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Tahajjud means specifically praying at night, and some scholars have said them to be prayers that are offered at night after sleeping. 

Al-Hajjaaj ibn ‘Amr al-Ansari (may Allah be pleased with him) said: One of you thinks that if he gets up at night and prays until morning comes that he has done tahajjud. But in fact tahajjud means praying after sleeping, then praying after sleeping. That is how the Messenger of Allah (blessings and peace of Allah upon him) prayed. 

The other view from different scholars is this Ibn Baaz (may Allah have mercy on him) said:

The time for tahajjud begins after ‘Isha’ and ends with the break of dawn. When the people have prayed ‘Isha’, the time for tahajjud begins and lasts until the break of dawn. The Sunnah is to pray qiyaam al-layl between the end of ‘Isha’ prayer and the break of dawn, even if it is one rak‘ah of Witr.

However, the best time for performing Tahajjud prayer is the last part of the night, because of what has been reported on the authority of 'Amr ibn 'Abasah that he said, I asked: O Messenger of Allah, in which part of night the supplication is more likely to be accepted?

He replied: In the last part: Pray as much as you like, for the prayer is attended by the angels and it is recorded till you offer the dawn prayer the Fajr prayer ..." (Sunan Abi Dawood)

Now the specific timing for this varies based on seasons and the actual region of the world you live in.

A simple way to calculate the exact time, would be to divide the night in three portions. It is calculated from sunset so Maghreb time, to the break of dawn before fajr prayer.

Let’s say there are 9 hours between the two, so each 1/3 of the night would be 3 hours.

Now, how many rakats, Generally, two rakats is seen as the minimum necessary for Tahajjud, but it's possible to pray as many as you wish to be prayed 2 by 2. There are hadiths which say the prophet (saws) prayed 8.

God willing In sha’a Allah this video was clear about the Tahajjud prayer.

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Alsalam Alaykum