There is a debate among the Muslim community on just how to calculate the beginning of the month of Ramadan (or indeed any month, but Ramadan takes on special importance). The traditional methods mentioned in the Qur'an and followed by the Prophet Muhammad, is to look to the sky and visibly sight the slight crescent moon (hilal) that marks the beginning of the month. If one sees the hilal at night, the next day is the first day of Ramadan and thus the first day of fasting. At the end of the month, when the community sights the hilal again, the Festival of Fast-Breaking (EID_UL_FITR) begins.
Over the years, various scholars and communities have answered this question in different ways. The prevailing opinion is that one should commit to a local moon-sighting, i.e. begin and end Ramadan based on the sighting of the moon in your local vicinity. Astronomical calculations can help us predict when the moon should be visible, but Muslims still tend to follow the traditional method of looking at the sky themselves and physically "sighting" the moon. Thus, the exact day of the beginning of Ramadan is not generally known until the night before the fast begins, when the moon is actually sighted and confirmed.
There are many hadith available regarding the moon sighting of Ramadan the messenger of ALLHA PROPHET MOHAMAD (PBUH) said : (HADITH_No 2378 SAHIH MUSLIM) Whenever you sight the new moon (of the month of Ramadan) observe fast, and when you sight it (the new moon of Shawwal) break it, and if the sky is cloudy for you, then observe fast for thirty days.
In another hadith PROPHET said: (SAHIH MUSLIM HADITH_No 2379) Observe fast on sighting it (the new moon) and break (fast) on sighting it (the new moon), but if the sky is cloudy for you, then complete the number (of thirty).
Islam has taught us when to start the holy month and when to end it and celebrate Eid_ul_Fitr and we can find these teaching through different hadith and Quran.