Duodecimal World | Lucy Lu Wang

A modest proposal for base twelve

I dream of a world where 10 ÷ 3 = 4 and 10 ÷ 6 = 2.

Twelve digits would have helped.

The rationale

Twelve has better properties.

0123456789∂β

Humans have been using base ten for millennia. It has mostly served its purpose, but it is not easily divisible by most numbers smaller than ten (only 1, 2, and 5).

Here, 10₁₀ is the decimal numeral ten, while 10₁₂ is the duodecimal numeral ten—equal to twelve in decimal. Base twelve has much cleaner divisions for most of these numbers (aside from 5, 7, and ∂):

Divide by	10₁₀	10₁₂
2	5	6
3	3.3	4
4	2.5	3
5	2	2.4972
6	1.6	2
7	1.428571	1.86∂351
8	1.25	1.6
9	1.1	1.4
10 a.k.a. ∂	1	1.2497
11 a.k.a. β	0.90	1.1
12 a.k.a. 10₁₂	0.83	1

Revising intuitions

Make room for ∂ and β.

Addition is relatively straightforward, but we need to recalibrate around the two new numerals:

1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = ∂

1 + ∂ = 2 + 9 = 3 + 8 = 4 + 7 = 5 + 6 = β

1 + β = 2 + ∂ = 3 + 9 = 4 + 8 = 5 + 7 = 6 + 6 = 10

This extends to two and three digits once the single-digit sums feel familiar:

13 + 16 = 2914 + 17 = 2β15 + 18 = 31

Subtraction

For subtraction, recalibrate around the same new numerals:

10 − 1 = β
10 − 2 = ∂
β − 1 = ∂
β − 2 = 9
β − 3 = 8
β − 4 = 7
β − 5 = 6
β − 6 = 5
β − 7 = 4
β − 8 = 3
β − 9 = 2
β − ∂ = 1

∂ − 1 = 9
∂ − 2 = 8
∂ − 3 = 7
10 − β = 1
11 − β = 2
12 − β = 3
19 − β = ∂
1∂ − β = β
1β − β = 10
10 − ∂ = 2
19 − ∂ = β
1β − ∂ = 11

The pattern continues normally: 20 − β = 11 and 20 − ∂ = 12.

Exercise · addition & subtraction

13 + 16 = ?

Multiplication

A new table to memorize.

We did not start out knowing our decimal multiplication tables. In a new base, we get to learn them again:

Useful anchors

2 × 6 = 10
3 × 4 = 10
4 × 3 = 10
6 × 2 = 10
6 × 6 = 30
8 × 9 = 60
∂ × ∂ = 84
β × β = ∂1

The new structure is cleaner: 10 is evenly divisible by 2, 3, 4, 6, giving 6, 4, 3, 2 respectively.

Exercise · multiplication

3 × 4 = ?

Division

Follow the multiples.

Division works as usual, only with two extra numerals. This lookup table is a long-division practice companion. Given the dividend in the table, its divisor is on the left and the two-digit quotient across the top.

Cleaner fractions

Many common fractions terminate after a single duodecimal digit:

1/2 = 0.61/3 = 0.41/4 = 0.31/6 = 0.22/3 = 0.83/4 = 0.95/6 = 0.∂1/5 = 0.2497

Exercise · division

10 ÷ 3 = ?

Telling time

The clock was already halfway there.

The familiar day already divides neatly into twelves. The simple option is to write conventional time in duodecimal.

Which means that a day has 20 hours: 10 hours AM and 10 hours PM. Each hour has 50 minutes, and each minute has 50 seconds. These quantities are unchanged; only their notation is different.

Conventional	Duodecimal notation
Midnight	0:00
6:00 AM	6:00
Noon	10:00
6:00 PM	16:00
11:00 PM	1β:00
Midnight	20:00