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=====The Julian Day Number and the Julian Date=====

The Julian Day Number only contains information on the calendar date and day of the week on which it occurs.  The JD Number does not contain any time-of-day information (see **Julian Date** below).

The Julian Day Number acts as a sequential serial number for the given date. Every date on our mathematical calendar system is assigned a unique integer serial number starting with 0, 1, 2, 3, ... up to the current date.  Since the mathematical origin of our calendar system is in the far distant past, JD Numbers are 7 digits long today.\\

<code>
 Calendar         Origin Date     Julian Date   JD Number
----------    ------------------  -----------   ---------
Julian        BC 4713-Jan-01-Mon     -0.5           0
Gregorian     BC 4714-Nov-24-Mon     -0.5           0
</code>

The JD Number for the first day of the year **AD 2023** is:
<code>for AD 2023-Jan-01-Sun       JDNum = 2459946</code>\\

In other words, the first day of the year AD 2023 was the **2459946**th day on the mathematical model of the Gregorian calendar system.  No other date on the entire Gregorian calendar has that unique serial number. \\

JD Number **2369916**  =  The date  **AD 1776-Jul-04-Thu**\\

Given the Julian Day Numbers of any two dates in history, the difference between them is the exact count of days between those dates, automatically accounting for any leap years in between.\\

Given any Julian Day Number on either the old Julian or modern Gregorian calendars, the day of the week (DoW) index can easily be found by:\\
<code>DoW = (JDNum + 1) mod 7</code>
**X mod 7**  simply means the integer remainder after dividing integer **X** by **7**.

Example for the DoW of  **AD 1776-Jul-04**  =  JD Number **2369916**\\
Where **DoW**:  **0**=Sun,  **1**=Mon,  **2**=Tue,  **3**=Wed,  **4**=Thu,  **5**=Fri  and  **6**=Sat

<code>
DoW = (2369916 + 1) mod 7
    = 2369917 mod 7
    = 4 = Thursday
</code>

-----
=====The Julian Date=====

Although they are related, the **Julian Date** and **Julian Day Number** are not exactly the same thing nor is the Julian Day Number simply the integer part of the Julian Date.\\
<code>
The relationship between the Julian Date (JD) and the Julian Day Number is:

JDNum = floor(JD + 0.5)
</code>

Unlike the **Julian Day Number**, which is always an integer, the **Julian Date** value can contain a decimal fraction part that gives us the time of day on the corresponding date.

For example:
<code>
Julian Date = 2459963.67717631 corresponds to date and time = AD 2023-Jan-19-Thu  at  04:15:08.033

JDNum = floor(JD + 0.5)
      = floor(2459963.67717631 + 0.5)
      = floor(2459964.17717631)
      = 2459964 (Corresponding to date  AD 2023-Jan-19-Thu)
</code>

To obtain the time of day elements (HH, mm, ss) from the Julian Date,  where HH = 00 to 24 hours:
<code>
hrs = 24*(JD - floor(JD + 0.5) + 0.5)
    = 24*(2459963.67717631 - floor(2459963.67717631 + 0.5) + 0.5);
    = 24*(0.177176310215145);
    = 4.25223144516349

HH  = floor(hrs)
    = floor(4.25223144516349)
    = 4

min = 60*(hrs - HH)
    = 60*(4.25223144516349 - 4)
    = 60*(0.2522314451635)
    = 15.1338867098099

mm  = floor(min)
    = floor(15.1338867098099)
    = 15

ss  = 60*(min - mm)
    = 60*(15.1338867098099 - 15)
    = 60*(0.1338867098099)
    = 8.03320258859401


So, the derived time of day elements are:
Time of Day = HH:mm:ss.sss = 04:15:08.033

So, the Julian Date value 2459963.67717631, equates to the
calendar date and time: AD 2023-Jan-19-Thu at 04:15:08.033
on the modern Gregorian calendar.

</code>