The Duodecimal System

by William H. Benson

October 3, 2019

     For centuries, the ancient Romans calculated sums with their clunky numerals: I, V, X, L, C, D, and M; or one, five, ten, fifty, one hundred, five hundred, and one thousand. They knew nothing better.

     Then, Western Europe’s Crusades to the Holy Land introduced Arabic-Hindu numbers to the Europeans. “The medieval world required nearly five hundred years to make the change from the archaic Roman numerals to the Arabic notation, with its miraculous 0.”

     Still, the Romans, like other civilizations, relied upon the Base 10 system for counting because of a biological accident. People stared at their hands and counted ten fingers. It made sense to attach a name to each of the ten digits: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

     But for measuring things, people tend to gravitate to a Base 12 system.

     For example, a foot has twelve inches, bakers sell donuts in lots of twelve’s, and a day has two twelve-hour segments, one for a day, another for a night.

     “A carpenter uses a ruler divided into twelve subdivisions, a grocer deals in a dozen eggs and a gross of cans, and apothecaries and jewelers divide a pound into twelve ounces.”

     The ancient Babylonians preferred a Base 60 system, or five twelve’s, a system that we still retain for time-keeping. An hour has sixty minutes, a minute sixty seconds. Mariners and geographers prefer the number 360, or thirty twelve’s, or six sixty’s, for their compass and for pinpointing locations.

     The number ten has only two factors, 2 and 5, but the number twelve has four factors: 2, 3, 4, and 6. Twelve is the smallest number with four factors. Twelve divides into exact thirds, but ten divided into thirds results in an unwieldy number, 0.33333…

     In the twentieth century, certain mathematicians, who argued for a Base 12 system, came together to form an organization that they named the Duodecimal Society of America, or the Dozenal Society. In the 1930’s, one of the society’s first presidents, F. Emerson Andrews, wrote a book, New Numbers: How Acceptance of a Duodecimal Base would Simplify Mathematics.

      In 1962, another mathematician, A. C. Aitken, wrote, “The duodecimal tables are easy to master, easier than the decimal ones. Anyone having these tables will do these calculations more than one-and-a-half times faster than in the decimal system.”

     How does Base 12 work? The society’s members assign a symbol to the numbers ten and eleven, such as X and E, for a total of twelve digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, and E. They pronounce the numbers ten, eleven, and twelve in Base 12 as “dek,” “el,” and “doh.”

     Instead of placing a decimal after the units, such as in “13.25” in the decimal system, society members distinguish their numbers with a semicolon, such as “13;25.”

     A member writes twelve as 10, because the second number to the left of the semicolon represents the number of twelve’s. For example, 20 represents twenty-four, 30 is thirty-six, and 40 is forty-eight.  

     The third number to the left of the semicolon represents 12², the number of grosses, or the number of 144’s. Thus, 723 in duodecimal = (7 x 144) + (2 x 12) + 3 units, or 1035. in the decimal system.

     The fourth number to the left of the semicolon represents 12³, or the number of 1728’s, the fifth digit represents 12 to the fourth power, or the number of 20,736’s, and so on.

     Because you and I are accustomed to thinking in terms of the decimal system, we feel a real need to convert duodecimal numbers back to decimal numbers. The good news is that mathematicians have created a conversion calculator that we can find on the internet.

     But how do you count on your fingers in Base 12? A very ingenious person noticed twelve segments on the four fingers: three on the index finger, three on the middle finger, three on the ring finger, and three on the pinkie. He or she then used the thumb to point to each segment when counting.

     One, two, three on the index finger; four, five, six on the middle finger; seven, eight, and nine on the ring finger; and ten, eleven, and twelve on the pinkie. A person can count on either hand up to twelve.

     Or a person can count on the left hand, and then use the thumb and finger segments on the right hand to indicate the number of twelve’s counted on the left hand. Thus, a person can count up to 144.

     Are people ready to add, subtract, multiply, and divide numbers in the duodecimal system? No. It is too complicated to convert. We are too ingrained in the way things are now. “Old habits die hard.”

      On November 23, 1793, during the French Revolution, clocks converted to ten hour days, one hundred minute hours, and one hundred second minutes. “The system though proved unpopular.”   

     The modern world will stick with the decimal system for counting and for mathematics, but retain certain portions of the duodecimal system for measurement and time-keeping.