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Today is the date marks the establishment of Republic of China. Many people from Taiwan might refer you that today is the “Double Ten Day”, but the term is not self explanatory, and today (1010) has really nothing to do with binary numbering. The correct term should be “Republic Day.”
Republic of China(ROC) was (is?) the first Chinese Republic, which ends Chinese monarchy in 1911. After World War II, Communists bets Nationalist who leads R.O.C. and form their second Chinese Republic – People’s Republic of China (PRC). Nationalists then bore the ROC name and fled to Taiwan. Before that, Taiwan was a Japanese colony for 50 years (1895-1945).
The ROC government continued acting as Chinese governments in exile until 1994, a document implicitly recognise the existence of PRC and place ROC as the government represents the Taiwanese people, willing to negotiate for peaceful unification (at the time.)
The new ruling party from 2000, which is pro independence though, push for the formal reorganisation of PRC and ourselves. Plans includes legally change the name of the state to Taiwan instead of ROC which suggested itself as a Chinese government in exile. To do that, with formally “giving up” the sovereignty over China, a constitutional change, even a new constitution has to be in place. PRC sees these reform as a move toward formal independence against China, and thread the use of force. Pro unification parties also rejects the idea.
For your information. I hope what I wrote above can clean things up for you, and answer the question “why Taiwan is formally named Republic of China?” in mind.
With legal point of view, de jule status of Taiwan can only be certain if a treaty was made by China, Japan and United States. But sadly, such treaty can never took place after a serious war, probably World War III. And nothing in the international law says the future if Taiwan should consider the will of Taiwanese residence (the US Taiwan Relation Act does though). Taiwan has operate herself independently for 57 years without a formal status, and is likely to do so for decades (or caught into a war which is the last thing I would like to see).