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# The Mathematics of Leap Years and Leap Seconds

Glenn Stok has studied fields of science, physics and philosophy, for over 40 years, since his college days with a Master of Science degree.

As a Computer Systems Programmer I once had to write an algorithm to determine the day of the week (Monday, Tuesday, etc) for any calendar day.

I had to thoroughly understand it to get it right, so I studied how the days of our calendar are calculated. Here is my explanation of the details of Leap Years and Leap Seconds.

In order to keep our Gregorian calendar in sync with mean solar time (UT1 time scale), we need to add a second once in a while in addition to adding a day every four years. However, there is more complexity that needs to be considered.

Everyone knows that we add an extra day, February 29th, every four years. However, most people don't know that we have to skip that addition once in a while.

If a year were exactly 365 and a quarter days, then adding a day every four years would work wonderfully. Unfortunately our Earth doesn't go around the sun so perfectly. More precisely, a year is 365.2426 days.

Source

## Extremely Accurate Time Measurements

We live in a time when we have the resources to do extremely accurate measurements. We have the technology to measure the rotation of the Earth so precisely that we can detect how it is slowing down. We use atomic clocks to keep an accurate account of time.

There are National Standards Agencies in many countries that maintain a network of atomic clocks. They are kept synchronized with extreme accuracy.

In addition, we have the master atomic clock at the U.S. Naval Observatory in Washington providing the time standard for the U.S. Department of Defense.

## Managing The Extra Fractions of Days

If it took exactly 365 days for the Earth to revolve around the Sun, then we would have a perfect calendar and there would be no need to make corrections. Since it takes 365.2426 days, adding a day every four years is adding too much. We need to skip adding some leap years, but how often?

If that additional fraction over 365 days were exactly .25 (an extra quarter of a day) then every four years adds up to a full day. If that were accurate, we would just add that extra day at the end of February every four years, but we don't. We need to skip some Leap Years.

Things are not that simple because it's not exactly an extra .25 day per year. That fraction I mentioned before, 0.2426, is a little less than a quarter of a day. Therefore, every 100 years we need to skip adding a leap year. Otherwise we would be adding too much.

But wait! That still isn’t perfect!

## We Need to Skip a Leap Year Every 100 Years

Skipping a leap year every 100 years only works if the extra time were exactly 0.25. However, we are still off by almost .01 from a quarter day. That .01 adds up to 1 in 100 years. Therefore we need to skip a leap year every 100 years. If we didn't skip it, we'd be adding too may days to the calendar.

We'll still get out of sync with solar time if we don't take it a step further.

As you can see, we still have that extra .0026 that we are off when skipping a leap year every 100 years. If you add that up, with some rounding error, that .0026 is a little over 1 day every 400 years (.0026 x 400 = 1.04) . That means that skipping a leap year every 100 years needs adjustment as well. We need to add a day back in.

## We Need to Add an Extra Day Every 400 Years

Remember that I said we normally skip a leap year every 100 years? However, we need to keep that leap year every 400 years in order to get that one extra day added back in.

The easiest way to add that missing day back in is to not skip a leap year if the year is a multiple of 400, even though it is also a multiple of 100. In other words, we keep February 29th on the calendar every 400 years.

## In Summary

Just to say it all in one sentence: We add a day every four years, but not every 100 years, unless it’s a 4th century year, at which point we do add that extra day anyway.

## Table of Leap Years and the Reasons For It

Year
Leap Year if Multiple of 4
But Skip if Multiple of 100
Unless it's a Multiple of 400
Leap Year?
1600
Yes
-
Yes
Yes
1700
Yes
Yes
No
No
1800
Yes
Yes
No
No
1900
Yes
Yes
No
No
2000
Yes
-
Yes
Yes
2004
Yes
No
-
Yes
2008
Yes
No
-
Yes
2012
Yes
No
-
Yes
2016
Yes
No
-
Yes
2100
Yes
Yes
No
No
As indicated in this table, we add a day for leap year every four years, but not every 100 years unless it’s a year divisible by 400.

## Why We Need to Add Seconds to Our Calendar

The algorithm for leap years still does not provide perfect accuracy. Adding a few seconds is also required. Climate and geological events can cause the Earth’s revolution around the Sun to fluctuate.

In addition to that, the Earth's rotation around its own axis is not consistent. It tends to slow down and speed up ever so little.

It has been reported that the 9.0 magnitude Earthquake in Japan in 2011 had shifted the Earth's axis by an amount between 10 cm (4 inches) and 25 cm (10 inches). Fluctuations such as this change the length of a day by an ever-so-tiny amount and it needs to be adjusted on our calendar.{1}

In order to improve the accuracy of our time clocks, we need to add a second or two every year. It’s called a leap second.{2}

Scheduling the addition of an extra second to a year is done to make these adjustments.

It's usually added, when needed, as an additional second at midnight, Coordinated Universal Time (UTC), on June 30th or December 31st.

## International Earth Rotation and Reference Systems Service

The International Earth Rotation and Reference Systems Service is the agency that decides when these time adjustments are made. They apply a leap second whenever necessary to keep our clock from being more than 0.9 of a second off.

The last several times we added a second was at midnight (UTC) on the following dates:

• December 31st, 2008
• December 31st, 2005
• June 30th, 2012
• June 30th, 2015
• December 31st, 2016

Now that you understand all this, you no longer have an excuse for being late when meeting a friend for dinner, or for keeping that appointment with your doctor.

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• Kristen Howe 2 years ago from Northeast Ohio

My pleasure Glenn. I always wondered about the leap year and never heard of leap seconds.

• Author

Glenn Stok 2 years ago from Long Island, NY

Kristen Howe - It sure is interesting. The Earth's rotation is slowing down, requiring the addition of leap seconds. Thanks for the vote up. Much appreciated.

• Kristen Howe 2 years ago from Northeast Ohio

Glen, this was real interesting to know we had a extra leap second last month and next year is a leap year. This was well-written and intriguing to know the history behind the leap year. Voted up!