## The Beauty of Base Twelve

*By Graham Everhart *

Suppose I have a collection of circles:

⬤

⬤⬤

⬤⬤⬤

⬤⬤⬤⬤

⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤

et cetera.

I want to say that I have ⬤⬤⬤⬤⬤⬤⬤ circles. Should I say ⬤⬤⬤⬤⬤⬤⬤ each and every time? Or is there a more efficient way to do that?

There is: “7.” *I have 7 circles.* Numbers are shortcuts for expressing quantity. We instantly know that 7 means ⬤⬤⬤⬤⬤⬤⬤.

But what if I have more than ⬤⬤⬤⬤⬤⬤⬤ circles? What if I have ⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤ circles?

Is it reasonable to have to learn ⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤ different symbols like 7 to express quantity?

Humans solved this problem by breaking large quantities into smaller groups:

⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤⬤

Notice how there are ⬤⬤ (two) whole groups of ⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤ (ten) circles each, and a partial group of ⬤⬤⬤⬤⬤⬤⬤ (seven) circles.

We keep the number of circles in each group constant, so we don’t need to indicate that. The important information is the ⬤⬤ and the ⬤⬤⬤⬤⬤⬤⬤, which we represent with a “2” for the number of whole groups and a “7” for the quantity of the partial group.

Hence, we represent ⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤ as “27.”

The fact that our group size is always ten means that our number system always uses ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. That’s because ⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤ is the smallest quantity to not have its own symbol—it has the fundamental symbol “1” boosted a place value to “10,” rather than a symbol beyond “9.”

But why ⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤ specifically? Why not ⬤⬤⬤ or ⬤⬤⬤⬤⬤ or ⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤?

Let’s take my “27” circles and group them in some different ways.

Our group size could be ⬤⬤⬤⬤⬤⬤⬤⬤ (eight):

⬤⬤⬤⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤⬤⬤

⬤⬤⬤

We now have ⬤⬤⬤ (three) groups of ⬤⬤⬤⬤⬤⬤⬤⬤ (eight) each, with a partial group of ⬤⬤⬤ (three). ⬤⬤⬤ and ⬤⬤⬤ are our important pieces of information, so we would represent this as “33.” This system would use eight symbols: 0, 1, 2, 3, 4, 5, 6, 7.

Our group size could be ⬤⬤⬤⬤⬤⬤ (six):

⬤⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤

⬤⬤⬤⬤⬤⬤

⬤⬤⬤

We now have ⬤⬤⬤⬤ (four) groups of ⬤⬤⬤⬤⬤⬤ (six) each, with a partial group of ⬤⬤⬤ (three). ⬤⬤⬤⬤ and ⬤⬤⬤ are our important pieces of information, so we would represent this as “43.” This system would use six symbols: 0, 1, 2, 3, 4, 5.

The number of circles has not changed. There were always ⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤⬤ circles, but we found different ways to represent them. “27” and “33” and “43” all refer to the same quantity—they just use different groupings and therefore different symbols.

The number system that uses ten symbols is called base ten and is also known as the *decimal system.* It’s the number system we use the most often.

Our first example used eight symbols, which would make it base eight. Our second example used six symbols, which would make it base six. Computers use base two, also known as *binary.* Some computer applications use base sixteen, also known as *hexadecimal.*

With such a wide array of options, is base ten really the best number system to use?

Mathematically speaking, base ten is not very useful. Ten is divisible by two and five. This means 10 ÷ 2 = 5 and 10 ÷ 5 = 2, but what about 10 ÷ 3? That becomes a very ugly 3.33333333333. ⅓ becomes 0.333333. ⅙ becomes 0.166667. Because of the limitations of the decimal system, fractions like those are really awkward to use in decimal form.

The number twelve, on the other hand, is divisible by two, three, four, and six. 12 ÷ 2 = 6 and 12 ÷ 3 = 4 and 12 ÷ 4 = 3 and 12 ÷ 6 = 2. Gene Zirkel cites this abundance of divisors as why humans based so many units of measurement not on ten, but on twelve:

- Twelve inches in a foot
- Twelve items in a dozen
- Twelve notes in a musical octave
- Twelve times five seconds in a minute
- Twelve times five minutes in an hour
- Twelve times two hours in a day
- Twelve months in a year
- Twelve times thirty degrees in a circle

If you need one-third of something, 4/12 of a foot is more intuitive than 0.333333 of a meter. If you need one-fourth of something, 3/12 of a foot is slightly more intuitive than 0.25 of a meter.

What about applying the divisibility of twelve to our numbers themselves?

The *dozenal system,* also known as the *duodecimal system,* is base twelve. It was first promoted by Sir Isaac Pitman in 1857. In dozenal, a rotated 2 (ᘔ) is the number after 9, and a rotated 3 (Ɛ) is the number after ᘔ.

Here’s a few side-by-side counting comparisons:

Decimal form | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

Dozenal form | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ᘔ | Ɛ | 10 |

Decimal form | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

Dozenal form | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1ᘔ | 1Ɛ | 20 |

Decimal form | 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

Dozenal form | 75 | 76 | 77 | 78 | 79 | 7ᘔ | 7Ɛ | 80 | 81 | 82 | 83 | 84 |

Decimal form | 133 | 134 | 135 | 136 | 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 |

Dozenal form | Ɛ1 | Ɛ2 | Ɛ3 | Ɛ4 | Ɛ5 | Ɛ6 | Ɛ7 | Ɛ8 | Ɛ9 | Ɛᘔ | ƐƐ | 100 |

There are many different ways of pronouncing these new numbers. This is one of them:

- The digit ᘔ is pronounced “ten.”
- The digit Ɛ is pronounced “eleven.”
- “10” in dozenal is pronounced “dozen.”
- “20” in dozenal is pronounced “twozen.”
- “30” in dozenal is pronounced “threezen.”
- “100” in dozenal is pronounced “hundzen.”
- “1000” in dozenal is pronounced “thouzen.”

Here are some pronunciation examples.

Number | Dozenal pronunciation |

1ᘔ | dozen-ten |

Ɛ8 | elevenzen-eight |

420 | four hundzen twozen |

666 | six hundzen sixzen-six |

1337 | one thouzen, three hundzen threezen-seven |

802,701 | eight hundzen two thouzen, seven hundzen one |

10,000,000 | dozen milzen |

Ɛᘔ9,000,000,000 | eleven hundzen tenzen-nine bilzen |

But dozenal’s real advantage is that fractions in dozenal form tend to be simpler than fractions in decimal form. Fewer dozenal-form fractions repeat infinitely, and dozenal-form fractions also tend to use use fewer digits than decimal-form fractions. Take a look:

Fraction | Decimal form | Dozenal form | Which form is simpler? |

1/2 | 0.5 | 0.6 | Tie |

1/3 | 0.333333… | 0.4 | Dozenal |

1/4 | 0.25 | 0.3 | Dozenal |

1/5 | 0.2 | 0.24972497… | Decimal |

1/6 | 0.166666… | 0.2 | Dozenal |

1/7 | 0.142857… | 0.186ᘔ35… | Tie |

1/8 | 0.125 | 0.16 | Dozenal |

1/9 | 0.111111… | 0.14 | Dozenal |

1/10 (1/ᘔ) | 0.1 | 0.124972497… | Decimal |

1/11 (1/Ɛ) | 0.090909… | 0.111111… | Dozenal |

1/12 (1/10) | 0.083333… | 0.1 | Dozenal |

Total |
2 |
7 |
Dozenal |

“But Graham, you can’t count to twelve on your fingers!”

https:/saintpat1985.wordpress.com/2017/04/23/why-we-should-have-been-born-with-twelve-fingers/

Checkmate, decimalists.

Have any questions or thoughts on the dozenal system? I’d love to hear them! Feel free to email me at everhgra000@edmonds15.org or talk to me in person.