O leap year had its origin in Julian calendar, which was established by Julius Caesar in 46 a. Ç. This calendar was based on the results of the studies carried out. by astronomer Sosigenes of Alexandria. This calendar had the following characteristics:
The twelve months of a year had a different number of days, totaling 365.25 days;
The first day of the year was January;
Every four years, a leap year was instituted, which would have 366 days.
O Julian calendar became Gregorian in the year 1582. The alteration of the calendar was suggested by Pope Gregory XIII, who was part of a commission of scholars formed by mathematicians such as the Jesuit Cristophorum Clavius, and astronomers such as Aloisius Lilius. This committee determined that:
Determinations for the Gregorian Calendar
The Julian calendar had a counting error, as there was an excess of days which, at the end of a century, resulted in ¾ of a day;
With the identification of the Julian calendar error, it was possible to determine that, every 400 years, there would be a difference of three days;
The excess three days should be introduced in later years. Those years would be leap years;
The Gregorian calendar should conform to the solar year, which is determined by the time period of the four seasons of the year (spring, summer, autumn and winter). A solar year has a duration of 365 days, 5 hours, 48 minutes and 46 seconds, having a total of 365.2422 days;
From the determination that a solar year has 365.2422 days, the Gregorian calendar determined that the leap year would happen every four years. Thus, in 400 years, we would have 100 leap years. In order for the counting of days to be in sync with the solar year, it was determined that three leap years should be eliminated. Thus, in 400 years, we would have only 97 leap years;
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The mathematical relationship described below determines a Gregorian year, which is approximately 365.2425 days.
365,2425 = 365 + 1 − 1 + 1
4 100 400 It was defined that, in the month of February, when the year is a leap year, one day will be added. Therefore, this month, in leap years only, will have 29 days.
Criteria for defining the leap year
For a better understanding, let's perform the leap year calculation to see which of the years described below fit into this category. Before that, we need to know what are the criteria that define it, that is:
For be leap, the year should be:
Divisible by 4. Therefore, the division is exact with the remainder equal to zero;
It cannot be divisible by 100. Thus, the division is not exact, that is, it leaves a remainder different from zero;
It may be divisible by 400. If it is divisible by 400, the division must be exact, leaving the remainder equal to zero.
According to the criteria set out above, we will determine whether the year 2015 or 2016 is a leap year. For this, there are three pre-established situations:
first situation: If the year 2015 or 2016 is an exact division with respect to 4, then we should check that it is not divisible by 100. If not, the year will be leap year;
Second situation: If the year 2015 or 2016 is not divisible by 4, then we should check if it is divisible by 400. If it is also not divisible, the year 2015 will not be a leap year;
Third situation: If the year 2015 or 2016 is not divisible by 4, then we should check if it is divisible by 400. If so, the year 2015 is a leap year.
Calculation to identify if the year is a leap year
1st) Let's check if the year 2015 is a leap year.
→ First moment: Check if 2015 is divisible by 4.
2015 |4
-200 503
15
-12
3
The division wasn't exact, that's because the rest of the 2015 division by 4 is 3.
→ Second moment: We must apply the second situation determined earlier. So let's divide 2015 by 400.
2015 |400
-2000 5
15
As the division of 2015 by 400 was not exact, we can conclude that the year 2015 is not a leap year. With that, the month of February has 28 days.
2nd) Let's now find out if the year 2016 is a leap year.
→ First moment:Check if 2016 is divisible by 4.
2016 |4
-200 504
16
-16
0
The 2016 division by 4 is exact, because the rest of the division was zero.
→ Second moment: Let's apply the first situation determined earlier, that is, divide 2016 by 100.
2016 |100
-200 2
16
The 2016 split by 100 was not exact; soon, the year of 2016 is leap and, consequently, February has 29 days.
Importantly, the Gregorian calendar is currently used in most Western countries. Of the eastern countries that do not use this calendar, we can highlight Chima, Israel, India, Pakistan, Iran, Algeria, among others.
By Naysa Oliveira
Graduated in Mathematics
