Date Arithmetic Methods


Directions : ILUAF and ISUAF

Date calculations can be evaluated from the top down, adding the larger units first - ILUAF and / or from the bottom up, adding the smaller units first  - ISUAF, as follows :

ILUAF = If Larger Units Added First = Top Down.

• Add Years, then Months, then Weeks, then Days to the "from event" date to arrive at the "to event" result date.

The ILUAF, top down method is the default and therefore will not be stated against date calculations on these pages.

ISUAF = If Smaller Units Added First = Bottom Up.

• Add Days, then Weeks, then Months, then Years to the "from event" date to arrive at the "to event" result date.

If the date components contain only, Days and / or  Weeks, then the ILUAF and ISUAF methods produce identical results.

If the date components contain, Days and / or  Weeks together with Months, then the ILUAF and ISUAF methods can produce different results.


Where the ISUAF, bottom up method is used in date calculations, it will be stated with a link to this page for it's description.


INClusive, Normal and INTerval type calculations.

Normal conventional date arithmetic = Date2 - Date1

Normal example : "Friday is 4 days after Monday"

Friday ( Day 5 ) - Monday ( Day 1 ) = 5 - 1 = 4

Normal is the default arithmetic type and so therefore when used in date calculations, will not be stated.


INClusive date arithmetic = ( Date2 - Date1 ) + 1

INClusive example : "Friday was the 5th day of the project that began on Monday"

Friday ( Day 5 ) - Monday ( Day 1 ) + 1 = ( 5 - 1 ) + 1 = 5


Note : In calcualtions where  more than one date component is used, the date calculator cannot show INClusive type date calcs using one URL,  so two URLs are required.


Where the INClusive arithmetic type is used in date calculations, it will be stated with a link to this page for it's description.


INTerval date arithmetic = ( Date2 - Date1 ) - 1


INTerval type date calculations do not count either the first date - Date1 or the last date - Date2.


INTerval type date calcs count only the whole number of ( WMD ) days, weeks, months & years between events.


Where the INTerval arithmetic type is used in date calculations, it will be stated with a link to this page for it's description.


INTerval example : "11 days were lost between the changeover from the Julian to Gregorian Calendars"

The last day of the old Julian Calendar was September 2, 1752

The first day of the new Gregorian Calendar was September 14, 1752

The intervening 11 days, were "lost" : September 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13 in 1752 do not exist, so quite a few folk did not have a birthday that year!


The lost 11 days of September 1752, gave rise to the urban legend of The English calendar riots of 1752.

Date1 = September 2, 1752
Date2 = September 14, 1752

INTerval = ( Date2 - Date1 ) - 1

INTerval = ( 14 - 2 ) - 1 = 11

Another good example of an INTerval type date calculation is the 911 day INTerval between 9/11 and the Madrid Train Bombings on March 11, 2004

The Normal date calculation produces a conventional date difference of 912 days ( Date2 - Date1 )

The INClusive date calculation produces a difference of 913 days ( Date2 - Date1 ) + 1


- because it counts both the start date and the end date in the calculation.

The INTerval date calculation ( Date2 - Date1 ) - 1 = 911 days, cannot be shown accurately using the date calculator because it always includes day 1 in the calculation, whereas, INTerval type date calcs only counts the number of whole days between the two dates.

The best that can done when using the date calculator to show INTerval type date calcs is to present the sum using the calculator function rather than the date difference function.

The result date this time is 1 day short, because it starts counting at day 1, so we need to be aware of this issue when checking the results of INTerval type date calcs.



The utility of INTerval type date calculations.


Since the INTerval date arithmetic method ( Date2 - Date1 ) - 1 = whole days ( WMD ) between dates, has provoked the most discussion, so in addition to the exemplar of the Julian / Gregorian calendar changeover described above, further illustration of it's utility is required.


Example :


You buy some shares at 1 minute to midnight on Monday 1st January.

( Monday, Date1 = 1 )


You sell your shares at 1 minute into the day on Friday morning, 5th January.

( Friday, Date2 = 5 )


What is the most accurate method to show the number of days which you held your shares?


Normal artithmetic = Date2 - Date1 = 5 - 1 = 4 days


INClusive arithmetic = ( Date2 - Date1 ) + 1 = ( 5 - 1 ) + 1 = 5 days


INTerval arithmetic = ( Date2 - Date1 ) - 1 = ( 5 - 1 ) - 1 = 3 days


So, what is the most accurate method in this example?


It certainly is not the Normal 4 days and neither is it the INClusive 5 days.


Clearly, it is the INTerval type arithmetic which is most accurate, since you held onto your shares for 3 whole days and 2 minutes.


The INTerval method is especially suitable when measuring durations between two events such as births, since the time of day for both births are usually unknown.


The effect of using the INClusive and INTerval date calculation methods, is to produce results which can be plus or minus,  ± 1 day from Normal.


Clearly over a time span of say, a week or two, 1 day is a significant margin of error, so care needs to be excerised in the interpretation of these near term results.


Whereas, over a time span of several decades or hundreds of years, a ± 1 day margin of error is negligible.


In any case, the expression of these various methods resolves to a semantic difference only.


Another aspect to consider is that there are always several root events to main events.  Therefore, it is simpy not possible for the design of every event relationship to yield results which all use Normal arithmetic.   Some relationships will be Normal, others will be INClusive and others will be INTervals.  That is simply a fact of life in date arithmetic, event scheduling and script design.




That's about it for the main set of numbers used, the date calculation direction, top down = ILUAF and bottom up = ISUAF and the INClusive, Normal and INTerval methods.


On these pages, the default direction and method are : Top Down - ILUAF - If Larger Units Added First using Normal arithmetic, so where Normal and / or ILUAF are used in date calculations, these defaults will not be shown.


Where date calculations use any of the other combinations, INClusive or INTerval or ISUAF, then the direction and method will be shown, with a link to this page for explanations.


The ILUAF and ISUAF methods can produce different results only when date calculation components include Days and / or Weeks along with Months.

The surprising outcome from using the combinations of the ILUAF, ISUAF, INClusive, Normal and INTerval methods, is that up to six different result dates can be achieved when adding the same date components to a single from event date.  

This was a welcome revelation because it explains why many of the suspect, nefarious chaos events we witness, occur in clusters on successive days, because each event on each day, all share the same root event and by adding the same date components to that root event date using the various method combinations, up to 6 different dates, often - but not always on consecutive days, can be produced.


Next : Scenario