We Need more Fingers

2010-07-20 Black windup alarm clock face

Image: Sun Ladder, via Wikimedia Commons.

Our base ten number system is so ingrained in us that it is difficult to imagine using anything else.  With our ten fingers to count on, it makes sense that we have ten symbols to represent numbers.  Despite this, other cultures have used extremely different number systems.  The Ancient Mesopotamians used a sexigesimal, or base 60, number system.  The Mayans used a vigesimal, or base 20 system.  Roman numerals are used to number the Rocky movies despite them being almost completely useless.  Most computer techies are familiar with binary and hexadecimal.  Many early peoples even used a system with only 5 symbols (Boyer 3).  Our current number system may be intuitive but it may not be the best one around.  What if we had an extra finger on each hand?  We would be using a much more useful number system.  We should move away from the decimal numbers we currently use, and switch to a base twelve, or dozenal, number system.

There are several reasons to seek more mainstream use of base 12.  The factors of a number, or the numbers that divide into it evenly, determine a lot about the number.  Twelve can be broken down into more factors than ten can be.  Ten is divisible by only 2 and 5, whereas 12 is divisible by 2, 3, 4, and 6.  This gives 12 an advantage over 10.  The additional factors make it easier to think of many fractions, such as fourths and sixths, since they will now have only have a single significant digit after the point.  This is particularly effective when dividing 1 into thirds because it will not leave us with an infinite series of 3s like it does in decimal.  The simple tricks that help us do arithmetic, such as the fact that in base ten if a number ends in an even number the whole number is even and thus is divisible by 2, depend on the factors that make up our number base.  Since 12 has more factors, similar tricks can be used for more numbers.  In base twelve, if a number ends in 0, 4, or 8 the entire number is divisible by 4, if it ends in a 0, 3, 6, or 9 then it is divisible by 3.  We will not even miss out on the trick for evens that base ten has since twelve is also divisible by 2.  The trick we now currently use for 11, where you alternate adding and subtracting the digits of a number and see if the resulting number is divisible by 11, will work for 13 when we make the switch, because now 13 will be the number that is one larger than our base.  Don’t be concerned about 11, because we will have a new trick for 11 in dozenal.  The trick we currently use for 9, where we just add up the digits of a number and then check if the sum is divisible by 9, will work with 11 once we change to dozenal.  It is easy to check the divisibility of far more numbers in dozenal than it is to check in decimal.

For a dozenal system, we would have to make some changes to the actual symbols we use.  We have 10 symbols to use for the numbers 0-9 and a base twelve number system would need 2 more symbols to represent ten and eleven.  There are many different sets of symbols we can use to fill the two new places.  Some number sets use *, and # to represent ten and eleven in order to correspond to the symbols on most phone number pads.  Others use X, and a backwards 3, and some use a backwards and upside down 2 and 3.  Some sets completely replace all the symbols we use for 0-9, along with adding two new symbols.  We could use any group of numbers that would help us acclimate to a base twelve system.

Some things that we already do everyday would assist in our transfer to a base twelve system.  We already have specific terminology for 12 and several of its powers.  We use the word dozen to refer to twelve, a gross for a dozen dozens, or twelve sets of twelve, and a great gross for a dozen gross, or twelve gross.  When looking at a clock, we already deal with twelves to determine the time.  Figuring out what the time is 5 hours after 10 pm is basically the same thing as adding 5 to 10 in base twelve.  As Professor James Monroe notes, thinking of egg cartons makes thinking of dozenal numbers easy.  If 1 egg carton holds 12 eggs, and 1 case holds twelve cartons, a number like 426 in base 12 can simply be thought of as 4 cases, 2 cartons, and 6 loose eggs.  Twelves seem almost as prominent in daily life as the number ten.

Despite the familiarity we already have with base twelve, switching to dozenal will still be incredibly difficult.  Because we would have more digits, kids would have to memorize larger multiplication tables.  Luckily, the tables will not be anywhere near as large as they would be if we still used Cuneiform.  However, the real difficulty in switching has little to do with what number base we want to use.  The trouble will be in converting all the numbers on everything that we use.  Every road sign, price tag, page number, and countless other places have numbers that will need to be converted to base twelve.  This will be more difficult than changing from using imperial units to metric units, and America still has not completely converted to metric.  Here in the U.S., some things are measured with metric units, but we still measure distances in miles and a sack of potatoes at the grocery store is measured in pounds.  Also, to add more difficulty, in metric we would have to come up with new prefixes that are based on powers of 12 instead of powers of 10.  Changing to dozenal numbers is such a monumental task we may not be able to accomplish it.

In spite of the difficulties, I believe we should do it.  The conversion will not happen overnight.  We must look further down the road.  Perhaps, start by teaching people how to do arithmetic in dozenal in addition to teaching them the usual decimal system that we use.  Then when people are comfortable with it, we could move on to using base twelve alongside decimal.  Eventually, our ancestors will be able to move on to a better number system than what we currently have.  Like the Kwisatz Haderach from Frank Herbert’s Dune, we must endure temporary struggles in order to achieve the Golden Path.

Source

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

Additional Material

http://www.dozenal.org/drupal/

http://www.dozenalsociety.org.uk/

http://youtu.be/U6xJfP7-HCc

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s