How will base-10 work in a world where duodecimals have never been thought of?

Question

In a world where duodecimals (base 12) have never been invented or thought of, how will decimals replace them?

We now know that we owe everything related to duodecimals to the ancient Egyptians and Sumerians. It was they who used base 12 in everything, such as telling time. The later civilizations simply builds up on it while also using the base-10 system.

But what if, there is a world where all these nonsense about base 12 never appeared in history? How will base 10 work in replacing base 12 applications?

I can imagine, these will change, but how will it work under base 10?

1. Time (I imagine a day will have 10 or 20 hours, instead of 60 minutes an hour, it will have 50 minutes and hour, or 10 months instead of 12)
2. Angle/geometry (how will 90 degrees translate into under base 10?)
3. Astronomy

I don't know if there are other use for base 12 other than this, but if you have any thoughts about it, feel free to add.

Answer 1

60 parts makes it easy to divide by 2, 3, 4, 5, 6, 12, 15, 20, and 30.

As a competitive advantage, people can work with angles, times, (as they still do) and currency, distance, etc. in early civilization where large complex buildings, complex finance, and dedicated investigation of math & scientific problems are first being made.

We don't have much trouble with decimal currency, and rounding off or approximating is not as much of an issue as with building something where you want the corners to line up and whatnot.

In order to easilyndo these things, they might have developed a head for rational arithmetic earlier, or even decimal fractions and long division.

What would change would be some other way of manipulating quantities and values that practitioners can handle with ease and efficiency. Even if you had the ability to work with fractions in clay tablets, how do you show that on your ruler? I'm thinking the instruments mightnbe pre-marked with fractional values that can be constructed geometricly, when making the primary markings. So, even though the ruker or compass is marked 1..10 units, there will be marks at 1/3 of its length, 1/4, etc. Then you need the same fractions marked with a denominator of 9, etc. You end up with a maze of marks... or do you? Systematically you end up with each unit divided into 12 subunits.

This is exactly how the modern Westwrn scale in music comes about. Start with the simple ratios and fill in all you can reach using what you made thus far.

So why didn't they invent it anyway, for professional tools even if not in popular use? Maybe the culture didn't care for these simple ratios, cared about others like 1/7, and was drawn to non liner-repeating patterns like geometric progressions, golden ratio, and other things that produce irrational values. Now, having easy divisions doesn't help much since those are all ugly unused values.

So my answer is that art and architecture would reflect these sensibilities instead, like fractals and immitations of nature. That would influence early scientific inquiries and they would use geometric progression, essentially logarithmic scales, rather than 60ths linear scales. Maybe they would discover $ e$ early on, and rulers would be based on vanishing points, and compasses would be marked with the sine of the angle rather than 60 degrees. For finance, they would understand compound interest intuitively.

Answer 2

Actually, the experience has been made to have everything on base 10 rather than base 12. It was made following the French revolution. In particular, there were

• 10 days weeks,
• 30 days months,
• (still) 12 months a year (plus a few extra days),
• twice 10 hours a day,
• 100 minutes an hour,
• etc.

You can read more about the French Republican Calendar and the Decimal Time on Wikipedia.

The complexity (adding a few days a year to keep adjusted, for example), and the resistance implied by the previous use of the older system meant that it was progressively abandonned until it was enacted in the law, due to two main reasons (officially):

• Issues with Leap Years;
• Beginning of year based on French history and thus not universal.

Of course if there were never another system used, the adoption of a 10-based system might be used, provided that the few issues were solved.

You should also note that through the Middle-Ages, different systems were used: 12-based, 10-based, as well as 20-based (and probably others). 12-based are advantageous for fractions: easy to compute a half, and third, a quarter, with only natural numbers.