N-day period

An “n-day period” is a period of n consecutive days, where n is some positive integer. For example, when n = 12, then it is a 12-day period. Appropriate calculation methods for finding the number of days when given the starting and ending date differ according to the case under consideration even if it is a　familiar problem. In some cases, it may also be necessary to take into account such things as “business days”, “holidays,” etc. In cases where provisions stipulated by laws or judicial precedents need to be considered, multiple calculation methods may exist, and this situation may lead to lawsuits in some cases.

Calculating the number of days in arithmetic

The calculation method of "n-day period" is strongly related to the definition of “n days later” and “n days before”. So, we started this section from the introduction of the definition of “n days later” and “n days before”.

The definition of “n days later” and “n days before”

Fig.1.n days before and n days later.

There are some different and unequal definitions for “n days later” and “n days before”. However, since calculation conventions for them are not the subject of this article, the following definitions are adopted for the purpose of this article.

Def.1 The definition of “n days later” and “n days before”

  • [n days later]:”n days later” is inductively defined so that the “tomorrow of the base date” will be the “one day later(one day after)”, inductively adding one day to it each passing day.( It corresponds to the red rectangles in the Fig.1.)
  • [n days before]:”n days before” is inductively defined so that “yesterday of the base date” will be “one day before,” inductively subtracting one day as you go back one day. (It corresponds to the black rectangles in the Fig.1.)
  • [Handling of negative numbers]:Definition is made so that “in the case of n > 0,” “−n day(s) later” will be “n day(s) before” and “−n day(s) before” will be “n days later.”

Examples:

  • “0 days later” and　“0 days before”:
    • Both 0 days later from Today and 0 days before from Today are Today.
  • “n days later”, If the period does not extend to the next month:
    • If your starting date is August 10, 2017, so that means that 3 days later would be August 13, 2017.
    • The above value is the same as the value obtained by calculating "=DATE(2017,8,10)+3" using Excel.
    • The date value is the same as the result of addition of day X and n-days (for example, day 10+3 days=day day 13).
  • “n days before”, If the does not back to the previous month:
    • If your ending date is August 7, 2017, so that means that 3 days before would be August 4, 2017.
    • The above value is the same as the value obtained by calculating "=DATE(2017,8,13)-3" using Excel.
    • The date value is the same as the result of addition of day-X and n-days (for example, day13 − 3 days = day 10).

Day count conventions in arithmetic

Fig.2.Number of days between specific two days. As an example, "Number of days between today and two days after tomorrow" are illustrated.

There are three primitive different and unequal methods of calculating the number of days between two specific dates as shown below. Even cases that require consideration of provisions by laws and judicial precedents will come down to the combination of the following primitive calculation methods and how to divide cases after all (though it may be concluded that multiple interpretations are possible.)

Def.2 Three primitive definitions of day counting.

  • [Calculation Method 1] The method that counts both the starting day and the ending day.
    • This is the calculation method by which the days between August 10, 2017 and August 13, 2017 are counted as four days.
    • It corresponds to the black triangles in the Fig.2.
    • If the starting day is the same as the ending day, the period in between is one day.
    • If the days are calculated by this method, the number of days matches “the number of cells needed when the dates are filled in each cell one by one from the starting day to the ending day.”
    • This calculation method is often used to determine the total number of days of a travel or an event.
    • If this calculation method is applied,
the number of days between ‘the starting day of a week’ and ‘the ending day of the week’ equals the number of days of the week,
the number of days between ‘the starting day of a month’ and ‘the ending day of the month’ equals the number of days of the month and,
the number of days between ‘the starting day of an year’ and ‘the ending day of the year’ equals the number of days of the year.
  • [Calculation Method 2] The calculation method that does not count either the starting day or the ending day.
    • This is the calculation method by which the days between August 10, 2017 and August 13, 2017 are counted as three days.
    • The above value is the same as the value obtained by calculating "=DATE(2017,8,13)-DATE(2017,8,10)" using Excel.
    • It is a calculation method exemplified by the number of red yellows or red triangles in the Fig.2.
    • This method is consistent with the calculation of above-mentioned “n days later” and “n days before.
    • If the starting day is the same as the ending day, the period between the starting day and the ending day is zero day.
    • This method is used when time required for the process is important.
  • [Calculation Method 3] The calculation method that excludes both the starting day and the ending day from the count.
    • This is the calculation method by which the days between August 10, 2017 and August 13, 2017 are counted as two days.
    • It corresponds to the pink triangles in the Fig.2.
    • It is a method that only counts the days in between (the days after the starting day and before the ending day).
    • This is a calculation method used to state "n days in between"(In Japanese, "中N日"(Naka-N-Nichi). This description is common in Japanese contracts or, designation of entry day in Japanese baseball player.

So, the relation of above-mentioned primitive methods is represented by following equation.

Eq1. The relationship between [Calculation Method 1] ,[Calculation Method 2],and [Calculation Method 3]

  • Suppose that, Numbers of days calculated by [Calculation Method 2] = n, then
    • Numbers of days calculated by [Calculation Method 1] = n + 1
    • Numbers of days calculated by [Calculation Method 3] = n − 1 (if n > 1)

Extending calendar

Extending calendar is a virtual calendar using "extending the days of the month", that is dates after the number of the days of the next month to be counted is added to the number of the days of the months to be counted. It will be useful to calculate if the period extends long teams. The definitions of the "extending the days" are as follows. Extending calendar is familiarly known as "お化け日暦(Obake Hireki)" in Japan.

Def.3 Extending calendar In the to define "extending the days of the month", we define following sequences M(n) and D(n) for specific date A. Here, the "n" shall be non-negative integer.

  • M(0) := the reference month
  • M(n) := M(0) + n month
  • D(n) := the date using if the date is included in month M(n).
    • In the positive case; D(n) = D(n − 1)-(number of days in M(n − 1))　　(for all n > 0, non-negative integer)
    • In the negative case; D(−n) = D(−n + 1)-(number of days in M(−n))　(for all n > 0, positive integer)

Following equations are satisfied.

Eq. 2
For the sequences M(n) and D(n) of Def. 3, using "month/day" notation, followings are satisfied. Here, n > 0 be positive integer.

:*In the positive case;

M(0) / D(0) th

= M(1) / (D(0) − (number of days in M(0))) th

= M(2) / D(1) − (number of days in M(1)) th

| ...

= M(n) / D(n − 1) − (number of days in M(n − 1)) th

:*In the negative case;

M(0) / D(0) th

= M(−1) / (D(0) − (number of days in M(−1))) th

= M(−2) / D(−1) − (number of days in M(−2)) th

| ...

=  M(−n) / D(−n + 1) − (number of days in M(n)) th

Examples:

  • (1).What is "August 32, 2017" in normal calendar?
In this case, M(0)=8 ,D(0) = 32, and
{|

|- | 8/32 th || = 9/ 32  − (31; number of days in August) th |- | ||= 9 /1 th. |}

"August 32, 2017" in extended calendar is "September 1, 2017" in normal calendar.
  • (2).What is the "September −1 th, 2017" in the normal calendar"
In this case, M(0)=9 ,D(0) = −1, and "9/−1 th" = "8/ −1 (31; number of days in August) th" = "8 /29 th", so,
"September −1 th, 2017" in extended calendar is "August 29, 2017" in normal calendar.
  • (3).What is the "August 85, 2017" in normal calendar？
In this case, M(0) = 8, D(0) = 85, and
{|

|- | 8/85 th || = 9/ 85th − (31; number of days in August)th |- | ||= 9 /54 th |- | ||= 10 /54 th − (30; number of days in September) |- | ||= 10 /24 th . |}

Therefore, "August 85, 2017" in extended calendar is　"October 24, 2017" in normal calendar.
  • (4). What is "October −51, 2017" in normal calendar?
In this case, M(0) = 10, D(0) = −51, and
{|

|- | 10/−51 th || = 9 / 9/ −51 +(30; number of days in September) th th |- | ||=9 /−21 th |- | ||= 8 /−21 + (31; number of days in August) th" |- | ||= 8 / 24 th. |}

Therfore, "October −51, 2017" in extended calendar is　"August 10, 2017" in normal calendar.

Advanced examples

In this section, we exemplify describe the calculation method, when the start date and the end date belong to different months.

Calculations of “n days later” and “n days before;

  • (1).If your starting date is August 30, 2017, so that means that 3 days later would be September 2, 2017.
    • The above value is the same as the value obtained by calculating "=DATE(2017,8,30)+3" using Excel.
    • By using "extending the days of the month",
Answer = "August 30 + 3, 2017" = "September 33 − (total number of days in September), 2017" = "September 2, 2017"
  • (2).If your starting date is September 2, 2017, so that means that 3 days before would be August 30, 2017.
    • The above value is the same as the value obtained by calculating "=DATE(20179,2)-3" using Excel.
    • By using "extending the days of the month",
Answer = "September 2–3, 2017" = "August −1 + (total number of days in August), 2017" = "August 31, 2017"
  • (3).If your starting date is August 10, 2017, so that means that 75 days later would be October 24, 2017.
    • The above value is the same as the value obtained by calculating "=DATE(2017,8,10)+75" using Excel.
    • By using "extending the days of the month", 75 days later would be calculated by following steps.

::*Step 1: What "75 days later" is the date in August?

→Answer = "August 10+75, 2017"　 = "August 85, 2017".
  • (4)..If your ending date is October 24, 2017, so that means that 75 days before would be August 10, 2017.
    • The above value is the same as the value obtained by calculating "=DATE(2017,10,24)-75" using Excel.
    • By using “extending the days of the month", 75 days before would be calculated by following steps.

::*Step 1: What the "75 days before " is the date in October?

→Answer = "October 24 − 75, 2017" = "October −51, 2017".

Calculations of numbers of dates between sstarting day and a specific date ending day. ;

  • (1).If your starting date is August 30, 2017 and, ending date is September 2, 2017, so, the total number of days between above-mentioned date are 3 days in the [Calculation Method 2].
    • The above value is the same as the value obtained by calculating "=DATE(2017,9,2)- DATE(2017,8,30)" using Excel.
    • By using "extending the days of the month"(positive case),
"September 2, 2017" =" August 33, 2017" therefore the answer is 33-30 =3 days.
  • (2).If your starting date is August 10, 2017 and, ending date is October 24, 2017, so, the total number of days between above-mentioned date are 75 days in the [Calculation Method 2].
    • The above value is the same as the value obtained by calculating "=DATE(2017,10,24)- DATE(2017,8,10)" using Excel.
    • By using "extending the days of the month"(positive case),
"October 24, 2017" = "August 85, 2017" therefore the answer is 85 − 10 = 75 days.

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