Why Does the Number 12 Occur so Frequently?
Eggs are packed in 12, there are 12 inches in a foot, there used to be 12 pennies in a shilling, the analog clock goes to 12. Why always 12?
One of the first things I remember moving to the US from Norway years ago for studies, was how awkward I found all the measurements: pounds, inches, feet, ounces and Fahrenheit. It seemed completely arbitrary and illogical.
Why was there 12 inches in a foot instead of 10? Why 16 ounces in a pound? I decided it was stupid and arbitrary and didn’t give it further thoughts.
That is until I started doing a programming exercise involving the old British way of dividing coins. The old pound sterling had 20 shillings and a shilling was divided into 12 pennies. And that was just the top of the crazy iceberg. There was farthings, halfpence etc.
It got me thinking that the sheer amount of crazy denominations for these coins, as well as equally odd measurement units, was too big to be a coincidence. E.g why did the number 12 keep popping up both with both money and measurements?
At the face of it, especially for somebody like me who grew up with the metric system, it looked like somebody had just tried to make life difficult for anybody doing math.
This got me curious. I wanted to find out how all this madness had come into place. Was it purely a coincidence, or was there some kind of deeper logic or reason behind it?
The Story of 12
To make a long story short, indeed there is. It has to do with division. Historically people have counted things in tens, hundreds and thousands, just like today. Even the ancient Egyptians and romans did that. But here is the interesting part. People have tended to divide things completely different.
People naturally arrived at 10 based counting due to our 10 fingers, but when they tried to divide things which were multiples of 10 they must have realized that was difficult. In every day life you want to frequently divide things into 3 or 4. That works rather poorly with 10. The romans must have realized this and decided to use 12 as their based number for division. 12 can easily be divided into 3 or 4 parts. Even further back in time the babylonians came to the same conclusion and divided our day into 12 hours.
Thus 10 and 12 has lived side by side for most of our history, and this is reflected in the names of our numbers. Ever wonder why elven and twelve are irregularly named? E.g. why are they not named one-teen and two-teen? It is because they have special significance, and why we also have names like dozen for 12 as well as gross for a dozen dozen.
The Many Ways in Which 12 Makes More Sense Than 10
There are an awful lot of cases from real life where the usefulness of 12 becomes apparent. What is important about a number base are the factors it contains. 10 contains just 2 and 5, while 12 contains 2, 3, 4 and 6. This has significant implications whether you are packing goods, tiling a floor, dividing up the time on a clock or the degrees in a circle, or creating different denominations of money.
Lets look at the detail of some of these examples.
Packaging in 12s vs 10s
People who do any kind of packaging have learned early that basing units on 12 makes sense. It is easier to pack that way, which is why things historically have been sold by the dozen. A grocer is simply somebody who sells goods by the gross.
If you want to pack 10 items you really only have two ways of doing it. Either put them in a long row of 10 or in two rows of 5. That is simply due to the limited number of factors in 10.
However 12 units can be packed in a number of ways:
- 3 x 4, three rows and four columns
- 2 x 6 two rows of six
- 2 x 2 x 3, two layers with two rows and three columns
Why does that matter? Lets say we are considering how much material is needed to create the boxes in which we package our items. If a 5x2 arrangements requires,
5 + 5 + 2 + 2 = 14 units of box material just for the sides, then a 3x4 arrangements actually requires just the same
4 + 4 + 3 + 3 = 14, yet allows you to fit two more items. Considering we need material for both the bottom and top of a box, we quickly realize that a 2 x 2 x 3 arrangement is even better. It is also naturally easier to carry and handle such boxes than long thin ones.
For the traditional grocer, items would not come prepackaged. The grocer would announce prices in useful units. E.g. you would naturally not sell meat by the gram, because most people would not want to buy such tiny quantities.
Selling eggs by the dozens e.g. made a lot of sense. That meant the buyer could easily pick half, a third or a quarter of those eggs, and determine the price accordingly. A quarter of 10 eggs is 2.5 eggs. The grocer can’t sell you that, as he can’t divide an egg. Nor can he sell you a third of 10 eggs. So say you want 3 eggs then. If eggs are sold by the dozen, that would mean 1/4 of the price posted per dozen. But if they were sold in tens the price would be 3.33333… of the price of 10. Not easy to work with.
This also gives a hint of why working with money divided into twelves made sense. Say a dozen eggs cost 3 shilling. If there was 10 pennies in a shilling then you would have to divide 30 pennies by 4. That gives 7.5 pennies. However if there are 12 pennies in a shilling, then a quarter of 3 shillings is 9. Thus we avoid having to do any rounding or introduce a smaller coin denomination.
Dividing a Day
A day is naturally represented as a circle, as anything we want to be able to divide whether it is a cake or pizza. You naturally want to be able to divide a pizza into 2, 4, 6 or 8 parts.
Dividing a pizza into 5 or 10 parts would be really awkward.
Likewise we like to be able to split our day into useable chunks. With 10 hours in a day a quarter of it becomes 2.5 hours. Thus we are forced to introduce decimals. With a 12 hour clock, this is simply 3 hours.
Dividing a Circle into Degrees
Our circles are divided into 360 degrees, which is 12 times 30. So we can see the number 10 pops up. But lets say we wanted to reconsider this historical division. Dividing it into 10 parts or a dozen would obviously not make sense, as we would too quickly need to handle decimals.
ten tens or 100 would make more sense. Lets compare that to a dozen dozens. Half the circle is 50 with metric degrees. 6 dozen degrees would be half the circle for our dozen dozens divided circle. So far no difference. A quarter would be 25 and 3 dozens respectively. That corresponds to 45 degrees, which is obviously useful to be able to express. 30, 60 and 90 degree triangles have very useful properties, so 30 and 60 degrees are also angles you want to be able to express easily.
With our metric angles that would become 8.33333… and 16.6666… which is awkward. For our 12 based on we would get 1 dozen and 2 dozen.
Both last examples should give a hint why the original French of creating a metric clock and metric degrees failed. It is simply very impractical for common operations.
The usefulness of the factors of 12 reveal themselves when trying to tile a floor with various geometric shapes, while the problems with the factors of 10 become apparent. You can’t tile a floor with pentagons.
However you can tile a floor with triangles,
rectangles and hexagons. All polygons where the number of edges are factors in 12. You can even tile it with octagons.
There are actually a lot more to say about 12 based number systems, e.g. how we write numbers in such a system and how we do 12 based arithmetic. But here I just wanted to give the rational or explanation for why 12 is such an important number. If we could have designed our number system over again we probably would have opted for 12 rather than 10.
So what are the uses for 12 based number systems if we can’t change the world to use them today? Twelve based arithmetic will still exist due to inches, feet, packaging, time and angles.
Especially for kids when they learn to use number it is useful for them to learn about different number systems. The problem with learning only our base 10 number system is that we start thinking that numbers are inherently 10 based, which robs us of an understanding of what numbers really are.
For teaching kids arithmetic, division and fractions, a twelve based system is far more useful. They can do more interesting things by cutting up a pizza into 12, 6, 4, 3 and 2 parts.