In terms of strict mathematics, they were perceived as incorrect by contemporaries. Given there was never a year '0' the first decade was argued to have begun in 1 AD and ended at the end of 10 AD. Similarly the first century began at the beginning of 1 AD and ended at the end of 100 AD, which the second century beginning at the beginning of 101 AD. The general mathematical rule as perceived at the end of the nineteenth century was that the last year of a century gives that century is numerical definition, i.e., the seventeenth century ended in 1700, the eighteenth century ended in 1800, the nineteenth century in 1900, the twentieth century at the end of 2000, etc.
However Zero-ists argued that strict mathematical accuracy was irrelevant, and that the new century should be celebrated at the time when the first two digits of the year changed; in other words, 1899 should be regarded as the last year of the nineteenth century, and 1900 as the start of the next century.
This argument was however largely ignored in 1899 and 1900, with January 1st 1901 being generally regarded at the time as the correct date to celebrate the change in century, as noted in the diaries of many figures of the time, who recorded the century change in their January 1st 1901 entries. Similarly Queen Victoria, who died early in 1901, was described as dying at the beginning of the new century (not a year into it), while Queen Elizabeth the Queen Mother, who was born in 1900, was regarded by her family and by herself as having been born in the last year of the nineteenth century.
In contrast, though following those apparently strict mathematical terms would suggest the twentieth century ended on December 31st 2000, the same date on which the second millennium ended, with both the new millennium and new century starting on January 1st, 2001, the late twentieth century opted to follow the perspective of the late nineteenth century Zero-ists by marking the change of century and millennium as taking place on December 31st 1999/January 1st, 2000.