May 2006Time and CalendarsSlide 1 Time and Calendars A Lesson in the Math + Fun! Series 2 0 0 6.

May 2006 Time and Calendars Slide 1

Time and CalendarsA Lesson in the “Math + Fun!” Series

2 0 0 6


About This Presentation

Edition Released Revised Revised

First May 2006

This presentation is part of the “Math + Fun!” series devised by Behrooz Parhami, Professor of Computer Engineering at University of California, Santa Barbara. It was first prepared for special lessons in mathematics at Goleta Family School during three school years (2003-06). “Math + Fun!” material can be used freely in teaching and other educational settings. Unauthorized uses are strictly prohibited. © Behrooz Parhami


A Brief History of Timekeeping

First atomic clock, (US National Bureau of Standards, 1949)

Ancient Sundial (1st or 2nd Century AD)

Modern $50 “Atomic Clock”

Two Grandfather Clocks(17th and 18th Centuries)

Pocket Watch and

Wristwatch(19th Century)


Activity 1: Making a Sundial Clock1. Fill up a small bottle with sand or soil and place a straight stick in it. Push the stick down until it is firmly planted and does not wobble.

2. Place a large piece of cardboard in a spot where the sun shines all day (you do need a sunny day for this activity), put the bottle and stick at the center of the cardboard, and draw a circle around the bottle’s base (to mark its exact location, in case it moves by accident).

3. Every hour, on the hour (9:00 AM, 10:00 AM, and so on), draw a line along the shadow cast by the stick and mark it with the time. By sundown, you will have a sundial clock!

If you want to be able to move your clock to a different location, use a compass to mark north on the cardboard and have the line point to north in the new location.

North


Activity 2: Making a Pendulum Timer1. Take a lightweight string that is about 1 meter (40 inches) long. Attach a heavy bob to one end and a key-ring to the other.

2. Hang the pendulum from a long nail, so that the bob is away from the wall and close to the floor or desktop.

3. Gently swing the pendulum and time 60 of its swings from one side to the other. The time should be about 1 minute. If it’s a bit less than 1 min, make the string a little longer; if it’s a bit more, shorten the string. Repeat this until each swing takes almost exactly 1 second.

The time for one swing (in seconds) is related to the length of the string (in meters) by the equation:

Side-to-side swing time Length

So to make a side-to-side swing take 2 seconds, the string length must be about 4 meters.


How Mechanical Clocks Work1. There is a lever that oscillates about an axis; much like a pendulum, but not hanging

2. Each oscillation causes a cogged (escape) wheel to rotate by a small amount

3. The movement of the escape wheel is transferred to hand movements via gears that reduce the speed

The oscillating element can be in the shape of a wheel (balance wheel), but the idea is the same

4. A spring is used to give the oscillating lever a gentle push so that it does not slow down


Converting Oscillations to Rotation

Oscillations of the red wheel cause the Y-shaped lever to move and to push the teeth of the escape wheel

Oscillations of the blue wheel turn the escape wheel by pushing its teeth, as shown in the animation to the right


A Brief History of Calendars

1. There are 40 different calendar systems in use today.

2. Our calendar system is called the “Gregorian Calendar”; there are also Chinese, Hebrew, Indian, Islamic, Persian, and many other calendars.3. Ancient Egyptians were the first to use years consisting of 365 days; early Chinese and Greek years were 354 days, as is the Jewish year; at various times, there have been both shorter and longer years (for example, 46 BC, nicknamed “year of confusion,” had 445 days, because Julius Caesar added 80 days to it so that the following year in his reformed calendar would conform to the solar year).

The purpose of a calendar system is to keep track of time periods that are linked to astronomical events (day = one rotation of the Earth around its axis, month = one revolution of the Moon around the Earth, year = one revolution of the Earth around the Sun). Week is the only calendar time convention that is not linked to astronomical events.


Why Do We Need Leap Years?1. One calendar year (Earth’s revolution around the Sun on average) is 365 days, 5 hours, 49 minutes (365.2424 Universal days).

2. After four 365-day years, we are left with 4 0.2424 = 0.9696 extra day, or almost one day. This is why we make every fourth year a leap year.3. But an extra day is a little bit more than 0.9696 day. The extra 0.0304 day adds up to a full day every 33 leap years, or in about 132 years.

4. It is rather inconvenient to make a correction every 132 years, so we drop the leap year every 100 years and put it back in every 400 years.

5. When we drop the leap year every 100 years, we remove one day instead of 0.0304 25 = 0.76 days every 25 leap years. This is ¼ day less than what it should be. So, every four centuries, or 400 years, we make a correction by using a leap year. This makes it just about right. Very infrequently (every 1000s of years), we need more corrections.

Interesting fact: Some years have a leap second added to them to correct for variations in timekeeping by atomic clocks around the world.


Finding the Day of the Week for a Date Between 1901 and 2099

Month Code

January, Leap Year –3

January, Nonleap –2

February, Leap Year 0

February, Nonleap 1

March 1

April –3

May –1

June 2

July –3

August 0

September 3

October –2

November 1

December 3

Example: On what day is July 4, 2006?

Divide year number by 4, ignore the remainder

Add the year

Add the month code from table to the right

Add the day of the month

Divide by 7, keep only the remainder

The remainder gives you the day in the table below

2006 / 4 = 501, with remainder 2

501 + 2006 = 2507

2507 + (–3) = 2504

2504 + 4 = 2508

2508 / 7 = 358 with remainder 2

0 1 2 3 4 5 6

Sun Mon Tue Wed Thu Fri Sat

Tuesday!


Activity 3: Your Birthday in Different YearsYear of your 18th

Birthday: _____

Divide the year number by 4, ignore the remainder

Add the year

Add the month code from table in the previous slide

Add the day of the month

Divide by 7, keep only the remainder

The remainder gives you the day in the table below

_____ / 4 = _____ with remainder ___

_____ + _____ = _____

_____ + _____ = _____

_____ + _____ = _____

_____ / 7 = _____ with remainder __

Month of birthday: _____Day of birthday: _____

0 1 2 3 4 5 6

Sun Mon Tue Wed Thu Fri Sat

Your birthday in the year 2050

_____ / 4 = _____ with remainder ___

_____ + _____ = _____

_____ + _____ = _____

_____ + _____ = _____

_____ / 7 = _____ with remainder __


Remember that a year is a leap year if it is divisible by 4, but not if it is also divisible by 100, except if it is also divisible by 400. For example, the year 1900 was not a leap year (divisible by 100 but not by 400) but the year 2000 was a leap year (divisible by 400)

Activity 4: Making a Calendar for the Year 2020

1. Figure out whether 2020 is a leap year.

2. Find the day of week for January 1, 2020; then complete the calendar.January

S M T W T F SFebruary

S M T W T F SMarch

S M T W T F SApril

S M T W T F SMay

S M T W T F SJune

S M T W T F S

JulyS M T W T F S

AugustS M T W T F S

SeptemberS M T W T F S

OctoberS M T W T F S

NovemberS M T W T F S

DecemberS M T W T F S


JanuaryS M T W T F S

FebruaryS M T W T F S

MarchS M T W T F S

AprilS M T W T F S

MayS M T W T F S

JuneS M T W T F S

JulyS M T W T F S

AugustS M T W T F S





JanuaryS M T W T F S

FebruaryS M T W T F S

MarchS M T W T F S

AprilS M T W T F S

MayS M T W T F S

JuneS M T W T F S

JulyS M T W T F S

AugustS M T W T F S






Perpetual CalendarsYear Type

1947, 1975, 2003 3

1948, 1976, 2004 4*

1949, 1977, 2005 6

1950, 1978, 2006 0

1951, 1979, 2007 1

1952, 1980, 2008 2*

1953, 1981, 2009 4

1954, 1982, 2010 5

1955, 1983, 2011 6

1956, 1984, 2012 0*

1957, 1985, 2013 2

1958, 1986, 2014 3

1959, 1987, 2015 4

1960, 1988, 2016 5*

0 1 2 3 4 5 6Sun Mon Tue Wed Thu Fri Sat

There are only 14 different calendar types: seven types depending on the day of the week for January 1, and two variations for leap and nonleap years. If you print these 14 calendars and call them 0, 0*, 1, 1*, . . . , 6, 6* (where the number is the day of week for January 1 and * indicates a leap year), the table to the right shows you which calendar to use for any given year.

Year Type

1961, 1989, 2017 0

1962, 1990, 2018 1

1963, 1991, 2019 2

1964, 1992, 2020 3*

1965, 1993, 2021 5

1966, 1994, 2022 6

1967, 1995, 2023 0

1968, 1996, 2024 1*

1969, 1997, 2025 3

1970, 1998, 2026 4

1971, 1999, 2027 5

1972, 2000, 2028 6*

1973, 2001, 2029 1

1974, 2002, 2030 2


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