In writing, I have always believed I can use day numbers without the "th" at the end. I know they are pronounced, but have been taught I can write them either with or without this "th" at the end. Is this correct? And if so, is one of them considered to be better?
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You can write either.
There is no real difference in meaning between the two.
To refer to the date without the month, however, you need to add the "-th".
You also need the "-th" to invert the order:
Although I believe in British English it is acceptable to write:
(British native speakers, please feel free to correct me on this. I am sure I have heard it and read this usage, but can't say for certain how common it is.)
I'm not so sure the "th" is even pronounced any more. I certainly hear people just saying "June fifteen" when referring to a date.
Can I write June 15 instead of June 15th?
Yes, you can. Both forms are equally correct. Some swear by one or the other, but those are personal or institutional dogmas.
You cannot use "June 15" in British English, it has no meaning. "June 15th" is correct although if you include the year then the day must come before the month of course.
In American English I think anything goes.