'A'ishah, may Allah be pleased with her, reported:the Quraish used to fast on the Day of 'Ashura' in the pre-Islamic days and the Messenger of Allah (may peace be upon him) also observed it. When he migrated to Medina, he himself observed this Fast (of 'Ashura') and commanded (others) to observe it. But when fasting during the month of Ramadan was made obligatory he said: He who wishes to observe this Fast may do so, and he who wishes to abandon it may do so. [Sahih Muslim 1897]
Ibn ‘Abbas, (RA), reported: “The Prophet, came to al-Madinah and saw the Jews fasting on the day of Ashura. He asked them about that”. They replied, “This is a good day, the day on which Allah rescued Bani Israel from their enemy. So Moosa observed fast on this day.” The Prophet said, “We have more claim over Moosa than you.” So the Prophet, observed fast on that day and ordered (the Muslims) to observe fast (on that day).” [Saheeh al-Bukhari, Muslim]
Ibn ‘Abbas also reported, “The Messenger of Allah fasted on the day of Ashura and ordered the people to fast on it. The people said, “O Messenger of Allah, it is a day that the Jews and Christians honour.” The Prophet, said, “When the following year comes - Allah willing - we shall fast on the ninth.” The death of the Prophet, came before the following year.” [Saheeh Muslim, Abu Dawood]
So it is recommended to fast on the 10th of Muharram and the 9th along with it as well.