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+200

%%{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 analog clocks🕓, bar📊charts, solar☀️terminator map🗺, Earth🌍orbit diagram, and daylight area chart below.

Fractional day time

Dec times are measured in fractional days. The shortest, longest, and thinnest clock🕓hands and 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📊.

Ticking analog clocks

Bar chart clocks
Longitude latitude map
Daylight area chart

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.

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 turn an yda into 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)
2025-07-17T18:14:41.900
hour(hms) / 24 +
minute(hms) / 1440 +
second(hms) / 86400 +
millisecond(hms) / 864e5
0.7602071759259259
datetime2unix(hms) / 86400 % 1
0.7602071759283717
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
0.7602234699537037
hms.timestamp() / 86400 % 1
0.7602234699552355
hms <- as.POSIXlt(Sys.time(), tz = "UTC")
hms$hour / 24 +
hms$min / 1440 +
hms$sec / 86400
[1] 0.7602237
(as.numeric(as.POSIXct(hms)) / 86400) %% 1
[1] 0.7602237

The equations below turn 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
1024.2920000106096
hsa = dsa % 3600
1024.2920000106096
map(x -> floor(Int, x), (dsa / 3600, hsa / 60, hsa % 60))
(0, 17, 4)
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]))
(0, 17, 5)
dsa <- (as.numeric(as.POSIXct(Sys.time())) / 86400) %% 1 * 86400
hsa <- dsa %% 3600
sapply(c(dsa %/% 3600, hsa %/% 60, hsa %% 60), as.integer)
[1]  0 17  5
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 or the five digits required to display beats (b).

When the current tod is shown with seven digits, the sixth digit changes too quickly to be read out loud and 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 National Oceanic and Atmospheric Administration (NOAA) provides yearly and daily eot values. To create a Dec version of the eot, we can convert the yearly NOAA eot values from minutes to centidays, combine dates and times into ydas, sort by yda, fit a curve to the sorted values, and then use the fitted model coefficients in the NOAA General Solar Position Calculations:

{0: 0.00014949317633417003,
 1: -0.4164553809136968,
 2: -0.2959657573651431,
 3: -0.46578504463469605,
 4: 0.5096179567359089}

The line📈chart above uses centiday eot(toy) values to show fewer digits on its y-axis. For time calculations, Dec always uses units that do not have metric prefixes. To calculate the tod of solar noon, the moment when the sun reaches its highest point in the sky, we keep only the decimal part after subtracting a longitude measured in λ and an eot(toy) value from the sum of 9.55 and a tzo.

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

Solar declination angle

Once we have eot(toy), we can find the Zone 0 tod of solar noon by subtracting a longitude measured in λ and eot(toy) from 9.55, a constant which shifts . Before we can find the sunrise and sunset tod, we first need to calculate the solar declination angle (sda). For simplicity, we can use the same math for the eot and sda, even though the NOAA General Solar Position Calculations propose the more complex model shown below.

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

Solar hour angle

If the “solar zenith angle” (sza) in the equation below is 0.2523 turns, the equation will yield a sunrise “solar hour angle” (sha) that we can use to get a sunrise or sunset tod. Dec measures angles in turns. The trigonometric functions of most — if not all – programming languages require radians. To convert turns into radians, we multiply by \(\tau\).

\[\text{radian} = \text{turn} \times \tau = \text{degree} \times \tau \div 360\]

\[\text{turn} = \text{radian} \div \tau = \text{degree} \div 360\]

\[\text{degree} = \text{radian} \div \tau \div 360 = \text{turn} \times 360\]

\[\text{sha} = \arccos\left( \frac{\cos(\text{sza}\times\tau)}{\cos(\phi\times\tau) \cdot \cos(\text{sda}\times\tau)} - \tan(\text{$\phi\times\tau$}) \cdot \tan(\text{sda}\times\tau) \right)\]

In the equation above, latitude is measured in turns called meridians (\(\phi\)). 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.

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

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

\[\text{duration} = \text{sunset} - \text{sunrise} = \text{sha} \times 20\]

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 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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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.