The traditional way of expressing the age of a person in Japan and other east Asian countries, "year-count" (kazoe-doshi 数え年), is different from that used in the west, called "full-year" (man-nenrei 満年齢). In the latter system, "She was 42-years old" means that at least 42 but less than 43 full years have passed since she was born. However, in the year-counting system, it could be paraphrased "She is in her 42nd (calendar) year since birth." When a person is born, he is in his first year of life. At the next new year, and every new year after that, one year is added to his age, regardless of when in the year he was born.
Hosokawa Tadaoki was born the 13th of the 11th month 1563. On that day he was 1 by year-count, and 0 by full-year
(1563/11/13; C:1, F:0). Other ages for example would be
(1564/1/1; C:2, F:0),
(1564/11/27; C:2, F:1),
(1565/1/1; C:3, F:1),
(1577/10/2; C:15, F:13),
(1577/12/28; C:15, F:14),
(1645/12/2; C:83, F:82).
For year-count, one calculates the age (A) in the current year (Y) from the year of birth (B) by A = Y-B+1 and the year of birth from the current age by B=Y-A+1. To determine the full-year age from the year-count age, subtract 1 or 2 from the year-count age, depending on whether the person's birthday that year has passed.
The year-count system has certain conveniences for the historian. If we read "X died in 1645 at the age of 83," we can give his dates as "1563-1645." However, for a Western age, unless we know the exact day of birth and the exact day of death, the birth year can only be "1561 or 1562."
The full-year (Western) system started to be used officially in Japan in 1902, but it was not used in general until after 1950. However, it has now virtually completely replaced the year-count system, though the fact that a full-year system is used is still sometimes stressed, as in a JR advertisement that "membership is limited to people who are sixty and over by the full count" (man roku-jû ijô no kata). Historians and historical fiction writers covering earlier periods usually use the year-count system, though.
The existence of two systems can create problems. For instance, when we read in English, "Hosokawa Tadaoki fought his first battle at the age of fifteen," does it mean the writer, writing in English, used the full-year system, or that he directly used a Japanese source, perhaps not realizing the existence of the year-count system? There is no way of telling without additional information. (In this particular case, since there is a letter written 1577/10/2 referring to his fighting, we can know that Tadaoki was 15 by the year-count but 13 by the full-year system.) Another problem is that even with Japanese authors, some intend to use the year-count system, but do not know how to calculate in it! In conclusion, if it is important to you to know the age of a person, do the calculations yourself, if possible.
On the other hand, when writing in English , you should make clear somehow which system you are using, as "15 (13 or 14 by the Western age count)," or put the date instead of the age.
Source:
Kôjien Dictionary personal observations