# Decimal time

Decimal time is the representation of the time of day using units which are decimally related. This term is often used specifically to refer to the time system used in France for a few years beginning in 1792 during the French Revolution, which divided the day into 10 decimal hours, each decimal hour into 100 decimal minutes and each decimal minute into 100 decimal seconds, as opposed to the more familiar UTC time standard, which divides the day into 24 hours, each hour into 60 minutes and each minute into 60 seconds.

The main advantage of a decimal time system is that, since the base used to divide the time is the same as the one used to represent it, the whole time representation can be handled as a single string. Therefore, it becomes simpler to interpret a timestamp and to perform conversions. For instance, 1:23:00 is 1 decimal hour and 23 decimal minutes, or 1.23 hours, or 123 minutes; 3 hours is 300 minutes or 30,000 seconds. This property also makes it straightforward to represent a timestamp as a fractional day, so that 2019-05-21.534 can be interpreted as five decimal hours and 34 decimal minutes after the start of that day, or 0.534 (53.4%) through that day. It also adjusts well to digital time representation using epochs, in that the internal time representation can be used directly both for computation and for user-facing display.

GMT at page generation ()
24-h time Decimal time
13:35:24 5:66:25
5h 66m 25s
0.56625d
French decimal clock from the time of the French Revolution. The large dial shows the ten hours of the decimal day in Arabic numerals, while the small dial shows the two 12-hour periods of the standard 24-hour day in Roman numerals.
Decimal Time clock reading 2.50 DT equivalent to 06:00 standard time

## History

### China

Decimal time was used in China throughout most of its history alongside duodecimal time. The midnight-to-midnight day was divided both into 12 double hours (traditional Chinese: 時辰; simplified Chinese: 时辰; pinyin: shí chén) and also into 10 shi / 100 ke (Chinese: ; pinyin: ) by the 1st millennium BC.[1][2] Other numbers of ke per day were used during three short periods: 120 ke from 5–3 BC, 96 ke from 507–544 CE, and 108 ke from 544–565. Several of the roughly 50 Chinese calendars also divided each ke into 100 fen, although others divided each ke into 60 fen. In 1280, the Shoushi (Season Granting) calendar further subdivided each fen into 100 miao, creating a complete decimal time system of 100 ke, 100 fen and 100 miao.[3] Chinese decimal time ceased to be used in 1645 when the Shixian (Constant Conformity) calendar, based on European astronomy and brought to China by the Jesuits, adopted 96 ke per day alongside 12 double hours, making each ke exactly one-quarter hour.[4]

### France

Astronomical table from the Almanach national de France using decimal time

In 1754, Jean le Rond d'Alembert wrote in the Encyclopédie:

It would be very desirable that all divisions, for example of the livre, the sou, the toise, the day, the hour, etc. would be from tens into tens. This division would result in much easier and more convenient calculations and would be very preferable to the arbitrary division of the livre into twenty sous, of the sou into twelve deniers, of the day into twenty-four hours, the hour into sixty minutes, etc.[5][6]

In 1788, Claude Boniface Collignon proposed dividing the day into 10 hours or 1000 minutes, each new hour into 100 minutes, each new minute into 1000 seconds, and each new second into 1000 tierces (Latin for "third"). The distance the twilight zone travels in one such tierce at the equator, which would be one-billionth of the circumference of the earth, would be a new unit of length, provisionally called a half-handbreadth, equal to four modern centimetres. Further, the new tierce would be divided into 1000 quatierces, which he called "microscopic points of time". He also suggested a week of 10 days and dividing the year into 10 "solar months".[7]

Decimal time was officially introduced during the French Revolution. Jean-Charles de Borda made a proposal for decimal time on November 5, 1792. The National Convention issued a decree on 5 October 1793:

XI. Le jour, de minuit à minuit, est divisé en dix parties, chaque partie en dix autres, ainsi de suite jusqu’à la plus petite portion commensurable de la durée.
XI. The day, from midnight to midnight, is divided into ten parts, each part into ten others, and so forth until the smallest measurable portion of duration.

These parts were named on 24 November 1793 (4 Frimaire of the Year II). The primary divisions were called hours, and they added:

La centième partie de l'heure est appelée minute décimale; la centième partie de la minute est appelée seconde décimale. (emphasis in original)
The hundredth part of the hour is called decimal minute; the hundredth part of the minute is called decimal second.

Thus, midnight was called either dix heures ("ten hours") or zero heures, noon was called cinq heures, etc. Units were either written out or abbreviated, such as 8 h. 72 m. Sometimes in official records, decimal hours were divided into tenths, or décimes, instead of minutes (e.g., 8.7 h.).[8][9] Although clocks and watches were produced with faces showing both standard time with numbers 1–24 and decimal time with numbers 1–10, decimal time never caught on; it was not officially used until the beginning of the Republican year III, 22 September 1794, and mandatory use was suspended 7 April 1795 (18 Germinal of the Year III), in the same law which introduced the original metric system. Thus, although decimal time is sometimes referred to as metric time, the metric system at first had no time unit, and later versions of the metric system used the second, equal to 1/86400 day, as the metric time unit. In spite of this, decimal time was used in many cities, including Marseille and Toulouse, where a decimal clock with just an hour hand was on the front of the Capitole for five years.[8] On the Palace of the Tuileries in Paris, two of the four clock faces displayed decimal time until at least 1801.[10] The mathematician and astronomer Pierre-Simon Laplace had a decimal watch made for him, and used decimal time in his work, in the form of fractional days. Because of this, astronomers still use decimal time.

Decimal time was part of a larger attempt at decimalisation in revolutionary France (which also included decimalisation of currency and metrication) and was introduced as part of the French Republican Calendar, which, in addition to decimally dividing the day, divided the month into three décades of 10 days each; this calendar was abolished at the end of 1805. The start of each year was determined according to the day of the autumnal equinox, in relation to true or apparent solar time at the Paris Observatory. Decimal time would also have been reckoned according to apparent solar time, depending on the location it was observed, as was already the practice generally for the setting of clocks. 13:35:24 GMT is 5h 72m 74s decimal time in Paris.

At the International Meridian Conference of 1884, the following resolution was proposed by the French delegation and passed nem con (with 3 abstentions):

VII. That the Conference expresses the hope that the technical studies designed to regulate and extend the application of the decimal system to the division of angular space and of time shall be resumed, so as to permit the extension of this application to all cases in which it presents real advantages.

In the 1890s, Joseph Charles François de Rey-Pailhade, president of the Toulouse Geographical Society, proposed dividing the day into 100 parts, called cés, equal to 14.4 standard minutes, and each divided into 10 decicés, 100 centicés, etc. The Toulouse Chamber of Commerce adopted a resolution supporting his proposal in April 1897. Although widely published, the proposal received little backing.[11]

The French made another attempt at the decimalization of time in 1897, when the Commission de décimalisation du temps was created by the Bureau des Longitudes, with the mathematician Henri Poincaré as secretary. The commission adopted a compromise, originally proposed by Henri de Sarrauton of the Oran Geographical Society, of retaining the 24-hour day, but dividing each hour into 100 decimal minutes, and each minute into 100 seconds. The plan did not gain acceptance and was abandoned in 1900.

### Switzerland

On 23 October 1998, the Swiss watch company Swatch introduced a decimal time called Internet Time, which divides the day into 1,000 decimal minutes (Swatch called them .beats), (each 86.4 seconds in standard time) counted from 000–999, with @000 being midnight and @500 being noon standard time in Switzerland, which is Central European Time (one hour ahead of Universal Time). A line painted on Swatch headquarters in the Swiss city of Biel (now Biel/Bienne) was declared to mark the Biel Meridian, and Central European Time was relabelled as "Biel Meantime" (BMT), even though it does not correspond to actual local mean time in Biel.[12]

## Conversions

There are exactly 86,400 standard seconds (see SI for the current definition of the standard second) in a standard day, but in the French decimal time system there were 100,000 decimal seconds in the day, so the decimal second was shorter than its counterpart.

Decimal unit Seconds Minutes Hours h:mm:ss.sss
Decimal second 0.864 0.0144 0.00024 0:00:00.864
Decimal minute 86.4 1.44 0.024 0:01:26.400
Decimal hour 8,640 144 2.4 2:24:00.000

## Decimal hours

Another common type of decimal time is decimal hours. In 1896, Henri de Sarrauton of the Oran Geographical Society proposed dividing the 24 hours of the day each into 100 decimal minutes, and each minute into 100 decimal seconds.[13] Although endorsed by the Bureau des Longitudes, this proposal failed, but using decimal fractions of an hour to represent the time of day instead of minutes has become common.

Decimal hours are frequently used in accounting for payrolls and hourly billing. Time clocks typically record the time of day in tenths or hundredths of an hour. For instance, 08:30 would be recorded as 08.50. This is intended to make accounting easier by eliminating the need to convert between minutes and hours.

For aviation purposes, where it is common to add times in an already complicated environment, time tracking is simplified by recording decimal fractions of hours. For instance, instead of adding 1:36 to 2:36, getting 3:72 and converting it to 4:12, one would add 1.6 to 2.6 and get 4.2 hours.[14]

## Fractional days

The time of day is sometimes represented as a decimal fraction of a day in science and computers. Standard 24-hour time is converted into a fractional day by dividing the number of hours elapsed since midnight by 24 to make a decimal fraction. Thus, midnight is 0.0 day, noon is 0.5 d, etc., which can be added to any type of date, including (all of which refer to the same moment):

As many decimal places may be used as required for precision, so 0.5 d = 0.500000 d. Fractional days are often calculated in UTC or TT, although Julian Dates use pre-1925 astronomical date/time (each date began at noon = ".0") and Microsoft Excel uses the local time zone of the computer. Using fractional days reduces the number of units in time calculations from four (days, hours, minutes, seconds) to just one (days).

Fractional days are often used by astronomers to record observations, and were expressed in relation to Paris Mean Time by the 18th century French mathematician and astronomer Pierre-Simon Laplace, as in these examples:[15]

... et la distance périhélie, égale à 1,053095 ; ce qui a donné pour l'instant du passage au périhélie, sept.29j,10239, temps moyen compté de minuit à Paris. Les valeurs précédentes de a, b, h, l, relatives à trois observations, ont donné la distance périhélie égale à 1,053650; et pour l'instant du passage, sept.29j,04587; ce qui diffère peu des résultats fondés sur cinq observations.

— Pierre-Simon Laplace, Traité de Mécanique Céleste

Fractional days have been used by astronomers ever since. For instance, the 19th century British astronomer John Herschel gave these examples:[16]

Between Greenwich noon of the 22d and 23d of March, 1829, the 1828th equinoctial year terminates, and the 1829th commences. This happens at 0d·286003, or at 6h 51m 50s·66 Greenwich Mean Time ... For example, at 12h 0m 0s Greenwich Mean Time, or 0d·500000...

— John Herschel, Outlines of Astronomy

Fractional days are commonly used to express epochs of orbital elements. The decimal fraction is usually added to the calendar date or Julian day for natural objects, or to the ordinal date for artificial satellites in two-line elements.

## Decimal multiples and fractions of the second

The second is the International System of Units (SI) unit of time duration. It is also the standard single-unit time representation in many programming languages, most notably C, and part of UNIX/POSIX standards used by Linux, Mac OS X, etc.; to convert fractional days to fractional seconds, multiply the number by 86400. Fractional seconds are represented as milliseconds (ms), microseconds (μs) or nanoseconds (ns). Absolute times are usually represented relative to 1 January 1970, at midnight. Other systems may use a different zero point (like Unix time).

In principle, time spans greater than one second are given in units such as kiloseconds (ks), myriasecond (mys), megaseconds (Ms), gigaseconds (Gs), and so on. (The myriasecond is based on the myria- prefix, which represented a multiple of 10000 but was made obsolete in the mid-20th century.) Occasionally, these units can be found in technical literature, but traditional units like minutes, hours, days and years are much more common, and are accepted for use with SI.

It is possible to specify the time of day as the number of kiloseconds or myriaseconds elapsed since midnight. Thus, instead of saying 3:45 p.m. one could say (time of day) 56.7 ks. There are exactly 86.4 ks or 8.64 mys in one day. However, this nomenclature is rarely used in practice.

## Scientific decimal time

Scientists often record (rather than observe) time as decimals. For example, decimal days divides the day into 10 equal parts, and decimal years divides the year into 10 equal parts. Decimals are easier to plot than both (a) minutes and seconds, which uses the Sexagesimal numbering system, (b) hours, months and days, which has irregular month lengths. In astronomy, the so-called Julian day uses decimal days centered on noon.

Seconds in a decimal minute

Since there are 60 seconds in a minute, a tenth part represents 60/10 = 6 seconds.

 Decimal minutes Second 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 6s 12s 18s 24s 30s 36s 42s 48s 54s 60s
Minutes in a decimal hour

Since there are 60 minutes in an hour, a tenth part represents 60/10 = 6 minutes.

 Decimal hours Minutes 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 6m 12m 18m 24m 30m 36m 42m 48m 54m 60m
Hours in a decimal day

Since there are 24 hours in a day, a tenth part represents 24/10 = 2.4 hours (2 hours and 24 minutes).

 Decimal days Hours/minutes 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2h 24m 4h 48m 7h 12m 9h 36m 12h 14h 24m 16h 48m 19h 12m 21h 36m 24h
Length of a decimal year

Since there are about 365 days in a year, there are about 365/10 = 36.5 days in a tenth of a year. Hence the year 2000.5 represents the day 2 July 2000.[17] More exactly, a Julian year is exactly 365.25 days long, so a tenth of the year is 36.525 days (36 days, 12 hours, 36 minutes).

 Decimal years Days DateTime 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 36.525 73.050 109.575 146.100 182.625 219.150 255.675 292.200 328.725 365.250 1 Jan0:00 6 Feb12:36 15 Mar1:12 20 Apr13:48 27 May2:24 2 Jul15:00 8 Aug3:36 13 Sep16:12 20 Oct4:48 25 Nov17:24 1 Jan6:00

These values, based on the Julian year, are most likely to be those used in astronomy and related sciences. A Gregorian year, which takes into account the 100/400 leap year rules of the Gregorian calendar, is 365.2425 days (the average length of a year over a 400-year cycle), resulting in 0.1 years being a period of 36.52425 days (3155695.2 seconds; 36 days, 12 hours, 34 minutes, 55.2 seconds).

## Other decimal times

Numerous individuals have proposed variations of decimal time, dividing the day into different numbers of units and subunits with different names. Most are based upon fractional days, so that one decimal time format may be easily converted into another, such that all the following are equivalent:

• 0.500 fractional day
• 5h 0m French decimal time
• @500.beats Swatch Internet Time (see below)
• 50.0 kes or cés (centidays)
• 500 millidays
• 50.0% time as a percentage
• 12:00 standard time

Some decimal time proposals are based upon alternate units of metric time. The difference between metric time and decimal time is that metric time defines units for measuring time interval, as measured with a stopwatch, and decimal time defines the time of day, as measured by a clock. Just as standard time uses the metric time unit of the second as its basis, proposed decimal time scales may use alternative metric units.