Q:

Are times after noon expressed in military time?

How are times after noon expressed when speaking? 
I assume that if it's obvious from context that it's after noon, then people will just say ５時.
But what if it's not obvious from context? My textbook (Japanese for Busy People I) suggests 午後５時, but I'm worried the textbook may be teaching how to say what English-speakers think about time in Japanese, rather than what a native Japanese speaker would say. As far as I have observed in Japan, times are often written down in military time, for example 23:59 rather than 11:59 pm. Do people write it down one way and say it in another, or do they sometimes use military time when speaking?
If military time isn't used when speaking, how are times after midnight expressed? Would ２５：００ be described using 午前 (午前１時), just like 9:00 would?

A:

Although the 12-hour clock system (12時間制) is much more common in general, the 24-hour system (24時間制) is used in some contexts.  Probably the most notable is train timetables.
25:00 is 午前1時 of the next day.  But in some cases, the speaker wants to treat it as part of the day before.  I do not think that there is a standard way to describe it, but I have seen both 25:00 and 深夜1時 for this purpose.

A:

From my personal experience, I find that people say the hour based on the 12 hour clock without the 午後 or 午前, and if the listener is not sure they will ask 夜？朝？ If they want to be specific they will say 朝７時 or 夜7時. 
I also find that the 24 hour clock is only used in writing and not in speaking. So, they do write it down one way and say it another. Even writing differs based on who is doing it. I think private individuals would write 午後7時, but organizations and companies would write 19:00.
To answer your last question, I think people would say 朝１時. I often hear this when people are asking what time an event will go to. 
I hope that answers your questions. 
I am a big fan of the 24 hour clock, but because both are used here, I make mistakes related to time. For example when I see 15:00, I will accidentally say 5:00 or some other kind of mistake.
(This is my first post, so forgive me if I am answering in an incorrect way.)

