Converting Julian Date to Gregorian

Date: 03/20/2001 at 06:13:06
From: max prenga
Subject: Converting Julian date to Gregorian date

Please help me to convert a Julian date into a Gregorian date. Assume 
that the base date is 01/01/0001 and that a Gregorian date is 
expressed as DD/MM/YYYY.

Cheers

Date: 03/20/2001 at 13:21:36
From: Doctor Rob
Subject: Re: Converting Julian date to Gregorian date

Thanks for writing to Ask Dr. Math, Max.

Your mention of a base date is confusing and apparently immaterial
to the calculation you ask for.

In the 21st and 20th centuries, just add 13 days to the Julian date
to get the Gregorian date.  In the 19th century, it was 12 days.
In the 18th century, it was 11 days.  In the 17th and 16th centuries,
it was 10 days.  In the Nth century, add [[3*N/4]] - 2 days.  Here
[[x]] means the greatest integer less than or equal to x, that is,
x rounded down to an integer.  The points where things hiccup are on
29 Feb 1900, 29 Feb 1800, 29 Feb 1700, 29 Feb 1500, and so on, which
exist in the Gregorian calendar but not the Julian.

Example:  Sept. 3, 1752 in the Julian calendar would be Sept. 14,
1752 in the Gregorian calendar, because 3 + [[3*18/4]] - 2 = 14.

For more on calendar calculations, see the following Web page from
our Frequently Asked Questions (FAQ):

  The Calendar and the Days of the Week  
  http://mathforum.org/dr.math/faq/faq.calendar.html   

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   

Date: 03/22/2001 at 03:48:44
From: Max Prenga
Subject: Re: Converting Julian date to Gregorian date

Thanks for your reply; however, there seems to be a wee bit of 
confusion, as I was not referring to the historical phenomena of 
Julian and Gregorian CALENDARS, rather I was referring to the 
abstract concept of a Julian and Gregorian DATE. A Julian date 
refers to the number of days from a base date as given by a Gregorian 
date.

For example, if a Gregorian date is 30/01/0001 and the base date is 
01/01/0001 then the Julian date is 30 days.

My query relates to working the other way. That is, given a Julian 
date of 10,457 days and given a base date of 01/01/0001 then what 
formula/logic can be used to obtain the Gregorian date of 08/08/0029 
(assuming no leap years .... which is an INCORRECT assumption).

Look forward to hearing from you soon.

Date: 03/22/2001 at 12:15:14
From: Doctor Rob
Subject: Re: Converting Julian date to Gregorian date

Thanks for writing back.

The confusion here is due to terminology.  What you describe is 
usually called the Julian day or Julian day number, not the Julian 
date.

To convert from Julian day number to Gregorian date, first divide by
365.2425, and round the quotient down to the nearest integer. Thus if
the Julian day number were 704876, the quotient would be 1929.8849...
which rounds down to 1929. The year is one larger than this, 1930.
Now the number of days in those 1929 Gregorian years is

   1929*365 + [[1929/4]] - [[1929/100]] + [[1929/400]]
     = 704085 + 482 - 19 + 4,
     = 704552.

The first term counts the number of days in those years not counting
leap years. The second term counts leap years every four years. The
third term subtracts for the nonleap years in centurial years, and
the fourth term adds for the leap years in quadricenturial years.
This follows the Gregorian rules for determining what years are leap
years.

That leaves 704876 - 704552 = 324 days in 1930 to be accounted for.
This amounts to about 324*12/365 = 10.65 months, or 10 months and some 
days. The ten months January through October in a nonleap year consist 
of 304 days. Then 324 - 304 = 20 days will be in November, and so the 
Gregorian date is November 20, 1930.

To sum this up in formulas, let the Julian day number be N. Then set

   y = [[N/365.2425]],
   YYYY = y + 1,
   z = 365*y + [[y/4]] - [[y/100]] + [[y/400]],
   d = N - z.

To find the month MM, start with d and subtract 31, 28 (or 29 in leap
years YYYY), 31, 30, 31, 30, 31, 31, 30, 31, 30, and 31, successively,
until you first get a result which is zero or negative. The month MM
is the number of subtractions you have done, and the last positive
subtraction result is the day DD.  The date is then DD/MM/YYYY.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/