Time Calculator

Three calculators sit above this. The first adds or subtracts two spans written in days, hours, minutes and seconds, and reports the total in every unit from years down to seconds. The second shifts a real date and time forward or back and shows the answer on a calendar. The third takes a whole expression such as 1d 2h 3m 4s + 4h 5s - 2030s and works through it in one go.

Why time arithmetic is not decimal arithmetic

Adding 50 minutes to 20 minutes gives 70 minutes, which is 1 hour 10 minutes rather than anything a decimal column would produce. Time is a mixed-radix system: seconds and minutes carry at 60, hours at 24, days at 7 for weeks, and months at nothing consistent at all. That is why a spreadsheet holding 1.5 hours as 1.5 is fine, while one holding 1 hour 30 minutes as 1.30 quietly breaks every sum it touches.

The calculators above convert everything to seconds, do the arithmetic there, and convert back. Doing it in a single base is the only way to keep the carries honest.

Common units of time

Unit	Definition
millennium	1,000 years
century	100 years
decade	10 years
year (average)	365.242 days, or 12 months
common year	365 days, or 12 months
leap year	366 days, or 12 months
quarter	3 months
month	28-31 days. Jan, Mar, May, Jul, Aug, Oct, Dec have 31; Apr, Jun, Sep, Nov have 30; Feb has 28, or 29 in a leap year
week	7 days
day	24 hours, or 1,440 minutes, or 86,400 seconds
hour	60 minutes, or 3,600 seconds
minute	60 seconds
second	the base unit
millisecond	10−3 second
microsecond	10−6 second
nanosecond	10−9 second
picosecond	10−12 second

Everything down to a week is exact. Months and years are not, which is why the first calculator labels those two lines as approximate and states what it assumed: one month as 30.4375 days and one year as 365.25. Those two agree with each other, since 365.25 divided by 12 is exactly 30.4375.

Why 60

Clocks count in sixties because the Sumerians did, in the third millennium BC, and the Babylonians carried the system forward. It survives in time, in angles, and in the way latitude and longitude are written.

Sixty earns its place. It has twelve divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. That means an hour splits cleanly into halves, thirds, quarters, fifths, sixths, tenths, twelfths, fifteenths, twentieths and thirtieths without a fraction appearing anywhere. Ten manages four divisors. For a civilisation doing arithmetic without decimals, that difference is the whole argument.

Where the 24-hour day came from

The Egyptians divided daylight into 12 parts using sundials, and since a sundial is useless after dark they used a set of 12 stars to mark the night. Two twelves side by side gave a 24-part day.

Those parts were not equal. Daylight hours stretched in summer and shrank in winter, because they always divided whatever daylight there was into twelve. Around 147 to 127 BC the Greek astronomer Hipparchus proposed fixing the hour at one twenty-fourth of an equinox day, when day and night are equal. These became the equinoctial hours, and they only came into everyday use in the 14th century, once mechanical clocks made it easier to keep a fixed hour than to track a varying one.

Hipparchus also worked with a 360-degree circle, later subdivided by Ptolemy into sixtieths and sixtieths again. The Latin names for those two subdivisions, pars minuta prima and pars minuta secunda, are where the words minute and second come from. The second is literally the second small division.

Calendars

Julius Caesar's calendar of 45 BC ran a 365-day year with an extra day every fourth year, making the average 365.25 days. The true tropical year is about 365.2422, so the Julian calendar ran roughly 11 minutes long each year, which is about a day every 128 years. By the 16th century the accumulated error had shifted the equinox by ten days.

The Gregorian reform of 1582 fixed it by dropping three leap years every four centuries: a century year is a leap year only when divisible by 400. So 2000 was a leap year and 1900 was not. That brings the average year to 365.2425, which is within about 26 seconds of the tropical year.

The second calculator above does calendar arithmetic properly, so adding 365 days across a leap year lands a day short of the same date, exactly as a real calendar behaves.

Measuring it

Early devices marked the passage of time rather than telling it. Oil lamps and candle clocks burned down at a known rate. The water clock, or clepsydra, was the most accurate instrument of the ancient world, timing the flow of water into or out of a vessel. Hourglasses appeared in the 14th century and were later used to calibrate specific intervals rather than to keep the time of day.

Christiaan Huygens built the first pendulum clock in 1656, the first regulated by something with a natural period of oscillation, and refined it to within about 10 seconds a day. That is roughly a hundred times better than anything before it.

Today the standard is atomic. Since 1967 the second has been defined as 9,192,631,770 periods of the radiation from a particular transition in caesium-133, which is why the second no longer depends on the Earth's rotation at all. The Earth is the part that drifts: leap seconds have been added to keep clock time in step with it, 27 of them between 1972 and 2016, and in 2022 the international community agreed to stop adding them by 2035.

What time actually is

Aristotle called time a number of movement in respect of the before and after, tying it to change: no change, nothing to count. Isaac Newton took the opposite view, treating absolute time as flowing uniformly regardless of anything happening in it. Gottfried Leibniz disagreed, arguing that time is only the order of events and has no existence apart from them.

Their dispute produced Newton's bucket. Water in a spinning bucket climbs the sides, and it keeps climbing after the bucket has stopped, so the curve cannot be caused by motion relative to the bucket. Newton concluded it must be motion relative to absolute space.

Einstein reframed the question. If the speed of light is the same for every observer, then time cannot be universal: clocks moving relative to one another tick at different rates, and space and time are one four-dimensional thing rather than two separate ones. General relativity answered the bucket too, replacing absolute space with the geometry of spacetime.

This is more than philosophy. GPS satellites carry atomic clocks in weaker gravity and at speed, so relative to the ground their clocks gain about 45 microseconds a day from gravity and lose about 7 from motion, a net gain near 38 microseconds. Left uncorrected the positional error would grow by roughly 10 kilometres a day. Every receiver you own depends on that correction being applied.

Reading your result

The first calculator gives the answer in every unit, and the ones down to weeks are exact. Take the month and year lines as approximate, because their length depends on which months and which years you mean. A stretch of 777 days is exactly 111 weeks, but calling it 25 months only holds on an average month.

Use the second calculator when the actual dates matter, since it counts real months and real leap years rather than averages. Use the third when you have several spans to combine, and mix the units freely: 2d 30h 90m is a valid input and normalises itself.

For the gap between two dates rather than a shift from one, use the Time Duration Calculator, and for hours worked across a week the Hours Calculator.