De La Salle University – Manila ©2014

MTH521M

Module 1: The Perpetual Calendar

PREPARED BY: De Castro, Kristine Joy P. Sabaybay, Rhea M. Santiago, Ericka M.

NUMBER THEORY APPLICATIONS OF CONGRUENCES

THE PERPRETUAL CALENDAR A Teaching and Learning Module

MODULE DESCRIPTION

This module is a scaffolding material for the students and teachers in the Senior High

School level to recognize and apply the notions of linear congruence in one of its relevant

applications – the perpetual calendar – identifying the day of the week for any given date. It also

provides some learning materials – including some illustrations, instructor’s guide for some

sections and examples and exercises - to enhance the said skill.

OBJECTIVES

By the end of the module, the learner will be able to:

1. describe and characterize the perpetual calendar

2. review the basic notions of modular arithmetic or congruence

3. recognize the concept of the congruence used in the perpetual calendar

4. determine the day of the week for any given date using the applied algorithm

CONTENTS

PREREQUISITE CONCEPTS Basic notions of Modular Arithmetic: Theory of Congruence

ABOUT THE TOPIC What is Perpetual Calendar

Gregorian Calendar: Historical development

PERPETUAL CALENDAR Preparatory Activity

Application of Congruence Examples and Illustrations

ENHANCEMENT EXERCISES

Learning Activity 1 Learning Activity 2 Learning Activity 3

MODULAR ARITHMETIC (Theory of Congruences)

Modular arithmetic is referenced in number theory, group theory, ring

theory, knot theory, abstract algebra, computer algebra, computer science, chemistry and the visual and musical arts.

It is also one of the foundations of number theory, touching on almost every aspect of its study and provides key examples for group theory, ring theory and abstract algebra.

Modular arithmetic is a system of arithmetic for integers, where numbers “wrap around” upon reaching a certain value – the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss, in his book Disquisitiones Arithmeticae, which was published in 1801.

For a positive integer n, two integers a and b are said to be congruent modulo n, and as written as nba mod , if their difference ba is an integer multiple of n

(or n divides ba , while integers congruent to a modulo n are creating a set

called congruence class, residue class or simply residue of the integer a, modulo n. Examples:

(1) 5mod318 since 15|5318|5

(2) 11mod537 since 32|11537|11

Complete Residue System modulo n

Let Nn . A collection of n integers n

aaaaS ,,,,321 is called a

complete residue system modulo n or a CRS if every integer b is congruent modulo n to exactly one of the elements in S. Examples:

(1) The set 7,22,13,16 forms a CRS modulo 5.

(2) Let 1,,2,1,0 nn . The set n is the set of least positive residues

modulo n.

Illustration of the Modular Arithmetic

A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods.

If the time is 9:00 now, then 4 hours later it

will be 1:00. Usual addition would suggest that the later time should be 9 + 4 = 13, but this is not the

answer because clock time "wraps around" every 12 hours; in 12-hour time, there is no "13 o'clock". The added 4 hours will then give the 1:00 time.

Using the notion of modular arithmetic, specifically the basic concept of the theory of

congruences: nba mod We let a be the result of adding 4 hours to the original time 9:00, hence:

a = 9 + 4 = 13

Since the hour number starts over after it reaches 12, this is arithmetic modulo 12. So we let n be equal to 12 and b be the equivalent time after the addition of hours.

1 is the remainder so it gives the desired 1:00 time

This congruence is true since 12 divides (13 – 1)=12.

Likewise, if the clock starts at 12:00 (noon) and 21 hours elapse, then the time will be 9:00

the next day, rather than 33:00. 12 is congruent not only to 12 itself, but also to 0, so the time called "12:00" could also be

called "0:00", since 12 is congruent to 0 modulo 12.This analogy is a little loose. The proper way to interpret this is that the group of integers modulo n act on the numbers of a clock, rather than the numbers on the clock being added together. Adding together two times on a clock is an example of a type error. However, it provides a useful way to understand the concept for the first time.

There are many other applications of congruence in different areas of living, may it be in sports, work-related tasks, security concerns or even the daily routine and expenses.

Time is money, as what other people say. Whether a person is setting appointments,

payment arrangements or planning holidays, determining the day of the week for any given date is always helpful – and this is one of the many practical and relevant uses of the theory discussed above.

12mod113

Instructor’s Guide: After making sure that the students understood this part of review, you may ask them some other illustrations of the modular arithmetic for intervention purposes.

Skip Counting to determine the day of the week for a given date may be very helpful

when dealing with small intervals between two dates. Whenever given with a very large number of days to count, this may not be of practical use anymore, what more if one is given a date in the past or a day in the future which is very far from the reference date. Try to answer the following questions below:

Do you know on what day you were born?

Do you know on what day we will be celebrating

Christmas of 2014?

Can you predict the day when 2020 would officially start?

There is actually a way for us to easily determine these days. Discover it yourselves with everything presented on the next pages!

Try this!

SUNDAY MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY SATURDAY

Suppose today is August 23, 2014 which is a Saturday, using skip-counting and with the

aid of the table above, determine the day of the week for the given dates/ number of days

below.

(1) After 20 days : ______________

(2) 31 days ago: ______________

(3) After 5 weeks: ______________

(4) October 15, 20114: ______________

(5) December 3, 2014: ______________

(6) January 1, 2015: ______________

PRELIMINARY ACTIVITY

Instructor’s Guide: You may modify the given dates and events for the preliminary activity i.e. to make it more relevant to your students. (may include their parents’ birthdays etc.)

PERPETUAL CALENDAR A perpetual calendar is a calendar valid for many years, usually

designed to allow the calculation of the day of the week for a given date in the future. In this section, an interesting formula will be developed to determine the day of the week for any date in any year. But before

accomplishing this goal, we must be given with a hint of the historical development of the popular calendar.

A hint of history…

Around 738 B.C. , Romulus, the legendar founder of Rome, is said to have introduced a

calendar consisting of 10 months, comprising a year of 304 days. His successor, Nauma, is credited with adding two months to the calenda. This new calendar was followed until Julius Ceasar changed the Egyptian calendar, which was based on a year of exactly 365 days, to a new calendar – the Julian calendar – with a year of average length of 365 ¼ days, with leap years every fourth year, to better reflect the true length of the year. The Julian calendar consisted of 12 months of 30 and 31 days, except for February, which had 29 days and every fourth yea, 30 days.

However, more recent calculations have shown that the true length of the year is

approximately 365.2422 days. As the centuries passed, the discrepancies of 0.0078 days per year added up, so that by the year 1582 approximately 10 extra days had been added unnecessarily as leap years. To remedy this, in 1582, Pope Gregory set up a new calendar. First, 10 days were added to the date, so that October 5, 1582 became October 15, 1582 (and the 6th through the 14th of October were skipped)

The Gregorian calendar designates those century yeas divisible by 400 as leap years; all

noncentury years divisible by 4 are also leap years. As an example, the year 1700, 1974, 1800, 1900 and 2100 are not leap years but 1600, 1776 and 2000 are. With this arrangement.

The Gregorian calendar, now used throughout the world, is so accurate that it differs to the

true year by 0.0003. The average length of 365.2425 days is rather close to the true year of 365.2422 days. The result is an error of 3 days per 10000 years.

In dealing with calendar dates for various parts of the world, we must also take into

account the fact that the Gregorian calendar was not adopted everywhere in 1582. In Britain, the Gregorian calendar was adopted only in 1752, and by then , it was necessary to add 11 days. Japan changed over 1873, the Soviet Union and nearby countries in 1917, while Greece held out until 1923.

Instructor’s Guide: This should not be an optional part of the discussion. Without this section, the students might find it difficult to understand some important concepts of the perpetual calendar.

PERPETUAL CALENDAR

The first century leap year occurred in 1600 which is 18 years after the introduction

of the Gregorian calendar; so the algorithm discussed below was developed to hold for years beyond

1600.

DEVELOPMENT OF THE ALGORITHM:

Also, since a leap year adds a day to February, we will count the New Year beginning with

March 1.

For example, January 3000 is considered the 11th month of 2999, whereas April

3000 is the 2nd month of year 3000.

Also, February 29 of 1976 is the last day of the 12th month of 1975.

We will assign these numbers for the months (m), and days (d) of the week.

For example, d = 3 denotes Wednesday and m = 5 denotes July.

Also, d = 0 denotes Sunday and m = 11 denotes January.

To compute d from d1600

We let dy denote the day of the week of March 1 (the first day of the year) in year y, where 1600y .

Because )7(mod1365 , dy is advanced from dy-1 by 1 if y is not a leap year and by 2 if

y is a leap year:

year leap a isy if2

year leap anot isy if1

1

1

y

y

y d

dd

d 0 – Sunday

1 – Monday

2 – Tuesday

3 – Wednesday

4 – Thursday

5 – Friday

6 - Saturday

m 1 – March 7 – September

2 – April 8 – October

3 – May 9 – November

4 – June 10 – December

5 – July 11 – January

6 – August 12 – February

We need to know the number of leap years k since 1600;

By division algorithm, ,1000,100 DwhereDCy so C denotes the number of

centuries in y and D the leftover.

100mod100

yDandy

C

Note:

remains.number whole theonly and omitted are remainders wherevaluesdenotessymbolThis

Example:

Therefore,

7modk1600-y

7mod1600 sinceyear leap

eachfor day extra one

1600 sinceyear

eachfor day one

1600

1600

d

dd y

Substituting y and k,

Again, this formula is used to identify dy, the day of March 1 in year y, given we know d1600; and to

identify d1600 from some known value of dy.

3884001004

yyyk

If y=2345, then C=23 and D = 45. Then,

7mod44

2

7mod44

5

7mod44

3342

7mod344

31600-D100C

1600

1600

1600

1600

DCDCdd

DCDCd

DCCDCd

DCCdd

y

y

To determine d1600

Extension of Formula B:

To generalize the formula, we need to know the number of days the first of the month is moved up from that of the previous month modulo 7.

This is the summary for the monthly additional values.

Example:

Because March 1, 1994 fell on a Tuesday, d1994=2. For y= 1994, C=19 and D=94

Thus, d1600 was a Wednesday. And substituting for d1600 in the formula A,

The day of the first of the month following a 30-day month is advanced by 2 days.

The day of the first of the month following a 31-day month is advanced by 3 days.

Example:

December 1, 1992 was a Tuesday. So January 1, 1993 fell on

day (2 +3) = 5, which is a Friday.

March 1 to April 1: 3 days April 1 to May 1: 2 days

May 1 to June 1: 3 days June 1 to July 1: 2 days

July 1 to August 1: 3 days

August 1 to September 1: 3 days

September 1 to October 1: 2 days October 1 to November: 3 days

November 1 to December 1: 2 days

December 1 to January 1: 3 days January 1 to February 1: 3 days

Instructor’s Guide: If not time-constrained, you may also ask the students to solve for themselves and identify the incremental values for each month, instead of giving them right away,

To find a function f that produces these increments:

As what can be observed from the summary given, the sum of the increments is 29 days, the average of which is days6.211/29 , so it was observed by Christian Zeller that the function

22.06.2 mmf can be employed to yield the above increments as m varies from 2 to 12.

Example:

To find the formula for the rth day of Month m:

The day d of the week for the rth day of month m is given by )7(mod)1('' rd , that is,

EXAMPLE 1

d = day of the week C = the first two digits of the year r = given day of the given month D = last two digits of the year m = number code for the given month (March as the 1st month)

This formula helps us to determine the day of the week of any

given date in the Gregorian calendar.

Determine the day of the week on which January 13, 2020, falls.

Solution: Since January is the 11th month of 2019, so y=2019, hence,

C = 20 , D = 19, m = 11, and r = 13

Thus, January 13, 2020 falls on a Monday.

EXAMPLES

EXAMPLE 2

EXAMPLE 3

Determine the day of the week on which August 12,1990 falls.

Solution: Since February is the 6th month of 1990, so y=1990, hence, C = 19 , D = 90, m = 6, and r = 12

Thus, August 12, 1990 falls on a Monday.

Determine the day of the week on which December 25, 2014 falls.

Solution: Since December is the 10th month of 2014, so y=2014, hence,

C = 20 , D = 14, m = 10, and r = 25

Thus, Christmas of 2014 will be celebrated on a Thursday.

Instructor’s Guide: Make sure to remind the students that the bracket symbol denotes the whole number value of the quantity and NOT the rounded off value.

Example: and NOT

Learning Activity No. 1

Group Work: Divide the class into 5 groups, of which the number of members depend on the total number

of students in class. Have them work on these problems together for 10 minutes and present their output for at most 3 minutes each group. Let them explain on how they came up with their answer. Group 1: Find the day of the week on which the following significant events in Philippine history, fall:

Group 2: Find the day of the week on which the following significant events in Philippine history, fall:

Group 3: Find the day of the week of the following historical events in US history:

Group 4: Find the day of the week on which the following world’s trivial events fall:

Group 5: Find the day of the week on which the following world’s trivial events fall:

Historical Event Date Day of the Week

Assassination of Senator Benigno ‘Ninoy’ Aquino August 21, 1983

Proclamation of Tagalog as Official Language June 7, 1940

Millenium Watch December 31, 2000

Historical Event Date Day of the Week

Declaration of Philippine Independence June 12, 1898

Declaration Martial Law September 21, 1972

Execution of Dr. Jose Rizal December 30, 1986

Historical Event Date Day of the Week

First man on the moon July 20, 1969

US Declaration of Independence July 4, 1776

George Washington’s birthday February 22, 1732

Significant Event Date Day of the Week Establishment of the United Nations October 24, 1945

First human travel in space April 12, 1961

First Atomic Detonation July 16, 1945

Significant Event Date Day of the Week

Sinking of the Titanic Ship April 15, 1912

First animal in space November 3, 1957

Beginning of Japanese Occupation in the Philippines December 8, 1941

Learning Activity No. 2

Individual Work: Provide each student with this work sheet and let them work on this individually for at most 15 minutes. Let some volunteer students to share their work in class.

1. Find the day of the week you were born, and of your birthday this year.

2. Which of your birthdays, until your one hundredth, fall on the same day of the week as the

day you were born?

3. Show that days with the same calendar date, in two different years of the dame century, 28,

56, 84 years apart,fall on the identical day of the week.

Learning Activity No. 3

This activity may be given to students as a group exercise or homework. The activity’s

instructions are presented below.

Divide the class into 12 groups. Each group will be assigned one (1)

unique month of the year 2050 to work on that said month’s schedule or

calendar, again by using the algorithm for the perpetual calendar.

Once done and asked to submit, all the 12 by-month calendar will be

compiled and will be a project of the class. The class must still check whether

each group’s work is accurate and correct, that is, all the months are in sync

with all the other months to have made a whole year round.

Instructor’s Guide: You may provide a blank calendar sheet for each group to make the calendar organized and more presentable.

De La Salle University – Manila ©2014

REFERENCES

Books: (Electronic pdf versions)

Koshy, T. (2007). Elemetary Number Theory with Applications: Second Edition. pp.282-

287 Academic Press Publication, Elsevier Inc., California, USA.

Rosen, K. (1986). Elemetary Number Theory and Its Applications. pp.134-138. Addison-

Wesley Publishing Company. California, USA

Other Electronic Sources

http://en.wikipedia.org/wiki/Perpetual_calendar.retrieved on August 8, 2014 http://mathworld.wolfram.com/Congruence.html.retrieved on August 10, 2014 http://www.crapsmath.com/perpet_box/algorithm.html.retrieved on August 9, 2014 http://www.henk-reints.nl/cal/gregcal.html.retrieved on August 10, 2014