The digits in the dozenal numeral system

Teaching Kids About Alternative Number Bases

We learn to count to 10 on our fingers as kids. The 10 based system becomes so ingrained in our way of thinking about numbers that it becomes synonymous with numbers.

A problem with that is that it becomes harder to fully understand what our numbers are, because one has not become accustomed to thinking that that is just one way of representing numbers.

Dozenal or Duodecimal Numeral System

One of the number systems I am a fan of is the dozenal number system. Dozenal means the base is 12 rather than 10. I’ve written an earlier article about what makes this number system so interesting.

Dozenal systems are all around us:

  • Clocks are divided into 12 hours.
  • There are 12 inches in a foot.
  • Eggs are sometimes packed in dozens (12).

So it is useful to know how to deal with a dozenal system even if it is not the normal system. One of the reasons I like to teach kids about it, is because you can do more interesting things with 12 than 10, because it has more factors. Meaning you can divide 12 by 2, 3, 4 and 6. 10 can only be divided by 2 and 5.

My approach to teaching about other number systems is to get kids to think about why we count to ten in the first place. We got ten fingers. Then you could ask how a dog would count using its legs. It could only count to 4. We could then call four “one dog”. So to represent the number ten, we would say that is 2 dogs and 2 legs:

2 dogs + 2 legs = 22₄ = 2×4 + 2 = 10₁₀

The small numbers (subscripts) such as the four in 22₄, indicates that this number is written in base ten. 10₁₀ means it is written base ten system, or our regular decimal number system.

Octal Numeral System

We could continue the experiment with other animals such as octopus,

which has 8 tentacles. It would then represent ten (10₁₀) as 1 octopus and 2 tentacles:

1 octupus + 2 tentacles = 12₈ = 1×8 + 2 = 10₁₀

A number base with 8, is called the octal numeral system. It is occasionally used in computer science. When we count up to 8, it means we only have the digits:

0 1 2 3 4 5 6 7

So when writing numbers with multiple digits we have to think different. For the decimal system we have to multiple each digit with some power of 10. Each when reading a number such as 74 using base 10 (decimal), we multiple 7 with 10 and 4 with 10 to the power of zero, which is 1.

74₁₀ = 7 × 10¹ + 4 × 10⁰ 
= 7 × 10 + 4 × 1
= 70 + 4

If we do the same for the octal system we need to multiply each digit with a power of 8.

112₈ = 1×8² + 1×8¹ + 2×8⁰ 
= 1×64 + 1×8 + 2×1
= 64 + 8 + 2 = 74₁₀

So you can see that 112 in octal corresponds to 74 in decimal numeral system.

While this might look odd we already do this with many measurements. In the American system my height is:

5 feet + 11₁₀ inches = 5'11'' = 5×12₁₀ + 11₁₀ = 71₁₀

Just like 1 octopus is made up of 8 tentacles, 1 foot is made up of 12 inches. We can thus write heights using the dozenal system. With the dozenal system we have added 2 extra digits E and X, to be able to count up to 12 without using multiple digits. So my heigh could be written as:

5 feet + E inches =  5×10₁₀ inches + E inches = 5E inches

So basically you would read 10 as 1 dozen and 0 ones. 5E is 5 dozens and E (11) ones.

Dozenal Counting

While base twelve numbers might be practical in a lot of ways, you’d probably object that it is highly impractical to work with as you can’t count to twelve on your fingers easily. Or can you?

The trick is to use your thumb as a number pointer. You move your thumb to the segment on your fingers representing a number from 1–12. I found it surprisingly easy to teach my kids to count this way. The advantage of knowing to count this way is that they then have easy access to far more numbers on their hands to do arithmetic. It also makes it easy to demonstrate multiplication and division with 2, 3, 4 and 6 easily using just fingers.

They can learn multiplying with 3 as simply counting whole fingers. 2 fingers represents 2 × 3 = 6, and 3 fingers represent 3 × 3 = 9.

A whole hand is 12 represented by four fingers, since we exclude the thumb for counting purposes. Dividing 12 by 3 can then be thought of this way:

One finger is 3. One hand is 12. How many fingers is there on a hand (excluding the thumb)? Likewise you could ask them to divide by 6, by asking how many double fingers there is on a hand.

Advantages of Dozenal Numbers over Decimal Numbers when Multiplying

Base twelve is actually a lot easier to learn the multiplication table for. Consider the decimal multiplication table. You will notice there are a lot of irregular number sequences.

If you look along either a column or row for the base twelve multiplication table you will see that the last digit will cycle between a few predictable ones, making it easier to remember.

Recap

The idea here is really just about learning to think about numbers. Even though you don’t really need to know about octal or dozenal numbers it helps your general number understanding, knowing about them. If you have to work with numbers which are dozenal in nature such as time, feet and inches then using the dozenal system would make it easier to perform calculations on them.

Next medium article I’ll show you how to write code to convert between dozenal and decimal numbers.