perpetual calendar
DESCRIPTION
A chart or mechanical device that indicates the day of the week, in the Gregoriam system, corresponding to any given date.TRANSCRIPT
Perpetual Calendar
Perpetual Calendar
Jaime J. Gutiérrez
Coloquios matemáticosDepartamento de Matemática
2 de mayo de 2010
Perpetual Calendar
Perpetual calendar?
Perpetual calendarA chart or mechanical device that indicates the day of the week,in the Gregoriam system, corresponding to any given date.
Perpetual Calendar
The Gregorian Calendar
The Gregorian CalendarThe Gregoriano calendar began in 1852.
The very first was 1600.
It was provided that years divisible by 100 would be leapyears only if they were divisible by 400 as well
For example, the years 1600 and 2000 were leap years,but 1700, 1800 and 1900 were not.
Perpetual Calendar
The Gregorian Calendar
The Gregorian CalendarThe Gregoriano calendar began in 1852.
The very first was 1600.
It was provided that years divisible by 100 would be leapyears only if they were divisible by 400 as well
For example, the years 1600 and 2000 were leap years,but 1700, 1800 and 1900 were not.
Perpetual Calendar
The Gregorian Calendar
The Gregorian CalendarThe Gregoriano calendar began in 1852.
The very first was 1600.
It was provided that years divisible by 100 would be leapyears only if they were divisible by 400 as well
For example, the years 1600 and 2000 were leap years,but 1700, 1800 and 1900 were not.
Perpetual Calendar
The Gregorian Calendar
The Gregorian CalendarThe Gregoriano calendar began in 1852.
The very first was 1600.
It was provided that years divisible by 100 would be leapyears only if they were divisible by 400 as well
For example, the years 1600 and 2000 were leap years,but 1700, 1800 and 1900 were not.
Perpetual Calendar
Numerical correspondence
Numerical correspondence or the days of the week.For the days of the week:
Sunday = 0Monday = 1Tuesday = 2Wednesday = 3Thursday = 4Friday = 5Saturday = 6
Perpetual Calendar
Numerical correspondence
Numerical correspondence for the months.
Januar = 11 July = 5February = 12 August = 6March = 1 September = 7April = 2 October = 8May = 3 November = 9June = 4 December = 10
Perpetual Calendar
Searching the formula
The first stepAssume that 1st March 1600 corresponds the number a0.Since 365 ≡ 1(mod 7), 1st March of the year 1660 + t
corresponds the number at, given by:
at ≡
(
a0 + t +
⌊
t
4
⌋
−
⌊
t
100
⌋
+
⌊
t
400
⌋)
mod 7
1st March 2001 was a Thursday, so for t = 401, we haveat = a401 = 4. From the congruence
4 ≡ a0 + 401 + 100 − 4 + 1(mod 7),
we obtain a0 = 3. 1st March 1600 was a Wednesday.
Perpetual Calendar
Searching the formula
The first approximationTill now, we have that the corresponding number to 1st 1600 + t
can be calculated by using:
at ≡
(
3 + t +
⌊
t
4
⌋
−
⌊
t
100
⌋
+
⌊
t
400
⌋)
mod 7
Write out the year in the form 100c + d, con d < 100, we havet = 100(c − 16) + d and the formula can be rewritten as:
at ≡
(
3 + 5c + d +
⌊
d
4
⌋
+⌊ c
4
⌋
)
mod 7
Perpetual Calendar
Searching the formula
On step more.We can calculate the corresponding numbers for the first daysof every month of the year t using the following relations:
(1 April)t ≡ (1 March)t + 3(mod 7)(1 May)t ≡ (1 April)t + 2 ≡ (1 March)t + 5(mod 7)(1 June)t ≡ (1 May)t + 3 ≡ (1 March)t + 1(mod 7)(1 July)t ≡ (1 June)t + 2 ≡ (1 March)t + 3(mod 7)(1 August
t≡ (1 July)t + 3 ≡ (1 March)t + 6(mod 7)
(1 September)t ≡ (1 August)t+ 3 ≡ (1 March)t + 2(mod 7)
(1 October)t ≡ (1 September)t + 2 ≡ (1 March)t + 4(mod 7)(1 November)t ≡ (1 October)t + 3 ≡ (1 March)t + 0(mod 7)(1 Decemmber)t ≡ (1 November)t + 2 ≡ (1 March)t + 2(mod 7)(1 Januar)t ≡ (1 December)t + 3 ≡ (1 March)t + 5(mod 7)(1 February)t ≡ (1 Januar)t + 3 ≡ (1 March)t + 1(mod 7)
Perpetual Calendar
Searching the formula
We come already almost.We can calculate the corresponding day of the way to the datenth day of the month m of the year 100c + d appliying
(n, m)100c+d ≡
(
n + rm + 5c + d +
⌊
d
4
⌋
+⌊ c
4
⌋
)
mod 7
For rm we have the correspondence:
m 1 2 3 4 5 6 7 8 9 10 11 12rm 2 5 0 3 5 1 4 6 2 4 0 3
rm =
⌊
13m − 1
5
⌋
Perpetual Calendar
The formula
The formula
(n, m)100c+d ≡
(
n + 5c + d +
⌊
13m − 1
5
⌋
+
⌊
d
4
⌋
+⌊ c
4
⌋
)
mod 7
As examples, we apply the formula to important days ofPanama’s history. ¿Sabe usted que día tuvo lugar laseparación de Panamá de Colombia?Aquí n = 3, m = 9, d = 3, n = 19 y
3 + 5 × 19 + 3 +
⌊
13 × 9 − 1
5
⌋
+
⌊
3
4
⌋
+
⌊
19
4
⌋
≡ 2 mod 7
¡El 3 de noviembre de 1903 fue martes!
Perpetual Calendar
The formula
The formula
(n, m)100c+d ≡
(
n + 5c + d +
⌊
13m − 1
5
⌋
+
⌊
d
4
⌋
+⌊ c
4
⌋
)
mod 7
As examples, we apply the formula to important days ofPanama’s history. ¿Sabe usted que día tuvo lugar laseparación de Panamá de Colombia?Aquí n = 3, m = 9, d = 3, n = 19 y
3 + 5 × 19 + 3 +
⌊
13 × 9 − 1
5
⌋
+
⌊
3
4
⌋
+
⌊
19
4
⌋
≡ 2 mod 7
¡El 3 de noviembre de 1903 fue martes!
Perpetual Calendar
The formula
The formula
(n, m)100c+d ≡
(
n + 5c + d +
⌊
13m − 1
5
⌋
+
⌊
d
4
⌋
+⌊ c
4
⌋
)
mod 7
As examples, we apply the formula to important days ofPanama’s history. ¿Sabe usted que día tuvo lugar laseparación de Panamá de Colombia?Aquí n = 3, m = 9, d = 3, n = 19 y
3 + 5 × 19 + 3 +
⌊
13 × 9 − 1
5
⌋
+
⌊
3
4
⌋
+
⌊
19
4
⌋
≡ 2 mod 7
¡El 3 de noviembre de 1903 fue martes!
Perpetual Calendar
The formula
The formula
(n, m)100c+d ≡
(
n + 5c + d +
⌊
13m − 1
5
⌋
+
⌊
d
4
⌋
+⌊ c
4
⌋
)
mod 7
As examples, we apply the formula to important days ofPanama’s history. ¿Sabe usted que día tuvo lugar laseparación de Panamá de Colombia?Aquí n = 3, m = 9, d = 3, n = 19 y
3 + 5 × 19 + 3 +
⌊
13 × 9 − 1
5
⌋
+
⌊
3
4
⌋
+
⌊
19
4
⌋
≡ 2 mod 7
¡El 3 de noviembre de 1903 fue martes!
Perpetual Calendar
The formula
The formula
(n, m)100c+d ≡
(
n + 5c + d +
⌊
13m − 1
5
⌋
+
⌊
d
4
⌋
+⌊ c
4
⌋
)
mod 7
As examples, we apply the formula to important days ofPanama’s history. ¿Sabe usted que día tuvo lugar laseparación de Panamá de Colombia?Aquí n = 3, m = 9, d = 3, n = 19 y
3 + 5 × 19 + 3 +
⌊
13 × 9 − 1
5
⌋
+
⌊
3
4
⌋
+
⌊
19
4
⌋
≡ 2 mod 7
¡El 3 de noviembre de 1903 fue martes!
Perpetual Calendar
Important dates in the panamanian History.
Fechas importanteSeparación de la Gran Colombia 3 de noviembre de 1903 martes
Independencia de España 28 de noviembre de 1821 miércoles
Gesta de los mártires 9 de enero de 1964 jueves
Llegada a la Luna 20 de julio de 1969 domingo
Fundación de la Universidad de Panamá 7 de octubre 1935 lunes