# My proposal for a new zero-based calendar

2013-04-14

The Anno Domini dating system and the Gregorian calendar that we now use are not zero-based numbering systems. There are no zeroth days, zeroth months or the zeroth year in these systems. That is because Europeans did not know zero as a number when they invented the AD dating system and the Roman calendar, the prototype of the Gregorian calendar. The numeral system we now use is, however, the zero-based decimal system. Identifying the time in reference to the current time system makes it difficult to calculate the functions whose variables include time. Although we should avoid unnecessary amendments, it is not desirable that we will continue to be bound by the numbering system which was formed when European mathematics was underdeveloped. Here I propose the best calendar that I can think of.

## 1. The reason the European calendars lack zero

The Anno Domini dating system and the Gregorian calendar that prevail in the current world are not zero-based numbering systems. There are no zeroth days, zeroth months or the zeroth year in these systems. So, every month starts on the day 1 and the year AD 1 immediately follows the year 1 BC. This is because there was no such number as zero when Julius Caesar introduced the Julian calendar, the prototype of the Gregorian, in 46 BC or Dionysius Exiguus (c. 470 – c. 544) devised the Anno Domini dating system in 525. The sign of zero as a place-value of the sexagesimal positional notation was used in the ancient Mesopotamia and Dionysius Exiguus used a word “nulla" meaning “nothing" alongside Roman numerals in 525, but the sign or the word was not treated as a number[1]. Therefore European time lines had no zero points. The ancient Greek, the source of the European Civilization, whose intellectual level was far higher than that of the ancient Roman, did not devise zero as a number either.

On the other hand the Mayan and the Hindu whose numerals had zero developed the calendar that starts in the year zero. The Mayan Long Count calendar identified August 11, 3114 BC in the proleptic Gregorian calendar as the day 0 of the year 0. It was a non-repeating calendar that ended on December 21, 2012. Their major repeating calendar is called Haab’, where 20 days/month × 18 months + 5 days = 365 days constitute a year. Day numbers of a month began with a glyph translated as the “seating of a month", namely the day 0 of that month, and ended with the day 19, according as their vigesimal positional numeral system. The Kali Yuga (Sanskrit: कलियुग), an era used with Hindu and Buddhist calendar, began with the day 0 of the year 0 that corresponds to January 14, 3102 BC in the proleptic Gregorian calendar. A zero-based day count of the Kali Yuga called Ahargana (Sanskrit: अहर्गण) meaning “heap of days" was also used in Indian calendric calculations.

Why did not the Greek mathematics adopt zero into their numeral system, although it was otherwise the most advanced in those days? That is because they did not believe in nothingness. Of course the Greek had an expression signifying nothing such as “μηδείς“, but they did not think the object of the word exists. In fact Parmenides of Elea (Greek: Παρμενίδης ὁ Ἐλεάτης; fl. early 5th century), Plato (Greek: Πλάτων, Plátōn; 424/423 BC – 348/347 BC) and most of the Greeks denied the existence of a vacuum. Aristotle, in his Physics, insisted that there should be no vacuums because the denser surroundings would soon fill up a thinner space[2]. This Aristotelian view later brought about a dictum, “Nature abhors a vacuum (horror vacui)" As nature has no emotions, we should think of this personification as the result of the projection of Europeans’ abhorrence of a vacuum onto nature.

Some Greek philosophers believed in a vacuum. The atomists such as Leucippus (Greek: Λεύκιππος, first half of 5th century BCE) and Democritus (Greek: Δημόκριτος, Dēmokritos; ca. 460 – ca. 370) assumed the empty (κενός) space where atoms (ἄτομος) could freely move. But their atomism was not accepted among the mainstream Greek philosophers. According to Diogenes Laërtius’s The Lives and Opinions of Eminent Philosophers, Plato disliked Democritus.

Aristoxenus in his Historical Notes affirms that Plato wished to burn all the writings of Democritus that he could collect, but that Amyclas and Clinias the Pythagoreans prevented him, saying that there was no advantage in doing so because the books were already widely circulated.[3]

Giving up obliterating the writings of Democritus, Plato tried to obliterate him from the history of philosophy by ignoring him. Plato, who mentioned almost all the early philosophers, never once alluded to Democritus, not even where it would be necessary to controvert him. Like the nature that was supposed to abhor a vacuum, Plato abhorred Democritus’s thought of a vacuum and tried to make his existence a vacuum by completely ignoring him. Owing to his efforts the Greek atomism was banished to oblivion. It was not until in 1643 that Evangelista Torricelli’s experiment on the column of mercury made Europeans realize the existence of a partial vacuum. In 1648 Blaise Pascal explained the vacuum in terms of the atmospheric pressure and denied the personified hypothesis “Nature abhors a vacuum".

As The Questions of King Milinda suggests, the Europeans believed in philosophy of substances, while the Hindus believed in the philosophy of nothingness. Jainists, for example, were atomists who assumed Paramānu as ultimate particles of all matter and Ākāśa as the infinite empty space that accommodates the Paramānu. The dates of Vardhamāna Mahāvīra (Sanskrit: महावीर), the founder of Jainism are unknown but he is supposed to precede Leucippus and Democritus. The dates of Siddhārtha Gautama Buddha (Sanskrit: सिद्धार्थ गौतम बुद्ध) is also unknown but he is supposed to precede Epicurus (Greek: Ἐπίκουρος; 341 BCE – 270 BCE) whose natural and moral philosophy is similar to that of Buddhism. So, Jainism might have influenced Leucippus and Democritus and Buddhism might have affected Epicurus.

Anyway the philosophy of nothingness could be the mainstream religion in India but was doomed to be excluded in Europe. Therefore it is not accidental that India accepted zero into their numerals advance in advance of Europe. The Hindu substantialization of zero went too far. Brahmagupta (Sanskrit: ब्रह्मगुप्त; 597–668 AD), in his book Brahmasphotasiddhaantasya (Hindi: ब्रह्मस्फुटसिद्धान्त; Brāhmasphuṭasiddhānta; Correctly Established Doctrine of Brahma) in 628, asserted not only 0+0=0, 0-0=0, 0×0=0, but also 0/0=0[4], which today’s mathematics does not allow. While Brahmagupta could not determine what quotient non-zero number divided by zero produces, other Hindu mathematicians assumed it to be zero, the unlimited and so on.

Why did Hindu mathematicians regard zero as the object of calculation? According to Hayashi Takao (1) they had to add, subtract, multiply and square zero so as to calculate in longhand without using abacuses (2) they had to formally operate the division or the extraction of square root of zero to solve linear or quadratic equations[5], but these reasons are not persuasive. Abacuses had been used in India since AD 1 and calculation in longhand was operated in other regions. The Babylonian who knew the quadratic formula did not treat zero as a number even when they calculated in longhand.

China had also a philosophy of nothingness. Laozi (Chinese: 老子) said in Tao Te Ching, “The Tao is like the eternal void filled with infinite possibilities. Its depth is like that of the Creator.[6]" and regarded the Tao as the eternal void and the source of everything. This philosophical background led to The Nine Chapters on the Mathematical Art (Chinese: 九章算术), the most important mathematical classic in China, which mentions the operation of no entry (無入) corresponding to zero.

When you subtract a number from another, subtract, if they have the same sign, and add, if they have different signs. Subtraction of a positive number from no entry produces a negative number and subtraction of a negative number from no entry produces a positive number.[7]

The last sentence says that 0-x=-x, 0-(-x)=x if x>0. The Nine Chapters on the Mathematical Art, however, did not make a sign for no entry or zero and expressed it in a blank. Unlike Chinese Hindu mathematicians idolized nothingness and devised a sign for zero as a number.

The Hindu mathematics was succeeded by al-Khwārizmī (عَبْدَالله مُحَمَّد بِن مُوسَى اَلْخْوَارِزْمِي‎; c. 780 – c. 850), a Persian mathematician and astronomer during the Abbasid Empire. His book, Al-Khwarizmi on the Hindu Art of Reckoning (Latin: Algoritmi de numero Indorum) in 825 was translated into Latin in the 12th century. Then Hindu numerals known as Arabic numerals were introduced into Europe (see the table below). The Sanskrit word śūnya, meaning “empty", was translated into the Arabic “ṣifr (Arabic: صِفر)" and then into the Italian zefiro and the English zero. But when zero was recognized as a number in Europe, the calendar without zero had been established as the de facto standard of the dating system.

Why did the ancient Greek or Europeans in general refuse zero? It was probably because they were perfectionists. As Euclid’s Elements indicates, the ancient Greek mathematicians sought the strict demonstration. It was not until the age of Pascal that the theory of probability as well as the theory of a vacuum was established, although the Hindu mathematicians developed a theory of probability in the 9th century[9]. If you can recognize the future only in terms of probability, it means your ability is limited and the ancient Greek did not want to acknowledge their inability. Because of their perfectionism they did not believe in nothingness, vacuums, zero or probabilities and tried to fill the empty place with substances. Their ideal might be praiseworthy but the result was ironical. Their desire to make their mathematics complete made it incomplete.

The European abhorrence of zero led to their preference of positional notation or place-value notation to sign-value notation. The former is superior to the latter in that the limited number of numerals can express the unlimited numbers. For example, in the case of decimal systems, 10 numerals from 0 to 9 can signify the unlimited numbers. But the ancient Greek developed an alphabetical numeral system that was akin to Egyptian sign-value notation rather than to Babylonian place-value notation. Some astronomers used the Babylonian Sexagesimal system but this positional notation did not come into general use. As a result Europeans had to use Egyptian fractions instead of decimal fractions.

An Egyptian fraction is a fraction notation to denote any given fractions by the sum of unit fractions whose numerators are 1, such as 5/6=1/2+1/3. As the figure below shows, the upper reciprocal glyph turns any natural number N into a unit fraction 1/N.

Egyptians, however, did not compose 2/3 of 2 and 3 as we do. They devised a special glyph that denotes only 2/3. The special glyph of 3/4 was also made but used less frequently. As this way of notation requires the unlimited number of special glyphs beyond human memory, Egyptians expressed a fraction as a sum of unit fractions. Europeans had used this inconvenient notation until the Middle Ages.

The lack of zero and positional notation confined the medieval European mathematics almost to the operations of natural numbers. Should we continue to be locked in the de facto standard of the numbering system that was formed when European mathematics was undeveloped? Next we will consider how to convert the discrete number system that identifies the span of time to the continuous coordinate system that identifies the point of time.

## 2. How to identify the point of time

The following figure shows the difference between the discrete span of time and the point of time on a number line of continuous real numbers.

The numbers of the Anno Domini dating system and the Gregorian calendar correspond to the red ones. The numbering system has no zero and natural numbers 1, 2, 3 … are allocated to the spans of time. As they have their own length 1, simple subtraction is not enough to get an interval between two time spans. For example, the time interval between the day 2 and the day 4 is 4-2+1=3. On the other hand the points of time correspond to the blue numbers. Like the point of the number line it has no own length. The interval between t1=1 and t2=4 is 4-1=3.

The continuum numbering system is superior to the discrete numbering system in that the former can identify any real number point of time, while the latter is superior to the former in that it is intuitively easier to understand. So I do not intend to dismiss it completely but we must consider how to convert one into the other. In order to correspond continuous points of time to discrete integers we should adopt the functions that Kenneth Eugene Iverson (17 December 1920 – 19 October 2004) proposed, namely the ceiling function and the floor function[10]. The year, month and day numbers of the Anno Domini dating system and the Gregorian calendar are the values of the ceiling functions and the numbers of hour, minute and second can be interpreted as the values of the floor functions. The former starts at 1 and the latter 0.

The ceiling functions map a real number to the smallest following integer.

$\lceil x \rceil=\min\,\{n\in\mathbb{Z}\mid n\ge x\}$

The graph below depicts that ceiling(x) is the smallest integer not less than x.

A month starts on day 1 instead of day 0, a year starts in month 1 instead of month 0 and the AD starts in year 1 instead of year 0. The century is also the value of a ceiling function whose unit is 100 years. For example the year 1945 is not the 19th century but the 20th century. This is because the years less than 101 belong to the 1st century instead of 0th century.

The floor functions map a real number to the largest previous integer.

$\lfloor x \rfloor=\max\, \{m\in\mathbb{Z}\mid m\le x\}$

The graph below depicts that floor(x) is the largest integer not greater than x.

Although the time system of hour, minute and second is not traditionally zero-based, it can be a zero-based numbering system, if you substitute 00:00 for 24:00 in 24 hour time notation.

It is inconvenient that the ceiling and the floor functions coexist on the same number line of time. While a new year starts in the 1st month instead of the 12th month, the change of a.m. and p.m. starts at 12 o’clock instead of 1 o’clock. While 00:01 belongs to the new day, the Sydney Olympic Games celebrated between 15 September and 1 October 2000 was not the first Olympics in the 21th century but the last in the 20th century. We should unify the function to avoid the confusion.

Which function is more preferable? The ceiling function puts zero before the start, that is to say, at the last of a period. Many computer programming environments, for example, use 31 December 1899 as default base date: 0th day 0th month 0th year. Such a trick might be good for computer programs that only experts would code, but not suitable for calendar that everyone uses. So, we should apply the floor functions to any level of the time identifying system. As I will show next, the floor function is easier to calculate when the unit of time has its subunits.

## 3. The merits of the zero-based numbering system

The floor function can approximately relate the discrete time to the continuous time. Let us recognize its merits by operating a simple calculation, subtraction. But before it we must determine the margin of error the approximation produces. Suppose the floor function values of t1 and t2 are T1 and T2 respectively.

$\lfloor t_{1} \rfloor=T_{1} \; \leftrightarrow \; t_{1}=T_{1}+\delta _{1} \; (0\leq \delta _{1} < 1)$
$\lfloor t_{2} \rfloor=T_{2} \; \leftrightarrow \; t_{2}=T_{2}+\delta _{2} \; (0\leq \delta _{2} < 1)$

Then the interval between T1 and T2 has the margin of error ±1.

$T_{2}-T_{1}= (t_{2}-\delta _{2})-(t_{1}-\delta _{1})=t_{2}-t_{1}+ \delta _{1}-\delta _{2}$
$\left | \delta _{1}-\delta _{2} \right |< 1$

In the case of the ceiling functions, the relation between the discrete value and the point of time is more complex, although the result of subtraction is equal.

$\lceil t_{1} \rceil=T_{1} \; \leftrightarrow \; t_{1}=T_{1}-1+\delta _{1} \; (0\leq \delta _{1} < 1)$
$\lceil t_{2} \rceil=T_{2} \; \leftrightarrow \; t_{2}=T_{2}-1+\delta _{2} \; (0\leq \delta _{2} < 1)$

Now let’s solve the following problem.

How long is the interval between 21:45 of day 1 and 3:15 of day 3? Ignore the error less than 1 minute.

If you make an hour a unit, you can calculate it as follows.

$2\times 24+3+\frac{15}{60}-\left ( 1\times 24+21+\frac{45}{60} \right )=5.5$

The answer is 5 hours and 30 minutes.

The hour is originally a floor function. Units longer than a day are the ceiling functions, but we can convert them into the floor functions. The expanded representations of the time format specified by ISO 8601[13] label years as positive or negative instead of BC or AD. The numerical value of years labeled Before Christ are reduced by one by the insertion of a year 0 before 1 AD, which can convert the AD dating system into a floor function. Now we can solve the following problem in a similar way.

How long did Augustus live, who was born on September 23, 63 BC and died on August 19, 14 AD? Ignore the error less than 1 day.

The year 63 BC is -62. The day September 23 is the 296th day and August 19 is the 261st day. They are day 295 and day 260 respectively in the floor function. If you make a day a unit, the lifetime of Augustus is calculated as follows.

$14+\frac{260}{365}-\left ( -62+\frac{295}{365} \right )=75+\frac{330}{365}$

It is 75 years and 330 days. Of course you cannot make out the day September 23 is the 296th day and August 19 is the 261st day at once. How can we make it easier? This is the topic of the next section.

## 4. The zero-based creation of a zero-based calendar

What kind of time identifying system would be the most rational, if we were allowed to ignore the tradition completely? So long as we live on the Earth, we cannot ignore the units of days and years, because the cycle of our lives is based on the diurnal rotation and the annual revolution of the Earth. But the month and the week in the solar calendar do not have the relation to the natural cycles, although the month synchronizes with the 30 day cycle of the lunar phase and the week corresponds to a quarter of the cycle of lunar revolution in the lunar calendar. A day is 24 hours and the unit less than an hour is sexagesimal. This tradition from the ancient Babylonia should be replaced with by the decimal notation.

This sort of rationalization was once challenged. So, if you are to make a completely new calendar ignoring the past, you should not ignore the past challenge to ignore the past. The French Revolutionary Calendar (French: calendrier révolutionnaire français) or French Republican Calendar (French: calendrier républicain français) which was created during the French Revolution and actually used by the French government for about 12 years from late 1793 to 1805 was the challenge of this sort. This literally revolutionary calendar decimalized all the units less than a month: a week is 10 days, a day is 10 hours, an hour is 100 minutes and a minute is 100 second. The picture below is the decimal clock created from the time of the French Revolution.

In China a day was once divided into 100 periods[16] and 10 days was a week called xún or huan (Chinese: 旬). So, France was not the first to adopt the decimal time system, but at least it was the first trial in Europe.

France also decimalized weights and measures. One meter was defined as one ten-millionth of the distance of a meridian and one gram was defined as the mass of one cubic centimeter. This meter-gram system is now adopted all over the world. Had France not proposed this new decimal system, the British system of weights and measures, 1 yard = 3 feet = 36 inches = 1/1760 mile and 1 pound = 16 ounces ＝ 7000 grains = 1/2240 ton, would be the de facto standard of the world.

The yard-pound system is far more complicated and inconvenient than the decimal meter-gram system. But the current system of time, 1 day = 24 hours = 1440 minutes = 1/7 week is as complicated and inconvenient as the yard-pound system. If the French Revolutionary Calendar as well as French meter-gram system had been adopted, the unit of time would be far more convenient to calculate.

The French Revolutionary Calendar, however, still retained irrational elements. First of all, it was not zero-based. It set 22 September 1792 of the Gregorian Calendar as day 1, month 1, year 1. It preserved 30 day month, although they abolished 7 day week. Why didn’t they decimalize every unit less than a year? If one month is 100 days and every year, month and day starts from 0, we can fully decimalize all units less than a year and we can operate the following calculation quite easily.

Suppose a machine produces 1000 products a day. If it produces them constantly from month 1, week 9, day 8, hour 7, minute 65 to month 2, week 0, day 0, hour 0, minute 0, how many products will it produce? Ignore the error less than 1 minute.

If you make a day a unit, you can express month 1, week 9, day 8, hour 7, minute 65 as 198.765 and month 2, week 0, day 0, hour 0, minute 0 as 200.000 and get the answer: (200.000-198.765)×1000=1235. This is a very simple calculation without using any non-decimal fractions. Of course if you make a year a unit you must deal with a fraction with 365 or 366 its denominator. But so long as you make a day unit, you do not need to use fractions except the decimal ones.

## 5. The zero-based World Calendar

In spite of its rationality the French Revolutionary Calendar did not take root because it utterly ignored the traditional custom. Once a de facto standard is locked in, it cannot be easily abolished. Though the cycles of the month and the week lose astronomical significance, they still have the cultural and social significance. So, we must respect the traditional custom as much as possible and reduce the amendment to the minimum.

One of the most famous proposed reforms of the Gregorian calendar of this kind is The World Calendar created by Elisabeth Achelis (January 11, 1880 – February 11, 1973) who founded The World Calendar Association on 21 October 1930 with the goal of its worldwide adoption. As the table below shows, the World Calendar is a 12-month calendar with equal quarters, each of which always begins on Sunday and ends on Saturday and has exactly 91 days = 13 weeks*7days/week or 3 months (31, 30, 30 days respectively).

The World Calendar has two off-week days in order to keep the calendar year synchronized with the seasonal year. Those are the “Worldsday" which is added at the end of the fourth quarter and the “Leapyear Day" which is added at the end of the second quarter in leap years. They are intended to be treated as holidays that are not assigned weekday designations.

The World Calendar has the following benefits over the current Gregorian calendar.

• The World Calendar is perpetual, with each day assigned an exact, repetitive date relative to week and month. Therefore we do not need to make a new calendar every year. We can use the same template of the calendar every quarter.
• The schedule of regular events does not need to be changed in relation to holidays or Sundays whose relation to the specific date of the events can change in the current Gregorian calendar. As the date of the events can be fixed, it is easy to remember.
• The current quarters are 90, 91, 92, 92 day long. There is a 2 day long gap between the longest and the shortest. The World Calendar reduces it to zero or at most 1 if off-week days are included. So, the result of quarterly settlement can be more evenly compared.
• There is a 3 day long gap between the longest month and the shortest one. The World Calendar reduces it to 1 even if off-week days are included. So, the monthly statistics can be more evenly compared. As weekdays in every month is 26 days, a monthly wage is paid for the same working hours[17].

The World Calendar was not zero-based. My proposal to reform the World Calendar is to make it zero-based and turn day numbers into consecutive numbers as the table below shows. The number becomes big but manageable because they are all two digits.

The zero-based World Calendar I propose has the following additional benefits.

• You can know the day and week numbers by dividing a day by 7. For example, since 73=7×10+3, the day 73 is the 3rd day, namely Wednesday, of the 10th week. In the current Gregorian calendar you must convert a day into a Julian Day to know what day of the week it is.
• The zero-based numbering system of the day, the day of the week, the week and the month enables us to use the number as the coordinate value and apply the function whose variable is time. I will give a simple example later.
• Friday the 13th is considered an unlucky day in Western superstition. The World Calendar created by Achelis lets it appear four times a year but this is not the case with mine. Mine has days numbered 13 but no weeks numbered 13.
• All the last days of months are multiples of 30, namely day 30, day 60 and day 90. Those in the current calendar are not fixed and therefore difficult to remember. As the last day of a month is often assigned as the deadline of contracts, it should be fixed and easy to remember.

If we convert day numbers into consecutive numbers, months become less important and quarters get more important. Just as we can give popular names to the days of the week, Sunday to the 0th day, Monday to the 1st day and so on, we can give popular names to quarters and the most popular names would be those of four seasons. Astrologically equinoxes and solstices divide the four seasonal quarters. A solstice occurs twice each year as the Sun reaches its highest or lowest excursion relative to the celestial sphere. Because the peak and the bottom of temperature are delayed by more than a month, the period from a solstice to an equinox can be appropriately called summer or winter, whether it is the northern or the southern hemisphere.

It differs from culture to culture when a year starts. In Egypt, the oldest civilization that adopted solar calendar, a year originally started in the middle of July when the inundation of the Nile began. The New Year’s Day of Ethiopian calendar is September 7. The lunisolar calendar of Babylonia starts on the new moon after the winter solstice. The first day of the year of Romulus, Iranian and Hindu calendar was around the vernal equinox. The beginning of the year of ancient China and Maya was originally the winter solstice.

To minimize the amendment of the current Gregorian calendar, we should set the starting point at the winter solstice. The winter solstice is usually December 21 or 22. Let’s set December 21 as the World Day and the next day as the 0th day of the 0th month. So the correspondence of the quarters of the zero-based World Calendar to the Gregorian dates is as follows.

The name of the quarter Winter (Oth quarter) The dates of the World Calendar The dates of the Gregorian Calendar 0/0 – 2/90 12/22 – 3/22 3/0 – 5/90 3/23 – 8/21 6/0 – 8/90 6/22 – 9/20 9/0 – 11/90 9/21 – 12/20

Although the winter solstice is 0/0, the vernal equinox (3/20-21), the summer solstice (6/20-21) and the autumnal equinox (9/22-23) are not exactly the 0th day of each quarter, because the revolutionary orbit of the Earth is an ellipse and owing to the Kepler’s second law, the period from the vernal equinox to the autumnal equinox which passes the aphelion is several days longer than the period from the autumnal equinox to the vernal equinox which passes the perihelion. They are, however, not so far from the boundaries of the quarters and the quarters are not far from our feelings of seasons.

What popular names to give to months should be decided locally, while the serial numbers should be common globally. In English the following proper names are given to months.

• 1st month: January
• 2nd month: February
• 3rd month: March
• 4th month: April
• 5th month: May
• 6th month: June
• 7th month: July
• 8th month: August
• 9th month: September
• 10th month: October
• 11th month: November
• 12th month: December

There were only 10 months in Romulus calendar, the prototype of the Gregorian calendar and the names were shifted from the number by two.

• 1st month: Martius
• 2nd month: Aprīlis
• 3rd month: Māius
• 4th month: Jūnius
• 5th month: Quīntīlis
• 6th month: Sextīlis
• 7th month: September
• 8th month: Octōber
• 9th month: November
• 10th month: December

The names of the first four months were named in honour of Greek or Roman gods: Martius in honour of Mars; Aprīlis in honour of Aphrodite; Māius in honour of not Greek Maia but an old agricultural god proper to Rome[18]; and Jūnius in honour of Juno. The names of the months from the fifth month on were based on their position in the calendar: Quīntīlis comes from Latin quinque meaning five; Sextīlis from sex meaning six; September from septem meaning seven; Octōber from octo meaning eight; November from novem meaning nine; and December from decem meaning ten.

There were no month between December and Martius, because they did not engage in agriculture in winter and did not need calendar during the period. As it became inconvenient, Numa Pompilius (753–673 BC; reigned 715–673 BC), the successor of Romulus, reformed the calendar of Romulus around 713 BC and added the following two months.

• 11th month: Ianuarius
• 12th month: Februārius

Later these two months became the first and the second month, January and February. Quīntīlis was renamed Julius (July) and Sextīlis Augustus (August).

Considering this history of Roman calendars, the following correspondence is faithful to the original.

• 0th month: February
• 1st month: March
• 2nd month: April
• 3rd month: May
• 4th month: June
• 5th month: July
• 6th month: August
• 7th month: September
• 8th month: October
• 9th month: November
• 10th month: December
• 11th month: January

January was named after Janus who is the god of beginnings and transitions, usually depicted as having two faces, since he looks to the future and to the past. So, his name is appropriate to the last month of the year that faces the past and the future. February was named after Februalia, the ancient spring festival to wash and purify the dead before the month of rebirth, Martius, today’s March. February is appropriate to the month starting on the winter solstice.

So much for names and let’s return to the rationality of the zero-based calendar. Now you can solve the following problem.

How many days are there from Tuesday of the week 4 in autumn 2050 to Friday of the week 10 in spring 2051? Ignore the error less than 1 day.

Autumn is the quarter 3, the day number of Tuesday of the week 4 is 4×7+2=30, spring is the quarter 1 and the day number of Friday of the week 10 is 10×7+5=75. Therefore the answer is 2051×365+91×1+75－(2050×365+91×3+30)＝365-91×2+45＝228.

In spite of this rationality many people would oppose to the World Calendars, whether the original or the zero-based. The main opponents of the World Calendars were adherents of a seven-day cycle, namely Jews, Christians and Muslims. The World Calendars count the intercalary days (Worldsdays and Leapyear Days) outside the usual seven-day week, which disrupts the traditional weekly cycle. They regard it as problematic that the religious day of rest would no longer coincide with the calendar weekend.

To be sure, the Bible says that on Mount Sinai God commanded Moses and his followers to keep the Sabbath (Shabbat) holy. Here is the fourth commandment.

Remember the Sabbath day, in its holiness. Six days you will work, and you will make all your craft. And the seventh day, rest for Yahweh your God. You will not do any craft, you and your sons and your daughters, and your slave, and your slave-woman, and your beasts, and the stranger who is within your gates. Because six days did Yahweh make the skies and the Earth, and the seas and all that is within them, and he rested on the seventh day. Because of this, Yahweh blessed the Sabbath day and made it holy.[19]

The fourth Commandment told them to rest after working six days but did not mention what to do on the eighth day nor did it tell them to treat seven days as a week to repeat strictly. The Bible says that “on the seventh day God ended his work which he had made; and he rested on the seventh day from all his work which he had made[20]“, but did not say what God did on the eighth day or before the creation of the world. If God had destroyed the world on the eighth day, re-created it thereafter, took a rest on the fourteenth day and thus repeated the same thing every seven days, the custom of having a rest every seventh day could be justified. But the Bible does not say such a thing. Therefore the intercalary days are not against the fourth Commandment.

The word Sabbath originally derived from the Babylonian sabattum or the Sumerian sa-bat meaning “mid-rest". The Babylonians and the Sumerians celebrated every seventh day as a holy day, counting from the new moon. Since the rest day was only the 28th day, the original Sabbath was monthly rather than weekly. Their 29 or 30 day lunation cycle consisted of three 7 day weeks and the final 8 or 9 day week, breaking the continuous 7 day cycle. It was probably because of this Babylonian custom that the Bible did not command the strict repetition of the 7 day week.

Among Abrahamic religions, only Jews and Seventh-Day Adventists are adherent to Sabbath. Jesus Christ said to the Pharisees, “The Son of man is Lord of the Sabbath[21]. It means the Sabbath is for men and not men for the Sabbath.

Jews observe Saturday as Shabbat, but the Sabbath of Christians except Seventh-Day Adventists is Sunday because Emperor Constantine combined Christianity with Mithraism, the Persian cult of the sun, and prescribed the day for rest to Sunday, the day of the Sun, in AD 321[22]. So, the Christian custom and obligation of Sunday rest originates in the pagan religion. Whether the Sabbath is Saturday or Sunday, the World Calendars allow double holidays once or twice a year, but they do not destroy the religious tradition. Muslims perform the congregational prayer (ṣalāt al-jumuʿah; Arabic: صلاة الجمعة‎) in Mosques on Fridays. As they use their original lunar calendar to perform it, the amendment of the Gregorian calendar does not affect their religious ceremonies.

The zero-based World Calendar I propose here is the best calendar that I can think of. A rational calendar system does not necessarily come into wide use because it is rational any more than Esperanto, the most rational artificial language, replaces the status of English as lingua franca. De facto standard, once locked in, can hardly be changed. Setting aside whether it is possible or not, when is the best time to switch from the Gregorian calendar to the zero-based World Calendar? The winter solstice in 2013 (22 December) is Sunday. If we set 22 December, 2013 as day 0 month 0 of 2014, we can seamlessly proceed to the zero-based World Calendar.

## 6. References

1. In Europe, not only 0 but also 1 was not treated as a number. The number 1 stands for a unit and the number was limited to a plurality of units. It was treated as a number for the first time in the work by Simon Stevin (1548 – 1620) in 1585, L’Arithmétique.
2. Αριστοτέλης. Φυσικής Ακροάσεως. Βιβλίο 4, Κεφάλαιο 6-9.
3. “Ἀριστόξενος δ’ ἐν τοῖς Ἱστορικοῖς ὑπομνήμασί φησι Πλάτωνα θελῆσαι συμφλέξαι τὰ Δημοκρίτου συγγράμματα, ὁπόσα ἐδυνήθη συναγαγεῖν, Ἀμύκλαν δὲ καὶ Κλεινίαν τοὺς Πυθαγορικοὺς κωλῦσαι αὐτόν, ὡς οὐδὲν ὄφελος. παρὰ πολλοῖς γὰρ εἶναι ἤδη τὰ βιβλία." Διογένης Λαέρτιος. Βίοι καὶ γνῶμαι τῶν ἐν φιλοσοφίᾳ εὐδοκιμησάντων Βιβλίον Θ’, 40.
4. Brahmagupta. Brahmasphotasiddhaantasya. Nāga Pabliśarsa; 1. saṃskaraṇa edition (1999). §. 18.34.
5. 林隆夫. 『インドの数学―ゼロの発明』中央公論社 (October 1, 1993). p. 41-43.
6. “道沖而用之 有弗盈也 潚呵 始萬物之宗" 老子. 『道德經』帛書校勘版. 第四十八章.
7. “正負術曰：同名相除，異名相益，正無入負之，負無入正之" 『九章算術』8. 方程, 3.4. 中國哲學書電子化計劃.
8. RTPking. “Numbers in languages like, Sanskrit, Telugu, English and Urdu“. 5 December 2012. Licensed under CC-BY-SA.
9. Ian Hacking. The Emergence of Probability. Cambridge University Press (August 31, 1984). p. 8.
10. Kenneth E. Iverson. A Programming Language. John Wiley & Sons (January 1962). p. 12.
11. Omegatron. “Graph of the ceiling function." 11 April 2006. Licensed under CC-BY-SA.
12. Omegatron. “Graph of the floor function." 30 March 2006. Licensed under CC-BY-SA.
13. The International Organization for Standardization. “ISO 8601:2004(E)." 4.1.3.3. Third edition 2004-12-01.
14. Cormullion. “A clock made in Revolutionary France, showing the 10-hour decimal clock“. Licensed under CC-BY-SA.
15. Rama. “Clock dial of the French Revolution." Licensed under CC-0.
16. Joseph Needham, Ling Wang, Derek J. De Solla Price. Heavenly Clockwork: The Great Astronomical Clocks of Medieval China. Cambridge University Press; 2 edition (November 28, 1986). p. 199.
17. When Elisabeth Achelis proposed it, the five-day workweek system was not common. If Saturday is counted as a day off or holidays are inserted unevenly, the monthly working hours are not even.
18. 岡田 芳朗, 後藤 晶男, 伊東 和彦, 松井 吉昭. 『暦を知る事典』東京堂出版 (May 1, 2006). p. 28-29.
19. “זכור את־יום השבת לקדשו ׃ ששת ימים תעבד ועשית כל־מלאכתך ׃ ויום השביעי שבת ׀ ליהוה אלהיך לא־תעשה כל־מלאכה אתה ׀ובנך־ובתך עבדך ואמתך ובהמתך וגרך אשר בשעריך ׃ כי ששת־ימים עשה יהוה את־השמים ואת־הארץ את־הים ואת־כל־אשר־בם וינח ביום השביעי על־כן ברך יהוה את־יום השבת ויקדשהו׃ ס" The Old Testament. Exodus. Chapter. 20. Verse. 8-11.
20. The Old Testament. Genesis. Chapter. 2. Verse. 2.
21. “κύριος ἐστιν τοῦ σαββάτου ὁ υἱὸς τοῦ ἀνθρώπου." The New Testament. Luke. Chapter. 6. Verse. 5.
22. Eviatar Zerubavel. The Seven Day Circle: The History and Meaning of the Week. University of Chicago Press; Reprint edition (March 15, 1989). p. 45.