# Base 12 Versus Base 16

I'm not good when it comes to math, so forgive me. I'm doing a personal study of is there a better base number for our culture to use? I have to consider factors like: the number of digits to write, ability to count visually(like using fingers), understanding fractions better(3's 4's are both even), and most importantly the time & calendar year. Note: symbols representing numbers is not an issue here.

So I decided between 12 instead of 16. I only considered 16 because of it's use in computer science really. 12 to me seemed superior in many ways, not too big, not too small, and especially since it's easy to factor into time.

Then I seen the Hexclock(http://www.intuitor.com/hex/hexclock.html), which uses base 16. How is this possible? I thought time was based on a 360 degree radius of the earth? 16 doesn't divide into 360 degree's.

Are there better arguments for using 16 instead of 12, besides computer science?

• It is possible because they go down to the second. Since $1$ hour = $3600$ seconds and $3600$ is divisible by $16$, it just becomes a matter of changing the display. – John Habert Mar 19 '14 at 14:46
• Base 12 is good if you use a lot of multiplication and division in your math because of easy divisibility by 2,3, and 4. Base 16 is good if you deal with a lot of divisions and multiplications of 2 like in binary. This is because of the exponents, 16 = 2^4. – Neil Apr 11 '15 at 0:12
• You need to be very good in math to study base systems application to computing. You need to know about floating point, rounding, irrational numbers, etc. – NoChance Jun 1 at 4:12

Each base has advantages and disadvantages. Base 10 works particularly nicely because you can immediately "see" divisibility by 2's and 5's (and 10's). It's also easy to detect other patterns as well (i.e. divisibility by 9 using casting out nines).

Those with a computer science background might argue that base 8 (octal) or 16 (hexidecimal) would be better choices. The Babylonians used base 60.

While using a large base (like 60) has some advantages (i.e. "seeing" more divisibility), it also has drawbacks. In particular, who wants to learn 60 basic numeral symbols! 10 seems to strike a nice balance. It's not that many symbols (0,1,$\dots$,9) and has some nice properties. Although part of our choice to use base 10 seems to be rooted in physiology (i.e. most people have 10 fingers).

Other than computing, 16 would be better than 12 because each digit could encode more information. However, while base 16 allows one to "see" divisibility by 2,4,8,16. Base 12 allows one to "see" divisibility by 2,3,4,6,12. This seem likely to be more useful. Another argument for 12 over 16 is that our children would only need to learn multiplication tables up to $12 \times 12$ instead of $16 \times 16$ (that's a big increase in basic multiplication facts to memorize)!

As for 360. That seems to be due to the fact that a year has approximately 360 days. ["Degrees" are not a natural measurement system. "Radians" are the natural mathematical choice.] This seems to have led the Babylonians to divide the orbit of the Earth into 360 degrees (each day ticks off one more degree until 360 gets you all the way around -- approximately). But 360 is too big for a base, so maybe 60 seemed like the best alternative. Base 60 then leads naturally to 60 seconds in a minute, 60 minutes in an hour. This is just a convention determined by the Babylonians.

• "divisibility by 9 using casting out nines", this is nothing special though. Octal would have divisibility by casting out 7's. You and I can't do it because we aren't used to. Apart from that, excellent answer. +1 – Guy Mar 19 '14 at 15:10
• Ok. Fair enough. With any base $n$ we get casting out $n-1$'s. But maybe I just have a special affinity for 9's! :) Also, here's a funny argument for switching to base 12: io9.com/5977095/… – Bill Cook Mar 19 '14 at 18:51
• Actually to be fair, you don't have a special affinity for 9's, you have a special affinity for base 10, which I whole hearted-ly support. – Guy Mar 20 '14 at 9:35
• "Base 10 works particularly nicely because you can immediately "see" divisibility by 2's and 5's (and 10's)." This is a misconception. 10 is how you represent 2 in base 2, 3 in base 3, 4 in base 4, etc... In any base 10 is the number for how you represent that base. In base pi, pi is represented as 10. So you can still do the moving decimal places thing for quick division and multiplication. In base 12 you would lose the easy divisibility by 5, but you gain easy divisibility by 3 and 4. – Neil Apr 11 '15 at 0:05
• in case you don't believe me, lets count a bit in base 2 to see how it works. 0 (zero), 1 (one), 10 (two), 11 (three), 100 (four), 101 (five), 110 (six), 111 (seven), 1000 (eight). So say I wanted to divide 1000.0 (eight) by 10 (two). I can use the Decimal trick and move the decimal over one to the left and get 100 (four). – Neil Apr 11 '15 at 0:09