10th birthday

Can I say,

(A) He celebrated his brother's 10th birthday.

Hi,

This is an interesting point. As a child, I was taught to use a period after an ordinal number.

eg Today is the 13th. of October. eg (on a letter) 13th. May, 2005 or May 13th., 2005

Unlike students today, we never asked 'Why?'  I assume it was because it is a form of abbreviation, eg 13th. May instead of thirteenth May

I looked on-line at a few style guides, and I must admit they said 'Don't use a period here'. However, that suggests to me that some people do use a period, and perhaps even correctly.

We live in times of simplification, where we tend to eliminate a lot of punctuation. eg I was taught to write eg as e.g. but I often don't. People often don't even write 13th but simply 13, so it does seem unlikely they'd worry about writing 13th.

I'm curious as to whether anyone else was taught the same as I was. Perhaps others may comment.

Best wishes, Clive