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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?

Xarcell
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    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 00:12
  • 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 01 '19 at 04:12

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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.

Bill Cook
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    "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: http://io9.com/5977095/why-we-should-switch-to-a-base-12-counting-system – 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 09:35
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    "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 00:05
  • 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 00:09
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    @Neil I'm not quite sure what your comment is addressing. In base 10, you can see divisibility by 10's, 5's, and 2's. This is no misconception. This is fact. And, yes, it is because of the base 10 "encoding" we've chosen for our numbers. What you've said about base 12, is mentioned in my original post (here you can "see" divisibility by 12's, 6's, 4's, 3's and 2's). – Bill Cook Apr 14 '15 at 00:16
  • Xarcell is asking for the reason why any particular base is best. You made it seem that base 10 is the only one that has the property of immediate visual divisibility. This is not true as I've shown. – Neil Apr 14 '15 at 04:15
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    I made no such claim. See my first sentence. – Bill Cook Apr 14 '15 at 11:55
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Look at base16 (hex). It has more factors than base10 (dec).

Factors of hex: 1, 2, 4, 8, 16 -five total Factors of dec: 1, 2, 5, 10- four total

Hex is superior to dec when it comes to factors.

16 is close enough to 10 that it's not too hard to memorise the multiplication table. So, base60, used by the Babylonians, is out of question.

Hex is used primarily in computing. Also, there are 16 ounces to a pound and 16 cups to a gallon, and the Roman foot was divided into 16 digiti or fingers.

Plus, the US and Canadian dollars have halves and quarters, as well as twos (the stella 4 dollar coin only made it to pattern phase, and the 8 dollar bill was only a proposal by Benjamin Franklin).

Add that to the fact that the US had gold coins that were multiplied or divided by two:

Double Eagle- $20 Eagle- $10 Half Eagle- $5 Quarter Eagle- $2.50

See the pattern?

There's also the fact that an average pizza is cut into not 10, but 8 slices. All you have to do to cut a pizza, is cut straight across four times. Or eight if you want 16 slices.

Try folding a paper into tenths. Then, see how much easier it is to fold it into eighths or sixteenths. Scaling is another thing. Many model cars are 1:64, and for good reason.

And try seeing who has 10 birth parents. No one does, they have 2. And 4 grandparents, 8 great grandparents, and so forth. Think of bloodlines. No one is exactly 1/10 anything. But 1/8, 3/16, 59/64? Sure.

Now look at the square roots of 10 and 16. For 10, it's 3.16... for 16, it's an easy 4. And now cube roots. For 10, it's 2.15... for 16, it's an easy 2.

More things are evently divisible by 2 than by 10 or even 5. Half of all numbers from 1 to 10 (or any power 10, such as a billion) are even as well. There is no special word for a number evenly divisible by 5 or 10, only by 2. And what would they be called, 5-even, 10-even?

Just think about why we use numbers in the first place. Base1 (un) is mostly used in tallying or counting fingers. So a million would literally be 1 repeated a million times. That's a bit much, so we have to group numbers, hence base systems with their own radix. 11 is a group of 10 and 1 left. 1F(hex) is a group of 16 and 15 left.

It's a lot easier to group things by 2 or its exponents than by 10 and the like. That's why computers use base2 (bin). If you forget that, then all you have is a number system that doesn't make any sense.

But 10 is perfect because you can move the decimal, count it on your hands, it's small enough to learn the arithmetic, and... well, what?

Comment if you have a question or want me to go on, or even just want to argue.

ColeDassWest
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