Draft

Dec Time

Introducing Declock, a timekeeping system that displays time in decimal days using math notation without the need for hours, minutes, or seconds.

Author

Martin Laptev

Published

2025+209

%%{init: {'theme': 'default', 'themeVariables': { 'fontSize': '32px'}}}%%
flowchart LR
   A[Dec]-->B[date]-->C[time]-->D[snap]-->E[span]
   click A "/dec"
   click B "/dec/date"
   click C "/dec/time"
   click D "/dec/snap"
   click E "/dec/span"

My website provides many examples of the Quarto publishing and the Dec measurement systems in action. I leverage Quarto support for the Observable data analysis and visualization system to create animated and interactive graphics like the bar📊charts clocks🕓, solar☀️terminator map🗺, Earth🌍orbit diagram, and daylight area chart below.

Dec times are measured in fractional days, often using metric prefixes like deci, centi, or milli. The top, middle, and bottom bars📊indicate the decidays, millidays, and centimillidays, respectively, of the time since the start, +, or until the end, -, of the day in the Dec time zone, , at the location of the red⭕️circle on the map🗺️beneath the bars📊.

To rotate the globe🌐in the Earth🌏orbit diagram, drag the red⭕️circle horizontally↔︎️or slide the red🔴dot on the daylight area chart vertically↕. The red⭕️circle’s vertical↕position determines the yearly daylight pattern visualized by the area chart. Shift the red—line on the area chart horizontally↔︎️to move the globe🌐along the ellipse of the Earth🌎orbit.

Bar chart clocks
Longitude latitude map
Daylight area chart
width
Yearly day aggregate (yda)

The red—line indicates a “day of year” (doy), , and the red🔴dot denotes a “time of day” (tod): . A doy identifies a day in a year like a Gregorian calendar month and “day of month” (dom). A tod specifies a point in a day like an “hour minute second” (hms) triplet. Together, a doy and tod can form a “yearly day aggregate” (yda): .

\[\text{yda} = \text{doy} + \text{tod}\]

\[\lfloor\text{yda}\rfloor = \text{doy}\]

\[\text{yda} - \text{doy} = \text{tod}\]

As their names suggest, doys and ydas are measured in days. The measurement unit of a tod can be a day or a submultiple of a day. By changing how a decimal tod is measured, we can shift its decimal separator or turn it into an integer. The tods along the y-axis of the area chart are integers because they have three digits and are measured in millidays.

Epochal day aggregate (eda)

We can replace an yda with a tod by keeping the remainder after dividing by one to isolate the decimal part of the quotient: mod 1 = . We can use this same approach to obtain a tod from an “epochal day aggregate” (eda): mod 1 = . The current eda tells us how many days have passed since the Dec epoch.

\[\text{tod} = \text{yda mod } 1 = \text{eda mod } 1\]

UNIX time equation

Similarly, UNIX time tallies the seconds since the UNIX epoch, which is exactly 719468 days after the Dec epoch. To get the tod in Zone 0, the Dec time zone that is in between the two leftmost vertical lines on the map🗺️, we can divide UNIX time by the number of seconds in a day and then keep the remainder after dividing the resulting days by one:

\[\text{tod} = \text{unix} \div 86400 \text{ mod } 1\]

Julian time equation

Julian dates track the days since the beginning of the Julian period and thus are akin to edas. We can produce a Zone 5 tod from a Julian date simply by keeping the remainder after dividing by one. If we want a Zone 0 tod instead, we should add 5 decidays to the Julian date before converting it to a tod to ensure that the final result is less than one day:

\[\text{tod} = (\text{julian} + 0.5) \text{ mod } 1\]

Hour minute second

We can also obtain a Zone 0 tod from a Coordinated Universal Time (UTC) hms triplet by summing its components after converting them to fractional days, as shown in the equation below. The computer programming code in the tabset panel beneath the equation compares tods derived from UTC and UNIX time as Quarto was rendering this webpage.

\[\text{tod} = \frac{\text{hour}}{24} + \frac{\text{minute}}{1440} + \frac{\text{second}}{86400}\]

using Dates
hms = now(UTC)
2025-09-26T02:29:05.001
hour(hms) / 24 +
minute(hms) / 1440 +
second(hms) / 86400 +
millisecond(hms) / 864e5
0.10353010416666666
datetime2unix(hms) / 86400 % 1
0.10353010416656616
hms = new Date();
hms.getUTCHours() / 24 +
hms.getUTCMinutes() / 1440 +
hms.getUTCSeconds() / 86400 +
hms.getUTCMilliseconds() / 864e5;
hms.getTime() / 864e5 % 1;
from datetime import datetime, timezone
hms = datetime.now(timezone.utc)
hms.hour / 24 + \
hms.minute / 1440 + \
hms.second / 86400 + \
hms.microsecond / 864e8
0.10355080748842592
hms.timestamp() / 86400 % 1
0.10355080748922774
hms <- as.POSIXlt(Sys.time(), tz = "UTC")
hms$hour / 24 +
hms$min / 1440 +
hms$sec / 86400
[1] 0.103551
(as.numeric(as.POSIXct(hms)) / 86400) %% 1
[1] 0.103551

The equations below convert UNIX time or a Zone 0 tod into the three components of an hms triplet: the “hour of day” (hod), “minute of hour” (moh), and “second of minute” (som), using a “daily second aggregate” (dsa) and “hourly second aggregate” (hsa). While both count seconds, dsas start at midnight and hsas begin at the top of the hour.

\[\text{dsa} = \text{tod} \times 86400 = \text{unix mod } 86400\]

\[\text{hsa} = \text{dsa mod } 3600\]

\[\text{hod} = \lfloor \text{dsa} \div 3600 \rfloor\]

\[\text{moh} = \lfloor \text{hsa} \div 60 \rfloor\]

\[\text{som} = \lfloor \text{hsa mod } 60 \rfloor\]

using Dates
dsa = datetime2unix(now(UTC)) / 86400 % 1 * 86400
8946.85700009577
hsa = dsa % 3600
1746.8570000957698
map(x -> floor(Int, x), (dsa / 3600, hsa / 60, hsa % 60))
(2, 29, 6)
dsa = Date.now() / 864e5 % 1 * 86400;
hsa = dsa % 3600;
[dsa / 3600, hsa / 60, hsa % 60].map(Math.floor)
from datetime import datetime, timezone
dsa = datetime.now(timezone.utc).timestamp() / 86400 % 1 * 86400
hsa = dsa % 3600
tuple(map(int, [dsa // 3600, hsa // 60, hsa % 60 // 1]))
(2, 29, 6)
dsa <- (as.numeric(as.POSIXct(Sys.time())) / 86400) %% 1 * 86400
hsa <- dsa %% 3600
sapply(c(dsa %/% 3600, hsa %/% 60, hsa %% 60), as.integer)
[1]  2 29  6
Universal time offset

The Global Positioning System, BeiDou, and Galileo global navigation satellite systems along with most — if not all — programming languages do not account for leap seconds, which appears to be for the best given that leap seconds will be abolished by 2035. The goal of leap seconds is to keep UTC within 25/24 centimillidays (cmds) of Universal Time (UT).

Instead of leap seconds, Dec matches UT using a “universal time offset” (uto). With the leap second insertion dates provided by the International Earth Rotation and Reference Systems Service, we can approximate the uto that yields UT when added to the Zone 0 tod on the Dec date chosen by the range🎚️inputs below: ÷ 8640 = .

Rounded offset decimal

Of the twenty eight utos that can be shown in the equation above, one is an integer, one is a terminating decimal, and the rest are repeating decimals. To express a repeating decimal uto, Dec uses an irreducible fraction that is called an “exact offset fraction” (eof) when by itself or a “rounding error fraction” (ref) if it follows a “rounded offset decimal” (rod).

In the equation below, the uto is the minuend, the rod is the subtrahend, and the ref is the difference. Dec uses the term minuend expansion to describe the replacement of a minuend with a subtrahend and a difference. By replacing a repeating decimal uto with a rod and a ref, we can show the initial digits of the uto as a decimal and the rest as a fraction.

\[\text{uto} - \text{rod} = \text{ref}\]

Use the first three range🎚️inputs below to select an hms triplet to be converted to decidays, plugged into the equation above as the uto, rounded to the number of digits chosen by the fourth range🎚️input, and inserted into the equation as the rod. Once the left-hand side of the equation is complete, we can solve it to get the ref: – = .

Time zone offset

In Dec, a uto can be any type of number, a “time zone offset” (tzo) is an integer, a “solar time offset” (sto) is a terminating decimal, and an eof is a repeating decimal. If we derived deciday offsets from all 86400 of the hms triplets that can be selected by the range🎚️inputs above, we would have 10 tzos, 3190 stos, and 83200 eofs or rod and ref pairs.

Coordinated Universal Time (UTC)

When we do the same to the 38 UTC offsets, we get only eofs or rod and ref pairs unless the number of leaps seconds included is zero or a multiple of 27. If the leap second count is zero or a multiple of 8640, we will get 3 tzos, 9 stos, and 26 eofs or rod and ref pairs. The 3 tzos will be stos if the number of leap seconds is a multiple of 27 but not 8640.

There are 14 negative and 24 positive UTC offsets. The UTC time zone with the most negative offset is completely uninhabited. The bar📊chart below visualizes Socioeconomic Data and Applications Center data from 2020 regarding the population of each UTC time zone. The vast majority of all people live in UTC time zones with positive offsets.

Negative UTC offsets only exist in the Americas and islands in the Atlantic and Pacific Oceans. Therefore, the bar📊chart above is essentially comparing the Americas to the rest of the world. According to 2021 United Nations Department of Economic and Social Affairs data, about one billion out of a total of almost eight billion people live in the Americas.

Whenever a negative offset is associated with a Dec date, a tod, or both a date and a tod, Dec will add one day to the date and ten decidays to the offset without modifying the tod. This typically occurs after the conversion of an hms triplet to a tod or a “year month day” (ymd) triplet to a Dec date. As a result, all Dec dates and tods have positive offsets.

Dec will not change a negative offset or its associated doy if the result of adding one day to the doy is uncertain. This uncertainly can only exist if we do not know whether a doy that is equal to 364 belongs to a common or leap year. The day after Day 364 of a common year is Day 0 of the subsequent year. In a leap year, Day 364 precedes Day 365.

Day of week

Even though it has no effect on the tod, adding one day to the doy also increments the “day of month” (dom) and “day of week” (dow) shown by Dec. The table below shows how someone accustomed to a negative offset could intrepret Dec dow numbers. From the perspective of a negative offset user, the dom and dow in Dec will be one day ahead.

Saturday Sunday Monday Tuesday Wednesday Thursday Friday
0 1 2 3 4 5 6
-7 -6 -5 -4 -3 -2 -1

The one day difference between positive and negative offsets may make Dec dow numbers more intuitive than POSIX dow numbers for people who consider Sunday to be the first dow. According to the Common Locale Data Repository and a 2023 population ranking, over 89% of people in the Americas live in a country that starts the week on Sunday.

Longitude and offsets

In Dec, offsets are closely related to longitude. Dec measures longitude in parallels (λ) or submultiples of λ like deciparallels (dλ). A tzo is essentially a dλ longitude that had its decimal part removed via rounding, flooring, truncation, or ceiling. Whereas tzos have one digit, deciday tods and stos typically have up to four digits after the decimal separator.

The fourth digit in the decimal part of any current deciday tod increments 105 times per day, 100 times per milliday, or once per beat (iob), which is the lower bound of the normal resting heart rate of an adult. For everyday life, we should limit the length of tods to the three digits needed to show millidays (md) or the five digits required to display beats (b).

When the current tod has seven digits, the sixth digit changes too quickly to be read out loud and the seventh changes so fast that it appears as a blur. Near the Equator, a longitude that has seven digits is accurate to within about ten zems (z) or four meters, which is roughly the length of a subcompact car or the width of a U-shaped living room layout.

The Equator is approximately (~) 103 millitaurs (mc), ~4 × 104 kilometers (km), or ~105 kilozems (kz) long. If we are near the Equator and move 1 mc, 40 km, or 100 kz to the East or West, our sto will change by ~1 md, ~1.44 minutes, or ~100 b and our longitude will shift by ~0.36 degrees, ~1 mλ, ~21.6 arcminutes, or ~100 arcbeats (ab).

For precise geopositioning, it may be helpful to show geographic coordinates in submultiples of ab, but we are unlikely to benefit from units smaller than md when displaying the solar time or estimates of what the tod will be when the Sun rises, reaches its zenith, or sets on a given day. By default, Dec uses three digits to show each solar time and sto.

Equation of time

The two types of solar time are “mean solar time” (mst) and “apparent solar time” (ast). To calculate mst, we keep only the decimal part of the sum of 0.95, the Zone 0 tod measured in days, and our longitude measured in λ. If we want ast instead of mst, the sum needs to include the result of plugging the “time of year” (toy) into the equation of time (eot).

\[\text{toy} = \text{yda} \div \text{n}\]

\[\text{mst} = (0.95 + \text{tod} + \lambda) \text{ mod } 1\]

\[\text{ast} = (0.95 + \text{tod} + \lambda + \text{eot(toy)}) \text{ mod } 1\]

To obtain the toy, we divide the yda by the number of days in the year (n). If we use trigonometric functions that are designed to work with radians, we will have to multiply the toy by \(2\pi\) or \(\tau\). We do not need to modify the toy before passing it to the eot() function defined below because its trigonometric functions expect turns instead of radians.

\[\begin{split} \text{eot(toy)} & = \beta_0 \\ & + \beta_1 \times \text{costau(toy)} \\ & + \beta_2 \times \text{costau(2} \times \text{toy)} \\ & + \beta_3 \times \text{sintau(toy)} \\ & + \beta_4 \times \text{sintau(2} \times \text{toy)} \end{split}\]

sincostau(turn) = sincospi(2 * turn)
sincostau (generic function with 1 method)
sincostau(0.25)
(1.0, 0.0)
sincostau(0.5)
(0.0, -1.0)
sincos(0.25 * 2 * pi)
(1.0, 6.123233995736766e-17)
sincos(0.5 * 2 * pi)
(1.2246467991473532e-16, -1.0)
sincospi = require(
  'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-sincospi@umd/browser.js'
  )
function sincostau(turn) {
  return sincospi(2 * turn);
}
function sincos(turn) {
  turn *= 2 * Math.PI
  return [Math.sin(turn), Math.cos(turn)]
}
sincostau(0.25)
sincostau(0.5)
sincos(0.25)
sincos(0.5)
from math import cos, sin, pi
from mpmath import cospi, sinpi

def sincostau(turn):
    turn *= 2
    return sinpi(turn), cospi(turn)

def sincos(turn):
    turn *= 2 * pi
    return sin(turn), cos(turn)

sincostau(0.25)
(mpf('1.0'), mpf('0.0'))
sincostau(0.5)
(mpf('0.0'), mpf('-1.0'))
sincos(0.25)
(1.0, 6.123233995736766e-17)
sincos(0.5)
(1.2246467991473532e-16, -1.0)
sincostau <- function(turn) {
  turn <- 2 * turn
  c(sinpi(turn), cospi(turn))
}

sincos <- function(turn) {
  turn <- 2 * turn * pi
  c(sin(turn), cos(turn))
}

sincostau(0.25)
[1] 1 0
sincostau(0.5)
[1]  0 -1
sincos(0.25)
[1] 1.000000e+00 6.123234e-17
sincos(0.5)
[1]  1.224647e-16 -1.000000e+00

The code above compares two functions which return both the sine and cosine of their input. The sincos() function assumes it will receive radians and the sincostau() works with turns. In general, working with turns is more convenient and intuitive. The examples above demonstrate that using turns instead of radians can yield more accurate results.

The values below are the coefficients of a model adapted from the National Oceanic and Atmospheric Administration (NOAA) General Solar Position Calculations. Before fitting the model to NOAA yearly solar data, we need to convert eot(toy) values from minutes to centidays, combine ymd triplet dates and hms triplet times into ydas, and sort by yda.

{0: 0.0011114386869002235,
 1: -4.155124400404918,
 2: -2.9710873225303756,
 3: -4.635570414590801,
 4: 5.116176940364264}

The line📈chart above uses md to display eot(toy) values as integers. There is little difference between mst and ast around Days 45, 103, 184, and 299. The difference ranges from about -9.8 on Day 244 to around 11.4 md on Day 350. We can calculate mst with just a tod and a longitude and we will be off by at most about a centiday compared to ast.

Apart from turning a mst into an ast, we can also use eot to more accurately estimate the tods of solar noon, sunrise, and sunset. The equation below creates a solar noon tod measured in days by adding 9.55 to a longitude measured in λ, subtracting a tzo and a eot(toy) value that are both measured in days, and keeping only the decimal part of the result.

\[\text{noon} = (9.55 + \text{tzo} - \lambda - \text{eot(toy)) mod 1}\]

Cambridge and Cambridge

To compare the solar noon tod in two cities, we can plug in the longitude of each city into the equation above. If all the variables in the equation other than longitude are set to zero, the result is almost five decidays for Cambridge, England in the United Kingdom and nearly seven decidays for Cambridge, Massachusetts in the United States.

\[4.99635 = (9.55 - 0.050365) \text{ mod 1} \times 10\]

\[6.97516 = (9.55 - 0.852484) \text{ mod 1} \times 10\]

The two homonymous cities are about two dλ apart and thus will always differ by around two decidays in solar time, regardless of what time zone we use as our frame of reference. England is in Zone 0 and Massachusetts is in Zone 8. If we change the tzo from zero to eight decidays, the solar noon tod for each city will be two decidays earlier.

\[2.99635 = (9.55 + 0.8 - 0.050365) \text{ mod 1} \times 10\]

\[4.97516 = (9.55 + 0.8 - 0.852484) \text{ mod 1} \times 10\]

Solar declination angle

The duration of daytime is the difference between the sunset and sunrise tods. To calculate the daytime duration, we only need a latitude and a “solar declination angle” (sda). For simplicity, we can use the same model for the eot and sda, even though the NOAA General Solar Position Calculations propose the more complex model shown below.

\[\begin{split} \text{sda(toy)} & = \beta_0 \\ & + \beta_1 \times \text{costau(toy)} \\ & + \beta_2 \times \text{sintau(toy)} \\ & + \beta_3 \times \text{costau(2} \times \text{toy)} \\ & + \beta_4 \times \text{sintau(2} \times \text{toy)} \\ & + \beta_5 \times \text{costau(3} \times \text{toy)} \\ & + \beta_6 \times \text{sintau(3} \times \text{toy)} \\ \end{split}\]

Dec measures latitude in turns called meridians (φ). Like λ, we can pass φ directly into trigonometric functions that accept turns. If the “solar zenith angle” (sza) in the equation below is 0.252314 turns, the equation will give us a “solar hour angle” (sha) that we can use to find out how long the Sun will shine or when it will rise or set on a given doy.

\[\text{sha} = \arccos\left( \frac{\text{costau(sza)} - \text{sintau($\phi$)} \times \text{sintau(sda)}}{\text{costau($\phi$)} \times \text{costau(sda)}} \right)\]

Before being plugged into the sha equation above, the mφ latitude selected by the first range input below is divided by 1000 to convert it to φ and the doy chosen by the second range input is turned into a toy and then into an sda. When the sza is 0.252314, we can divide the result of the sha equation by \(\pi\) to obtain the daytime duration: .

\[\text{sunset} - \text{sunrise} = \text{sha} \div \pi\]

daytimeDuration
latInput / 1000

Like other trigonometric functions, the arcosine function returns radians. We can divide by \(\tau\) to convert radians to turns. After converting the sha to turns, we can sum it with the solar noon to get the sunset tod or subtract it from the solar noon tod to obtain the sunrise tod. The sunrise and sunset tods are equidistant from the solar noon tod.

\[\text{sunset} = \text{noon} + \text{sha} \div \tau\]

\[\text{sunrise} = \text{noon} - \text{sha} \div \tau\]

Before we insert the latitude selected by the second range input, we convert mφ to φ.

\[\text{sha} \div \tau = \text{noon} - \text{sunrise} = \text{sunset} - \text{noon}\]

When we divide the sha by \(\tau\), we get the difference between the solar noon and sunrise tods or the sunset and solar noon tods.

Sunrise and sunset
Solar hour angle

The first range input above sets the doy that will be used to obtain the sda that along with the φ selected by the second range input will be plugged into the sha equation above. Once we have the sha, we can double it to get the daylight duration in negative days, add it to the solar noon tod to get the sunrise tod, or subtract it from the solar noon tod to get the sunset tod.

The difference between a solar noon and sunrise sha is a sunrise tod. Likewise, the sum of a solar noon and sunrise sha is a sunset tod. To get the daylight duration in decidays, we can either subtract the sunset tod from the sunrise tod or multiply the sunrise sha by twenty.

\[\text{sunrise} = \text{noon} + \text{sha}\]

\[\text{sunset} = \text{noon} - \text{sha}\]

\[\text{duration} = \text{sunset} - \text{sunrise} = \left|\text{sha} \times 2\right|\]

The sunrise and sunset times in the equations above depend on the time zone of solar noon. To select a time zone may be strange to consider The range🎚️inputs below set the geographic coordinates that are used for the equations above. The line chart beneath the inputs visualizes sunrise and sunset for every doy, similar to the daylight area chart beneath the map🗺above. The main difference between the charts is that the area chart shows local tods and thus is independent of longitude, whereas the line chart displays tzos and therefore changes with longitude.

To find the UTC tzo of a given longitude and latitude, we could use an application programming interface (API) or a database. If we only have longitude, we need to first round degrees to zero or the nearest multiple of fifteen for whole hour tzos, 7.5 for half hour tzos, or 3.75 for quarter hour tzos and then divide by fifteen to convert degrees to hours.

Next

The next article in the Dec section of my site shows how we can combine Dec dates and times into Dec snaps🫰, which are analogous to the combined date and time representations in the ISO 8601 international standard for dates and times. The final article in the Dec section demonstrates how Dec dates, times, and snaps🫰can be paired up to express time intervals called Dec spans🌈.

%%{init: {'theme': 'default', 'themeVariables': { 'fontSize': '32px'}}}%%
flowchart LR
   A[Dec]-->B[date]-->C[time]-->D[snap]-->E[span]
   click A "/dec"
   click B "/dec/date"
   click C "/dec/time"
   click D "/dec/snap"
   click E "/dec/span"

Cite

Please spread the good word about Dec using the citation information at the bottom of this article. You may also want to cite the Observable notebooks that I adapted into the clock🕓, bar📊chart, map🗺️, and daylight☀️plot visualizations in this article or the 2014 blog post which proposed a system of 20 decimal time zones, each 5 centidays wide, based on the Greenwich Meridian:

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Reuse

Citation

BibTeX citation:
@online{laptev2024,
  author = {Laptev, Martin},
  title = {Dec {Time}},
  date = {2024},
  urldate = {2024},
  url = {https://maptv.github.io/dec/time},
  langid = {en}
}
For attribution, please cite this work as:
Laptev, Martin. 2024. “Dec Time.” 2024. https://maptv.github.io/dec/time.