my question is about new years day in what century was January 1st made official for new years day?

Answer 1

I can answer this because...I Know (tm)!
It started very informally and it started in the Dark Ages (roughly 5th-10th Century). If you ever read the 'Anglo Saxon Chronicle' you will notice it is little more than a series of diary entries. In actual fact the ASC is a series of entries jotted alongside abbey accounts. When the new year started (for economic reasons) often varied. Easter was a popular time for the start of the year (and April is still popular as the beginning of the 'financial' year), as was various other religious festivals and dates.
And one of those dates was January 1st.
It was no more or less popular/valid as any other, but over time it became increasingly more popular. By the year 1000AD it was the most popular (but still not the only one!).
So that is the answer to your question. I'm afraid there is no definite answer. There was no 'official' ruling - it came to be over a period of several centuries.

Answer 2

In the US and UK it was adopted along with the Gregorian calendar in 1751-1752.