In the show the doctor sort of dies and regenerates as a new iteration of the same character. Has all the memories, and retains certain key character traits. The same person, but also a different person. Each has a different personality, with different aspects coming to the fore. So, in some Doctors, the element of the trickster is more apparent, some are more whimsical, others more abrupt and terse, some more romantic in outlook, others very pragmatic.
Very occasionally, the Doctor will meet one of his earlier incarnations (often around anniversary specials - mostly not the best episodes in a lot of respects, but fun for the fans :p
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