Confusion at noon and midnight
It is not always clear what times "12:00 a.m." and "12:00 p.m."
denote. From the Latin words meridies (midday), ante (before) and post
(after), the term ante meridiem (a.m.) means before midday and post
meridiem (p.m.) means after midday. Since strictly speaking "noon"
(midday) is neither before or after itself, the terms a.m. and p.m. do
not apply. However, since 12:01 p.m. is after noon, it is common to
extend this usage for 12:00 p.m. to denote noon. That leaves 12:00
a.m. to be used for midnight at the beginning of the day, continuing
to 12.01 a.m. that same day.
However, because practical confusion is still possible, some style
guides recommend replacing "12:00 p.m." with "12:00 noon" and "12:00
a.m." with "12:00 midnight".
The American Heritage Dictionary of the English Language (Fourth
Edition, 2000) has a usage note on this topic: "Strictly speaking, 12
a.m. denotes midnight, and 12 p.m. denotes noon, but there is
sufficient confusion over these uses to make it advisable to use 12
noon and 12 midnight where clarity is required."
Many U.S. style guides, and NIST's "Frequently asked questions (FAQ)"
web page, recommend that it is clearest if one refers to "noon" or
"12:00 noon" and "midnight" or "12:00 midnight" (rather than to "12:00
p.m." and "12:00 a.m."). Some other style guides suggest "12:00 n" for
noon and "12:00 m" for midnight.
The Canadian Press Stylebook (11th Edition, 1999, page 288) says,
"write noon or midnight, not 12 noon or 12 midnight." Phrases such as
"12 a.m." and "12 p.m." are not mentioned at all.
The use of "12:00 midnight" or "midnight" is still problematic because
it does not distinguish between the midnight at the start of a
particular day and the midnight at its end. To avoid confusion and
error, some U.S. style guides recommend either clarifying "midnight"
with other context clues, or not referring to midnight at all. For an
example of the latter method, "midnight" is replaced with "11:59 p.m."
for the end of a day or "12:01 a.m." for the start of the next day.
That has become common in the United States in legal contracts and for
airplane, bus, or train schedules, though some schedules use other
conventions.
My advice is to use "12:00 noon" instead of "12:00 p.m." and "12:00 midnight" instead of "12:00 a.m." Realize that readers may be confused which day 12:00 midnight belongs to, so make sure context makes it clear.
Best Answer
It's not particularly common for expressions of time.
It's similar to degrees-minutes-seconds: instead of decimal degrees (38.897212°,-77.036519°) you write (38° 53′ 49.9632″, -77° 2′ 11.4678″). Both are derived from a sexagesimal counting system such as that devised in Ancient Babylon: the single prime represents the first sexagesimal division and the second the next, and so on. 17th-century astronomers used a third division of 1/60th of a second.
The advantage of using minute and second symbols for time is that it obviously expresses a duration rather than a time.
From the time 01:00:00 to the time 02:34:56 is a duration of 1 hour, 34 minutes and 56 seconds (1h 34′ 56″)
Prime markers start single and are multiplied for susbsequent appearances, so minutes use a single prime ′ and seconds use a double-prime ″. They are pronounced minutes and seconds respectively in the case of durations like this.
Note that a prime ′ is not a straight-apostrophe ' or a printer's apostrophe ’, although straight-apostrophes are a reasonable approximation and printer's apostrophes do occur as well.