The number on your yearly "birthday" is not the number of birthdays you had (you only have one birthday) but the number of years you survived since your birth.
So the day of your fifth birthday is the first day of your sixth year alive.
Jewish law sometimes counts from 1 instead of 0 for birthdays, so a newborn is not 0 years old, they are in their 1st year. (But it's written as "when they are 1", so you have to know which counting method is being used.)
Chinese traditional age counting does the same. My mother had to constantly explain to our relatives when I tell them my age -- I use Western conventions.
