If a child was born today, which is July 18th and the third Monday of July, would he or she become one month old on August 18th or the third Monday in August (August 22)?
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This is a trick question. Each month does not consist of a whole (integer) number of weeks, except February, or equal number of days.
Just choose what you like best and stick with that. I would choose the n'th of the month and say my birthday is on the 18th, because that's the easiest for everybody.
The only exception is if your birthday is February 29th. Then you need to know that the 29th is not actually the leap day, the 25th is! So in non-leap years your birthday is Feb 28th, not March 1st. If your birthday is on the actual leap day (25th) then I guess the 26th will do as birthday in non-leap years.
Counting weeks is only relevant for infants, when you see the pediatrician. He'll know anyway, so don't bother about that. Once the child is 8 weeks old, switch to counting months. Once he's 24 months old, switch to counting (half) years.
I calculate month birthdays like I calculate year birthdays: By date.
Ie, the 29th of April, our daughter was one year and one month old. As she is born the 29th, this means she wouldn't have an actual birthday most Februaries, but so what? It's not like you have a party for monthly birthdays anyway. It's just a matter of how many months you say she is, and for that you usually switch a couple of days before anyway. Nobody expects you to say that your kid is "One year, 5 months and 28 days old". You say "18 months".
But if you want to be exact: http://howoldismykid.com/
Being an engineer who works on things like recurring payments, I tend to over-engineer the answer for my wife's liking. A month it turns out is really really close to 30.5 days. So I tend to use that. Which means each month is either 30 or 31 days... alternating. (note that this is much easier to use when using a date calculator that will let you just say "2007-08-07 + (30.5 * 18)".)
The only real world numbering system is days from start of year, where that point is arbitrarily chosen as 10 days after winter solstice (typically in the western world)
Weeks are an affectation defined to work with the way humans traditionally work and rest. Months are also just traditional, and have changed over time - see the Julian and Gregorian calendars for example.
However the generally accepted practice is that you use the day of the month - eg born on 18 June, 1month old on 18 July etc.