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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.

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    $\begingroup$ You may be trying to reach deeper than necessary. Base 12 makes for some convenient easy division by 3 and 4, but it is not a fundamentally earth shattering concept. I would expect base 10 to work exactly the same as it does in the current world, except for a handful of places where we have historical connections to 12, and they'd be replaced with the nearest 10. What sort of implications are you looking for? (consider that 99.99% of all data is now in base-2, and it hasn't changed us all that much!) $\endgroup$ – Cort Ammon Aug 2 '15 at 16:30
  • $\begingroup$ I believe you may be confusing "number" with "numeral". The numerals we choose to represent numbers is, in a sense, arbitrary. $\endgroup$ – Epsilon Aug 2 '15 at 17:25
  • $\begingroup$ Maybe so, but one could read deeper into it. How would things be such that 12 and 60 were not found to be useful and efficient for portioning? $\endgroup$ – JDługosz Aug 2 '15 at 18:05
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    $\begingroup$ I'm not denying that base 12 is convenient or useful. My point is that there is nothing natural about base 12. I've heard some claim base 12 originated from 12 sumerian gods. I don't know where this belief come from, but I'll just accept it. What if this world's belief system has nothing to do with gods or their star system is totally different? What if their belief system is reincarnation? Since base 12 does not occur naturally, it will have to be invented. Why would anyone complicate matters by inventing a different system that is not intuitive at all? $\endgroup$ – Ifree Contractors Aug 3 '15 at 8:28
  • $\begingroup$ There's nothing natural about base 10 either (although you probably know this, right?) $\endgroup$ – Erik Aug 3 '15 at 13:21
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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.

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  • $\begingroup$ Thanks for your view. Can you explain further on the sine of angle instead of 60 degrees? For example, assume that you are on a ship and you want to tell your crew to align the cannons to 90 degrees port, how will you give such an order to your cannon crews? $\endgroup$ – Ifree Contractors Aug 3 '15 at 8:33
  • $\begingroup$ 90 degrees would be "all the way", or 1. So just turn to face port. For intermediate angles, imagine the length of the canon as a diagonal line. If you look how much length it takes in the port direction: place boards parallel to "full port" on either end of the canon like book ends, and see how far apart they are. Practically, thkuse would be marks on the floor, and you turn the canon so the front and rear are over marks a specified distance apart. They might have intuative names based on the sun's shadow for a time of day, which also follows similar rules. $\endgroup$ – JDługosz Aug 3 '15 at 13:47
  • $\begingroup$ I think "in order to easily do these things" there is a path of lesser resistance, that doesn't involve handling decimal places (invented in 1500s AD, whilst base 60 Babylonian system was around 2000 BC). Just have a few extra words, e.g. for $3\frac{1}{3}$ and $2\frac{1}{2}$ . . . and have these measure points shown on rulers/scales etc. $\endgroup$ – Neil Slater Aug 3 '15 at 14:37
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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.

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