q_2 + \dots + q_k}{2^k \cdot 5^k}$$
'O ia ho'i: Inā hiki ke ho'onui 'ia ka denominator i \(2^k \cdot 5^k\) no ka hakina ma'amau i loko o ka 'ano pōkole loa \(\frac{z}{n}\) , he hakina pauka hope. . Inā mākou e noʻonoʻo i ka helu kumu nui o ka denominator \(n = p_1^{l_1} \cdot \, \dots \, \cdot p_j^{l_j}\) , a laila e like me ka theorem kumu o ka helu helu, hiki ke hōʻike ʻia kēia me he mea lā. \(f = 2^{km} \cdot 5^{kn}\) i \(2^k \cdot 5^k\) ina \(n = 2^m \cdot 5^n\) . Pili kēia:
ʻO nā hakina wale nō i loaʻa ʻole nā helu kumu kumu ma mua o nā 2 a i ʻole 5 i ka wā i hoʻopau piha ʻia e hoʻopuka i kahi hakina decimal palena.