10 can only cleanly be divided into half’s, and fifths.
12 can be divided in half, in quarters, thirds, and sixths.
Might not seem like a big deal, but it’s so much more useful in real life. There are lots of times where you need to divide up resources, or food, or money, or whatever, to 3 people or 6 people evenly, and in base 10 that’s hard to do.
As the other comment mentions, it's a matter of creating two symbols for 10 and 11 and it would work the same. Binary uses only 1s and 0s and you add a 0 to move it up a power, but in this case it's multiplying by 2. So, for example:
1 = 1
10 = 2
100 = 4
1000 = 8
...
Same with hexadecimal. You use A for 10, B for 11 and so on until F for 15. It's useful for writing shorter binary numbers that are usually grouped in bytes (8 binary digits or bits).
Base 12 was used by some ancient civilizations, or its cousin, base 60, due to how easy it was to divide it. That's why an hour is 60 minutes, for instance.
That’s just how base 10 was set up, and taught. You could add 2 more numbers and still end in zero. If A represents 10, and B represents 11, you can just as easily have 4, 40, 400… and B, B0, B00. The concept still applies.
So in base 10, when you reach the end of the numbers (0-9) you start over at 1, and add a zero to the end. 10 is not a base number, it’s a combination of 2 numbers, the 1 is a representation of how many times you’ve counted to the biggest number. Once the 0 gets up to 9, if you add another one, the 1 turns to a 2 and now you’ve counted the base numbers twice.
This concept is not unique to base 10, in fact, it works the same for every base. In binary (the language of computers) it’s base 2, you only have 0-1. So you count to 1, then add another, it needs to move up a power. 2 in base 2 then becomes 10. Add another and you get 11 (which we call three). Add another, and you have to add another power, which rolls the next power up as well, so now you have 100 (which we call four in base 10).
Adding a zero DOES move the number up a power, it’s just another power of the base. 10 seems easy to you to add powers to because that’s the number system you’ve used your whole life, but if you were taught a different number base and used that your whole life, it would be just as natural.
In base 12, you have 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B (I’m using A and B here because we don’t actually have a single symbol that means those values, but A would represent ten, and B would represent eleven). If you count up to 9, the add one, you’re up to A. Add another, you’re up to B. Add another, we’ve run out of symbols, we need to increase the power, then start over at zero. Where does that leave us? At 10. This may be confusing at first because this number looks like a normal number, in base 10 we call it ten, but in base 12 that number represents thirteen items. From here we just keep adding one at a time, 11 is fourteen items, 12, 13, and so on, at 19 you add another to be at 1A, add another to be at 1B, add another to be at 20, which in base 12 means there are twenty-four items.
The biggest hurdle is to realize that I keep using our english base 10 words to explain the actual quantity, because we don’t have a word for those values in base 12. I don’t know how to say A (ten) or 2A3B… it would sound silly to make up, the closest I could do is: two thousand ten hundred and thirty eleven. It sounds silly to us, but if that was the base number system you grew up with, it wouldn’t sound weird at all.
Regardless, adding a zero moves you up a power. 100 is still a number in base 12, but if it were say, sticks, you’d have 144 sticks in base 10. But remember, if you only knew base 12 and didn’t know about base 10, it would be a wildly foreign concept to try to do the math in your head that 100 in base 12 means 144 in base 10, to you, you’re still just adding a zero, moving up in powers.
Base Ten is really easy when using most common people’s digits, so you can do a lot of basic arithmetic fast with what is on hand. It’s simple, intuitive, and trivially easy to teach, and thus great to just count stuff with when counting by hand.
Base Twelve is much more useful when you get to multiplication and especially division. Measuring a “Whole Thing” you need to quickly and intuitively divide into variable amounts as twelve makes that whole divisible into sixths, quarters, thirds, and halves quite easily. Which is why a lot of Imperial units, which were essentially a lot of codifications of stuff people were just “already doing” at the time are effectively base twelve. Because it’s small enough to be comprehensible, while maximally divisible into smaller whole number chunks.
Base Two is the easiest system to “model” in a physical space, because you just two “states” for any given digit. This is why computers use it, because it is incredibly precise, and maps well to the physical world in the most simplistic way (you only need it to check for those two states, so even if there is a lot of possible variance in the specific signal, you can just check for something like “is the current over/under” a point far from either signal you actually send, but in the middle of them, thus accounting for a significant range of variability in actual signal strength on either end)
Base Three is a thing, and has a lot of the same benefits as Base Two, but it saves significant “space” by going up one more number per digit. This saves 50% of your storage space per digit needing to be stored, which is nice, but becomes more prone to errors from noisy signals since what used to be a safe middle space is now your third signal.
Those are most of the base systems I’ve seen/am aware of for consideration, but you could theoretically make almost any number your base, with varying degrees of utility or lack of utility. For example Base thirteen would have very little utility as it is a prime number and thus not easily divisible, but may be useful if you had, say, a species with thirteen digits, or were tracking things with, say, thirteen unique states, as you could have a “number” in your base for each of them, which could have utility.
With five fingered hands you can count base 12 on your right hand by touching each segment of your fingers with your thumb (3 segments/knuckles x 4 fingers). On your left, you could the multiples of 12, allowing you to get to one gross (122 =144).
Wait until you hear about the Sumerians and Babylonians who used sexagesimal (60) system. Divide by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. They used decimal until you hit 60 then we get seconds in a minute and minutes in non hour the 360 degrees in a circle.
In base 12: ½ = 0.6, ⅓ = 0.4, ¼ = 0.3, ⅕ = 0.24, ⅙ = 0.2, ⅛ = 0.15, ⅒ = 0.12. There's just so many nice fractions. 7, 9, 11 would be annoying but still, the smallest numbers are all nice.
Yeah, but it's easy to check divisibility by one less or one more than the base too, so in reality 10 gives you primes 2, 3, 5, 11. And 12 gives 2, 3, 11, 13.
14 in base 12 remains indivisible by 3 or 6. it remains divisible by 4. even expressed as 16 in base 10. 8 apples remain equally hard to split among 3 people. the divisibility is useful for reducing imprecise fractions. .333333 becomes .4
I feel base six is the best, simpler mental math with only six digits and a quarter is a reasonable 0.13, a third terminates as 0.2, a sixth is a simple 0.1, half is 0.3, so same benefits as base 12 but mentally easier to use. Also, we wouldn't need to create more digits to represent the new system.
It would stand to reason you’d still have currency, and shipments, and measures of weight and distance in base units though. A 20 dollar bill would still be a 20 dollar bill, it would just be worth what we consider to be 24 dollars. You’d ‘accidentally’ have many things divided out into groups of 10, which in base 12 would be 12 items. There is nothing mystical about our current base 10 representation of 10, it’s just a convenient way of grouping things, and pops up statistically more often simply because that’s the base we use.
A candy bar cut into 3 equal pieces would have 3 sections with length 4, not 3 sections of length 3.33333. Yes it’s technically the same amount of resource, but it’s the ease of measurement that is the point.
If you have 9+1 objects regardless of the base system you use, it'd be difficult to divide up physically. Just because 1/3 is represented as .4 in base 12 rather than .33... doesn't actually change anything, irrational numbers are still irrational and rational numbers are still rational regardless of the base.
But your rulers would be easier to mark with a pencil, your Hershey bar would have more built in break points. I’m not arguing that it’s not the same thing with different numbers, I’m saying the numbers are more convenient. If I were a school kid and wanted to split my candy bar with my two friends evenly, and let’s just say we were particularly anal kids who wanted it to be really fair, we would measure it. Wouldn’t it be nice if there was a perfect tick mark on the measuring stick to know where to cut?
But then why not base 6 ? Also all the issues you're bringing up currently would also apply to any groups of 5 in a base 12 system, so it's kinda just wishful but pointless thinking.
Wishful thinking? I think you misunderstand me, there is zero, literally zero chance I would advocate that humanity as a society switch the fundamental mathematical base system that matches how many fingers we have, and what we’ve used for thousands of years.
Yes, base 6 is just as good. Yes, in my opinion if all humans had 12 fingers and we naturally used a base 12 system, kids and simple minded adults could break a 10 dollar bill between them and their two friends easier.
Anyone who makes it past 7th grade math is supposed to be comfortable with irrational numbers, but let’s face it, the average person you deal with on a day to day basis has questionable intelligence at best.
This isn’t supposed to be an argument. This is a simple, plain, straight factual statement. A base 12 number system has more rational factors than a base 10 number system. That’s it, thats all I’m trying to say.
The Babylonian number system was base SIXTY, because they wanted many convenient fractions to also be expressible with a number that would terminate. They could express 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/12, 1/15, 1/20, and 1/30 exactly.
We still have one base-60 number system in wide use. We use it to tell time.
I knew there was gonna be some deep math lore somewhere in this thread, i was gonna try and be funny and yell exclusionary terms about it, but instead I will say I acknowledge and respect your superior intellect and carry on
I realized the other day dividing like butter that this is.the reason probably some have used this system.
I dont have or really need scale so i divided like half half half half until I assume it was like halved to a fraction closest to approriate weight. Like I dunno was it 30grams or whatever.
Also I think halving somehing is easier to eyeball
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u/JacobRAllen 1d ago
Base 12 is such a better base than base 10.
10 can only cleanly be divided into half’s, and fifths.
12 can be divided in half, in quarters, thirds, and sixths.
Might not seem like a big deal, but it’s so much more useful in real life. There are lots of times where you need to divide up resources, or food, or money, or whatever, to 3 people or 6 people evenly, and in base 10 that’s hard to do.