In Joseph Smith history he states he was in his fifteenth year (JSH 1:7)… Though he was 14 when he had the first vision. Why is this the case I’m just a little confused by it.
This seeming discrepancy is caused by the difference between cardinal and ordinal numbers. Cardinal numbers count a quantity or amount. I have three children. I’ve seen seventeen plays. I ate five tacos. Ordinal numbers put those objects in a position, or orders them. This is my third child. I’m watching my eighteenth play. I felt sick eating my sixth taco.
Sometimes the numbers match up: if I introduce you to my third child, you can be certain I have at least three children. Sometimes they do not match up — often when dealing with fractions. When I step out during intermission, I call my son and tell him I watching my eighteenth play, but I’ve only seen seventeen of them. That’s because I’m rounding down. I’ve actually seen seventeen and a half, but no one wants to sound pretentious. When the show is complete, I will have seen eighteen plays and the numbers will again match. Similarly, I ate 5 whole tacos, and some fraction of a sixth. It is only when the last one is completed (it wasn’t) that the numbers match
When we speak of ages, we most often use cardinal numbers, so it sounds foreign to hear ordinals used to describe it. Historians get good at navigating the two and seamlessly transition from the 1950’s to the mid-20th century (same period). New parents often describe their child’s age in terms of the largest whole quantity permissible. The baby is 11 months, or the infant is 3 weeks, or perhaps 6 days, or the infant may be only 8 hours old. This is subtle, but it’s a way of dealing with the fractional age. The baby is 0 (whole) years old, but in her first year. If you think of the child’s life as a video recording where 1 memory card can record 1 year of life, that first year you have no (zero) memory cards to show for it but you are filling your first memory card. Continuing on to when the child is 1 year 11 months, you have 1 memory card (the child is a 1 year old) but you are recording the second memory card (the child is in her second year). And so on, until we come to 14 year old Joseph Smith, who has lived 14 (whole) years of life but is in his 15th year.
A practical takeaway from all this (besides understanding history better), is that when your friends arrive at milestone years of 10, 20, 30, and 40, you can ask them what it’s like to be in the second, third, fourth and fifth decades (respectively) of life. It leads to an early onset of the midlife crisis but seems to lose effect after 40.