Base 12 vs Base 10
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.
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.
Comments (20)
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.
Because 3 and 4 are divisors of 12 but not divisors of 10.
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
I really dont 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 isnt. What else is there to say?
I suggest questioning the idea of "its premise of even 1/2s and 1/4s at higher orders". Why think there is really anything to "defy"?
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?
:up:
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/
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.
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 2? 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:
Euler's Identity works regardless of the base number system used.
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.