The short answer to why we use it is that we inherited it - base 12 of hours/months from the Egyptians and base 60 of second and minutes from Mesopotamians (who got it from the Sumerians).

Egyptians used base 12 a lot for a similar reason that we use base 10 a lot. We use 10 because we have ten fingers, and they used 12 because one hand has 12 knuckles (they'd count on one hand). But it was handy because there are 12 lunar cycles, so it helped keep things more consistent.

Base 60 is also handy because 60 is first number divisible by the first six counting numbers and by 10, 12, 15, 20 and 30. If you use 60, you have options! Note that we also use 60 for angles and dividing up the globe.

The use of base-60 for time and base-10 for most other things originates from different historical and mathematical practices. Base-60, or the sexagesimal system, is thought to have originated from the ancient Sumerians and Babylonians who used it for their astronomical calculations. This system has 12 divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, allowing it to be easily divided into halves, thirds, quarters, fifths, sixths, tenths, twelfths, and more, without resulting in fractions. This versatility has made it useful for timekeeping and angle measurement, both of which continue to be used to this day.

Conversely, the base-10 system, or decimal system, likely derives from the fact that humans have 10 fingers, making counting in this system intuitive and easy to learn. The base-10 system has fewer divisors: 1, 2, 5, 10, meaning it can only be cleanly divided by 2 and 5. While this system is less flexible for representing certain fractions, it’s more straightforward for counting and calculation, leading to its wide use in daily life, mathematics, and scientific calculations.

The integration of base-60 and base-10 systems hasn’t been widely adopted due to the significant societal change required. During the French Revolution, a decimal timekeeping system was attempted, with a day consisting of 10 hours, each hour containing 100 minutes, and each minute made up of 100 seconds. However, this system lacked popularity and was soon discarded.

Despite this, decimal watches exist today. These watches follow the concept of decimal time, dividing a 24-hour day differently, with an “hour” on a decimal watch being longer than a traditional hour, a “minute” being longer than a traditional minute, and a “second” being slightly shorter than a traditional second. They are typically seen as novelty or specialty items, appreciated by those who prefer the mathematical simplicity of a base-10 system. But due to the wide prevalence and deep entrenchment of the base-60 system in societal, technological, and legal systems, traditional timekeeping remains dominant.

(Source: I saw it on QI once - a British pub quiz style tv show)

I really want a decimal watch, just as a talking piece.

But I’m sure it would be a nightmare for actually keeping track of time.

While it would he handy to have everything integrated, it's not always vital. We don't need to convert kilograms to seconds very often. Artificially enforced systems have always had trouble replacing older anachronistic ones that are otherwise still useful. Ask people in the UK about their cars' efficiency and you'll often get an answer in miles per gallon.

There's nothing magical about 10, either, other than the accidents of evolution that left us with ten fingers. Base 12 is also extremely convenient, and comes from Sumerians counting with their thumb against each of the three joints on 4 fingers. Go through that process once for each finger on the other hand, and you get 60. And of course, in any industry where things are packed into packages, like nearly everything we buy, dozens fit better than tens. 60 divides very neatly into many convenient and geometrically simple fractions, and a lot of what we do with circles benefits from this.

We probably would have been better standardizing on a base unit that's a power of two, which has more mathematical weight than ten does.

Decimal time was introduced in Revolutionary France, just like the metric system and a decimal calendar. They made it non-mandatory after 17 months, partly because of the enormous costs to replace all clocks. The decimal calendar survived longer and was abolished by Napoleon as part of his reconciliation strategy with the Catholic church.

Decimal time is still used by the way. Astronomers use fractional days because it's easier to do calculations with. And that very same ease of use is why Microsoft Excel uses fractional dates to calculate dates, as it requires less calculations.

Pilots keep their logbook in decimal hours, and the tach meter (engine hours at a certain speed) and Hobbs meter (aircraft operated hours) are decimal too.

Each decimal hour (0.1) is equal to 6 minutes. So if I 'fly' for 1 hour and 13 minutes I log 1.2 hours. I don't do the mental math though, I just record what the Hobbs says.

IIRC, people have tried to create, and enforce, metric time in the past but it always fails. Basically it seems like that ever since the Mesopotamians invented base 60 mathematics for keeping track of time, some geometry, and related concepts it is stuck for that purpose because it really does seem to be the best number set for the job. It did not stick around for anything else though because it's basically garbage for those tasks. Base 10 works pretty good; it's easy to move around zeroes and decimal places, and since most people have 10 fingers it's pretty intuitive.

I think it's because there are important, naturally occuring units of time that simply don't divide well - that is, the day and the year. Having it standardized to metric would still leave us with 1:365.24 conversion. Using metric time would require us to stop being metric beyond the day, or just have a cumbersome conversion number to talk about years.

On the other hand, things like weight, length, and temperature are completely arbitrary and there's no natural standard unit, so changing those to another completely arbitrary unit is easy.

The French tried doing decimal time using base 10 during the French Revolution. People have thought about it in the past and tried

https://en.wikipedia.org/wiki/Decimal_time

12 and 60 divide nicely. A quarter of a 12-hour clock is 3 hours, but in decimal time it'd be 2.5 hours. A third is 4 hours in base 12, but some gross 3.33 repeating in decimal.

I just don't like it.

Unix time is base 10 and I’d say it is pretty widely used. Not for wristwatches but by all kinds of software on the device you’re using to read this right now.

You can blame the babalonians for that. They have a 60 base number system and was used for time and dates.

We have 10 fingers and the ancient Babylonians liked the number 60.

The French tried.

Dont forget we use base 2 and base 16 in computing ;)

It has been tried unsuccessfully.

https://en.m.wikipedia.org/wiki/Decimal_time

If we go below 1 second it seems we use base 10 when using milliseconds

Check out swatch beat time as well, another form of decimal time that was attempted

Here is the Wikipedia article on decimal time.

https://en.wikipedia.org/wiki/Decimal_time

Conceptually it does exist! Metric time

There actually have been a couple of times, the most recent one I can think of is the Swatch “Beat” from the late 80s. There are also a couple sci-if books that use it (Stross, I think).

The problem is inertia, and what problem does it solve.

Yeah, but honestly i think both are kinda stupid. we should use something sensible like base 16(easy conversion to binary but better readability) or base 12(easily divisible by 2, 3 and 4).