calc: Date Forms
5.9 Date Forms
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A “date form” represents a date and possibly an associated time. Simple
date arithmetic is supported: Adding a number to a date produces a new
date shifted by that many days; adding an HMS form to a date shifts it
by that many hours. Subtracting two date forms computes the number of
days between them (represented as a simple number). Many other
operations, such as multiplying two date forms, are nonsensical and are
not allowed by Calc.
Date forms are entered and displayed enclosed in ‘< >’ brackets. The
default format is, e.g., ‘<Wed Jan 9, 1991>’ for dates, or ‘<3:32:20pm
Wed Jan 9, 1991>’ for dates with times. Input is flexible; date forms
can be entered in any of the usual notations for dates and times. 
Date Formats.
Date forms are stored internally as numbers, specifically the number
of days since midnight on the morning of December 31 of the year 1 BC.
If the internal number is an integer, the form represents a date only;
if the internal number is a fraction or float, the form represents a
date and time. For example, ‘<6:00am Thu Jan 10, 1991>’ is represented
by the number 726842.25. The standard precision of 12 decimal digits is
enough to ensure that a (reasonable) date and time can be stored without
roundoff error.
If the current precision is greater than 12, date forms will keep
additional digits in the seconds position. For example, if the
precision is 15, the seconds will keep three digits after the decimal
point. Decreasing the precision below 12 may cause the time part of a
date form to become inaccurate. This can also happen if astronomically
high years are used, though this will not be an issue in everyday (or
even everymillennium) use. Note that date forms without times are
stored as exact integers, so roundoff is never an issue for them.
You can use the ‘v p’ (‘calc-pack’) and ‘v u’ (‘calc-unpack’)
commands to get at the numerical representation of a date form.
Packing and Unpacking.
Date forms can go arbitrarily far into the future or past. Negative
year numbers represent years BC. There is no “year 0”; the day before
‘<Mon Jan 1, +1>’ is ‘<Sun Dec 31, -1>’. These are days 1 and 0
respectively in Calc’s internal numbering scheme. The Gregorian
calendar is used for all dates, including dates before the Gregorian
calendar was invented (although that can be configured; see below).
Thus Calc’s use of the day number -10000 to represent August 15, 28 BC
should be taken with a grain of salt.
Some historical background: The Julian calendar was created by Julius
Caesar in the year 46 BC as an attempt to fix the confusion caused by
the irregular Roman calendar that was used before that time. The Julian
calendar introduced an extra day in all years divisible by four. After
some initial confusion, the calendar was adopted around the year we call
8 AD. Some centuries later it became apparent that the Julian year of
365.25 days was itself not quite right. In 1582 Pope Gregory XIII
introduced the Gregorian calendar, which added the new rule that years
divisible by 100, but not by 400, were not to be considered leap years
despite being divisible by four. Many countries delayed adoption of the
Gregorian calendar because of religious differences. For example, Great
Britain and the British colonies switched to the Gregorian calendar in
September 1752, when the Julian calendar was eleven days behind the
Gregorian calendar. That year in Britain, the day after September 2 was
September 14. To take another example, Russia did not adopt the
Gregorian calendar until 1918, and that year in Russia the day after
January 31 was February 14. Calc’s reckoning therefore matches English
practice starting in 1752 and Russian practice starting in 1918, but
disagrees with earlier dates in both countries.
When the Julian calendar was introduced, it had January 1 as the
first day of the year. By the Middle Ages, many European countries had
changed the beginning of a new year to a different date, often to a
religious festival. Almost all countries reverted to using January 1 as
the beginning of the year by the time they adopted the Gregorian
calendar.
Some calendars attempt to mimic the historical situation by using the
Gregorian calendar for recent dates and the Julian calendar for older
dates. The ‘cal’ program in most Unix implementations does this, for
example. While January 1 wasn’t always the beginning of a calendar
year, these hybrid calendars still use January 1 as the beginning of the
year even for older dates. The customizable variable
‘calc-gregorian-switch’ (Customizing Calc) can be set to have
Calc’s date forms switch from the Julian to Gregorian calendar at any
specified date.
Today’s timekeepers introduce an occasional “leap second”. These do
not occur regularly and Calc does not take these minor effects into
account. (If it did, it would have to report a non-integer number of
days between, say, ‘<12:00am Mon Jan 1, 1900>’ and ‘<12:00am Sat Jan 1,
2000>’.)
Another day counting system in common use is, confusingly, also
called “Julian.” Julian days go from noon to noon. The Julian day
number is the numbers of days since 12:00 noon (GMT) on November 24,
4714 BC in the Gregorian calendar (i.e., January 1, 4713 BC in the
Julian calendar). In Calc’s scheme (in GMT) the Julian day origin is
-1721422.5, because Calc starts at midnight instead of noon. Thus to
convert a Calc date code obtained by unpacking a date form into a Julian
day number, simply add 1721422.5 after compensating for the time zone
difference. The built-in ‘t J’ command performs this conversion for
you.
The Julian day number is based on the Julian cycle, which was
invented in 1583 by Joseph Justus Scaliger. Scaliger named it the
Julian cycle since it involves the Julian calendar, but some have
suggested that Scaliger named it in honor of his father, Julius Caesar
Scaliger. The Julian cycle is based on three other cycles: the
indiction cycle, the Metonic cycle, and the solar cycle. The indiction
cycle is a 15 year cycle originally used by the Romans for tax purposes
but later used to date medieval documents. The Metonic cycle is a 19
year cycle; 19 years is close to being a common multiple of a solar year
and a lunar month, and so every 19 years the phases of the moon will
occur on the same days of the year. The solar cycle is a 28 year cycle;
the Julian calendar repeats itself every 28 years. The smallest time
period which contains multiples of all three cycles is the least common
multiple of 15 years, 19 years and 28 years, which (since they’re
pairwise relatively prime) is 15*19*28 = 7980 years. This is the length
of a Julian cycle. Working backwards, the previous year in which all
three cycles began was 4713 BC, and so Scaliger chose that year as the
beginning of a Julian cycle. Since at the time there were no historical
records from before 4713 BC, using this year as a starting point had the
advantage of avoiding negative year numbers. In 1849, the astronomer
John Herschel (son of William Herschel) suggested using the number of
days since the beginning of the Julian cycle as an astronomical dating
system; this idea was taken up by other astronomers. (At the time, noon
was the start of the astronomical day. Herschel originally suggested
counting the days since Jan 1, 4713 BC at noon Alexandria time; this was
later amended to noon GMT.) Julian day numbering is largely used in
astronomy.
The Unix operating system measures time as an integer number of
seconds since midnight, Jan 1, 1970. To convert a Calc date value into
a Unix time stamp, first subtract 719163 (the code for ‘<Jan 1, 1970>’),
then multiply by 86400 (the number of seconds in a day) and press ‘R’ to
round to the nearest integer. If you have a date form, you can simply
subtract the day ‘<Jan 1, 1970>’ instead of unpacking and subtracting
719163. Likewise, divide by 86400 and add ‘<Jan 1, 1970>’ to convert
from Unix time to a Calc date form. (Note that Unix normally maintains
the time in the GMT time zone; you may need to subtract five hours to
get New York time, or eight hours for California time. The same is
usually true of Julian day counts.) The built-in ‘t U’ command performs
these conversions.
Info Catalog
calc: HMS Forms
calc: Data Types
calc: Modulo Forms