The Appropriation Books held at the Registry of Shipping and Seamen in Cardiff are the central records of the allocation of official numbers for British ships.
The six volumes cover the period from the first official number allocation on 16 April 1855 to the 1950s. All the British ports of registry are covered, including the ports of registry in the former British colonies.
There are more details about using the viewer in a larger help file on the image display page and more detail about the Appropriation Books themselves through this link:
The official number should be a number between 1 and 199550.