Base 12 vs Base 10

Mp202020

Let me preface with the adamant statement that when it comes to mathematics, especially the theoretical aspects of it, I am an absolute layperson. I am only suggesting this question to this community because it has kept me up at night trying to understand what I am missing. I speculate that I am making some sort of logical flaw within the realm of numbers, I just need someone to point it out to me and explain to me what I am getting wrong.

I recently looked at some study that was explaining how if we had been born with 6 fingers/toes (instead of 5) how different certain aspects of our day-to-day lives would be. Primarily the inconvenience that working/building off of a base-10 number system (likely due to our having 10 fingers, and 10 toes) gives us that we would not experience had we shaped our metric systems off of a base-12 system (likely if we had been born with 6 fingers/toes, we may have gone that route).

It was explained that certain conveniences would exist off of a base-12 system that wouldn't necessarily change us in a profound way but would offer advantages that we don't enjoy with our base-10 system. For example, base-12 can always be divided into 1/4th, 1/2, but also 1/3 evenly. There was a slew of other advantages explained, but they all chalk up to creating a more "convenient" system we could have worked off of. I found it very interesting.

Anyway, my mind started racing about it and can't stop. Anything that is base-10 can always be divided into halves (1/2) however it can never be divided into thirds evenly. Base-12 always can be divided into halves, quarters, and thirds.

My unrelenting and annoying thorn is: why can base-10 numbers be divided into quarters and halves, only unless that base-10 number does not represent it's smallest unit (the number 10). 10 can become 5, but trying to split it into quarters yields 2.5, and thirds obviously 3.333~. However if that base 10 number becomes 100, it can at least be halved and quartered evenly, 50/25.

Why can the lowest base-12 number (12) be split evenly into halves/thirds/quarters, while the lowest base-10 cannot be split into 1/2 and 1/4?

Can anyone here give any more insight as to what the true power of base-12 is? I am just a confused, I'm sorry if this post comes off as naive. I may not be expressing my troubles with this accurately.

unenlightened

Anything that is base-10 can always be divided into halves (1/2) however it can never be divided into thirds evenly. Base-12 always can be divided into halves, quarters, and thirds. — Mp202020

Hmm. It's quite tricky to talk about, and with this way of expression you are misleading yourself a bit. First off, there is nothing you can do in base 12, that you cannot do in base ten, and vice versa. The amounts don't change, only the notation. Either a number can be divided by 3 without remainder, or it cannot, and the notation makes no difference to that.

Let's construct a base 12 notation using the familiar digits 0-9 and use 'a' and 'b' to represent the two extra fingers. Then our three times table will ru like this in the number line:

1 2 3 4 5 6 7 8 9 a b 10 11 12 13 14 15 16 17 18 19 1a 1b 20 21 22 23 24 25 26 ...

You can see that in base twelve, any number ending with a 3, 6, 9, or 0 is divisible by 3. (This is confusing us who are used to base ten, because "10" in this notation represents "a dozen" and "26" represents "thirty" , or better, "twodoz-six" and so on.)

Extending very briefly to fractions and "dozimals", 1/3 would become "0.4" - no longer a recurring endless expression. But 1/5 on the other hand would become recurring "0.2497..."

And at that point, my old brain protests and refuses to go further, except to mention that the ancient Egyptians used a base of sixty, and that is where we get our measures of time and angles from. And don't expect any calculations from me in base sixty.

Michael

Why can the lowest base-12 number (12) be split evenly into halves/thirds/quarters, while the lowest base-10 cannot be split into 1/2 and 1/4? — Mp202020

Because 3 and 4 are divisors of 12 but not divisors of 10.

Mp202020

↪Michael

yes, I guess my struggle is with the fact that the smallest unit of base 10 cannot be divided evenly except into 1/2, unless that base 10 unit exceeds its minimum value (10). However base-12 always adheres to even splits whether 1/2, 1/4, or 1/3, whether it be the minimum value (12) or 1200. 10 only seems to create the possibility of 1/2 and 1/4 when exponentiated on itself, while 12 can conform to its own rules regardless.

Mp202020

↪unenlightened

im not sure how else to digest this, I would simply say that why do higher orders of base-10 divide evenly into 1/2 and 1/4 while the number 10 itself does not? However base-12 always divides evenly, exactly into 1/2. 1/4, and 1/3 even at its lowest value (12) or any higher order of it? Is there some metaphysical substantiality that 12 has?

Mp202020

Perhaps this is a “why is the sky blue” kind of question. I don’t fully understand why, but for some reason this discrepancy bothers me. It feels like two types of numbers (base-10 and base-12) are playing by a different set of rules, simply because the lowest form of base10 (10) doesn’t split the same way it’s own higher orders can, however base-12 (even at its lowest value 12) sticks to the same rule.

wonderer1

why do higher orders of base-10 divide evenly into 1/2 and 1/4 while the number 10 itself does not? — Mp202020

It's a simple matter of 4 being a factor of 12 but not of 10.

Consider:

10 = 2 * 5
100 = 10 * 10 = (2 * 5) * (2 * 5) = (2 * 2) * (5 * 5) = 4 * 25

Mp202020

↪wonderer1

I think my question pervades the specific maths. It’s almost a metaphysical question- how can the pure simple number 10 defy it’s premise of even 1/2’s and 1/4’s at higher orders, while 12 follows the same rules of its higher orders?

Michael

↪Mp202020

I really don’t understand your confusion.

12 divided by 4 is 3.
10 divided by 4 is 2.5.
20 divided by 4 is 5.

3 and 5 are whole numbers and 2.5 isn’t. What else is there to say?

Mp202020

↪Michael

is 10 a different type of number compared to 12? Simply by way of it doesn’t split evenly in the same way it’s higher orders do, the way 12 does?

wonderer1

I think my question pervades the specific maths. It’s almost a metaphysical question- how can the pure simple number 10 defy it’s premise of even 1/2’s and 1/4’s at higher orders, while 12 follows the same rules of its higher orders? — Mp202020

I suggest questioning the idea of "it’s premise of even 1/2’s and 1/4’s at higher orders". Why think there is really anything to "defy"?

wonderer1

is 10 a different type of number compared to 12? Simply by way of it doesn’t split evenly in the same way it’s higher orders do, the way 12 does? — Mp202020

12^2 (144) is evenly divisible by 9, whereas 12 is not evenly divisible by 9.

So 12 doesn't split evenly the way its higher orders do. Is there any particular significance to this?

Mp202020

↪wonderer1

I think this might be the kind of thing I was looking for!!

wonderer1

↪Mp202020

:up:

Mp202020

↪wonderer1

when speaking about base numbers, what advantages do you posit would come from utilizing base 12 over 10?

unenlightened

Is there some metaphysical substantiality that 12 has? — Mp202020

12 is the lowest abundant number. An abundant number, A, is one whose divisors (including 1 but excluding A itself) add up to more than A. Factors of 12 are 1 + 2 + 3 + 4 + 6 =16

Here is an article that might be of interest:
https://www.quantamagazine.org/the-mysterious-math-of-perfect-numbers-20210315/

wonderer1

↪Mp202020

Different number bases have different convenience advantages depending on context. I use base 2 and base 16 frequently.

Base 12 would probably provide some convenience advantage over base 10, in that 12 can be divided evenly by five smaller integers (1, 2, 3, 4, 6), whereas 10 can only be divided evenly by three smaller integers (1, 2, 5).

Perhaps using base 12 would produce greater social harmony and result in world peace. For example, pizzas should always be sliced into 12 pieces to maximize the odds of harmony in pizza sharing.

Mp202020

↪wonderer1

Agreed. Do you think the relative conveniences a base-12 system would offer could have possibly lubricated our understanding of mathematics/physics to have potentially progressed meaningfully quicker in these disciplines?

SophistiCat

We still subdivide time into 12, 24 and 60 units, even though we use decimal notation. A remarkable legacy of ancient Sumerian mathematics, which used the sexagesimal system. When you wonder whether an event that takes place in two and a half hours will be in the morning or the afternoon, you are effectively thinking in base 12. Many older length, weight and monetary units, including the English units that still persist in the US, are also rooted in subdivisions of 6 or 12, rather than 10. The main reason base 10 won out in most other spheres was due to the near-universal adoption of Arabic/Indian numerals for positional notation - a historical contingency.

wonderer1

Do you think the relative conveniences a base-12 system would offer could have possibly lubricated our understanding of mathematics/physics to have potentially progressed meaningfully quicker in these disciplines? — Mp202020

I'm not a mathematician, and don't have sufficient basis for any strong opinion on the matter. As @SophistiCat pointed out there is a convenience to using integer multiples of 12 when thinking about geometry involving lines and circles, so we use 360 degrees to a circle. On the other hand it can be convenient to consider circles in terms of 2π radians instead of 360 degrees.

Regardless of what number base came to dominate in ancient history, advances in mathematics were dependent on the ability of mathematicians to shift between different ways of enumerating things.

Perhaps it is worth considering Euler's Identity:

$a7464809a40f9e486de3a454745f572fbf8bb256$

Euler's Identity works regardless of the base number system used.

Leontiskos

The main reason base 10 won out in most other spheres was due to the near-universal adoption of Arabic/Indian numerals for positional notation - a historical contingency. — SophistiCat

Right - the decimal system was not a foregone conclusion, historically speaking. The fingers/toes claim is interesting, and having the source would be worthwhile.

It is also worth noting that computers make use not only of binary, but also octal and hexadecimal.

Base 12 vs Base 10

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