Calendars and dates

Why this exists

Alper is planning his family’s summer vacation, and Mirri and Coby are reading the school year calendar that came home in their backpacks. The vacation is “thirty days”, but which thirty? School starts again “in August”, but on which day, and is it a Monday?

Counting days one by one works for a few days and breaks down for a few hundred. A calendar fixes this. It groups the days into weeks, months and years, gives each group a name, and so gives every single day its own name: its date. With dates, two people can agree on a day months ahead, count how far apart two days are, and work out the day of the week without a long count.

Definition

A calendar groups days in three ways at once.

• A week is 7 days, named Sunday, Monday, Tuesday, Wednesday, Thursday, Friday and Saturday. The names repeat in the same order forever.
• A month is 28, 29, 30 or 31 days. There are 12 months in a year.
• A year is 365 days, or 366 days in a leap year.

A date names one day by its month, its day of the month and its year: July 4, 2027. In the United States it is written in numbers as month/day/year, so July 4, 2027 is 7/4/2027. Many other countries write day/month/year, where 7/4/2027 means the 7th of April. Writing the month as a word removes any doubt.

A year is a leap year when its number is a multiple of 4, except a century year (one ending in 00), which is a leap year only when it is a multiple of 400. A leap year gets one extra day, February 29.

Intuition

Remembering the months

The month lengths look random, and nobody works them out; people remember them. One way is the old rhyme: “Thirty days hath September, April, June and November. All the rest have thirty-one, except February alone.” Another uses your knuckles. Make two fists side by side and touch the knuckles and the dips between them from left to right, naming the months. January lands on a knuckle, February in a dip, March on a knuckle, and so on. July lands on the last knuckle of the first fist and August on the first knuckle of the second. Every knuckle month has 31 days; every dip month has 30, except February. Both are memory aids only. The table above is the fact.

Weeks are rows

A calendar page is a grid with one column for each weekday. Each row is one week.

Going down one row moves 7 days later and lands in the same column, so the same weekday. That is why the 1st, 8th, 15th, 22nd and 29th of July 2027 are all Thursdays: each is 7 more than the last. Any jump by a whole number of weeks keeps the weekday. Only the leftover days, the ones that do not make a whole week, move you along.

So to find the weekday a number of days later, divide by 7. The quotient counts whole weeks, which change nothing, and the remainder says how many weekdays to step forward. The remainder here has a meaning of its own, as in Remainders and what they mean.

Why leap years

A year is meant to be one trip of Earth around the Sun, and that trip takes about 365 and a quarter days. A calendar of 365 days would slip a quarter of a day behind each year, a whole day every 4 years, and after a few hundred years summer would fall in the months of spring. Adding one day every fourth year catches the quarters up.

The trip is in fact a little shorter than 365 and a quarter days, so a leap day every 4 years adds slightly too much. Skipping it in three century years out of every four fixes most of that. Calendars with a leap day every four years are very old; the century rule came later, with the calendar most of the world uses now, often called the Gregorian calendar.

Results

Worked examples

Boundaries

• “Days after” and “how many days”. August 24 is 76 days after June 9; that count does not include June 9 itself. A camp that runs from June 9 to June 11 lasts 3 days if you count the first and the last day both, though June 11 is only 2 days after June 9. Say which you mean.
• Months cannot be turned into days by one fact. “Three months” is 89 to 92 days, depending on which months. Count with the real month lengths.
• The leap year rule has its exception. A multiple of 4 is not enough for a century year: 1900 and 2100 are not leap years.
• Number dates depend on the country. Month/day/year is the US order. When the reader may use day/month/year, write the month as a word.
• The weekday rule needs whole days. Dividing by 7 works for counts of days, not for months or years directly, since those have no fixed number of weeks.

Where this goes next

Days, weeks and years are the large end of Units of time. How long it is from one clock time to another is Elapsed time, and Twenty four hour time and crossing midnight gives each hour of the day its own number. Different places on Earth are on different dates at the same moment, and the International Date Line is where the date changes. The remainder trick for weekdays is an early taste of counting in a cycle, where only the remainder matters, an idea that grows much later into Modular arithmetic.

Exercises

Months and dates

1. Write March 14, 2028, in US numbers.
2. Write 11/7/2027 with the month as a word.
3. Name the four months that have 30 days.
4. How many days are in June, July and August together?
5. How many weeks and days is a 31-day month?
6. How many days are in January, February and March of 2028, which is a leap year?
7. In the United States, 6/5/2027 is June 5. What day would a person who writes day/month/year read it as?

Leap years

8. Is 2036 a leap year?
9. Is 2026 a leap year?
10. Is 1800 a leap year? Is 1600?
11. List the leap years from 2041 to 2060.
12. How many days are in the year 2040? In the year 2300?

Weekdays and counting days

13. Today is a Tuesday. What day of the week is 10 days from now?
14. Today is a Wednesday. What day of the week is 25 days from now?
15. Today is a Monday. What day of the week was it 9 days ago?
16. Coby’s family leaves for vacation on Saturday, July 10, and comes home 16 days later. On what date and weekday do they come home?
17. How many days after May 20 is June 15?
18. How many days after July 4 is September 1?
19. Coby’s day camp runs from June 21 to July 2, and he goes every day, the first and the last included. How many days of camp is that?
20. The year 2026 is not a leap year and begins on Thursday, January 1. On what weekday does it end, on December 31?
21. What date is the 200th day of a year that is not a leap year?
22. Mirri’s birthday is October 2. One year it falls on a Monday. What weekday is it the next year, if no February 29 comes in between? And the year after that, if a February 29 does?

Reasoning

23. Explain why a date falls one weekday later from one year to the next, when no February 29 comes in between.
24. Coby says 1900 was a leap year because 1900 is a multiple of 4. Explain what is wrong.
25. In a year that is not a leap year, February 1 is a Sunday. Explain why February fills exactly four rows of the calendar, and why no other month ever can.