# How to Convert to Base-12

The base-12 numbering system is also known as duodecimal and dozenal. Because 12 is divisible by the prime factors 2 and 3, one of the advantages of the duodecimal system is that fractions over 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, etc (i.e., integers whose prime factorization contains only 2s and/or 3s) can all be expressed as terminating decimals. By comparison, in base-10, any fraction whose denominator is divisible by 3 yields a repeating decimal.

Base-12 uses the digits 0 through 9 as well as the digits A and B to represent ten and eleven respectively. There are other symbols you can use for ten and eleven; A and B are arbitrary but probably more natural to those familiar with hexadecimal notation.

The easiest way to convert a decimal (base-10) number to a base-12 number is with the remainder method. It is also possible to convert a decimal number to base-12 by expressing it as the sum of multiples of powers of 12, but this method is more cumbersome. Below is the general method of conversion and several examples of converting in both directions.

## Remainder Method

To convert a number from base-10 to base-12, successively divide the number by 12, rounding the quotient down to the nearest whole number and noting the remainder at each iteration. Stop when the quotient is 0. Then concatenate the remainders in reverse order, starting from last to first, to create the base-12 representation of the number. Let's work out an example using the decimal number 45097.

45097÷12 = 3758 remainder 1

3758÷12 = 313 remainder 2

313÷12 = 26 remainder 1

26÷12 = 2 remainder 2

2÷12 = 0 remainder 2

Reading the remainders backward gives us the string 22121. Thus, the base-12 representation of the decimal number 45097 is 22121.

## How to Check Your Work

To check if you computed the base-12 conversion correctly, you need to understand what the base-12 string represents in terms of powers of 12. Let's start by breaking down what a decimal string means in terms of powers of 10. For example, the number 45097 is equivalent to

4*10^4 + 5^10^3 + 0*10^2 + 9*10^1 + 7*10^0

Then the duodecimal number 22121 must be the same as

2*12^4 + 2*12^3 + 1*12^2 + 2*12^1 + 1*12^0

If you compute this expression in base-10, you get exactly 45097, so our base-12 calculation is consistent and correct.

## Another Example

Let's convert 131880971_{dec} to a base-12 integer. We start by computing the list of remainders:

131880971÷12 = 10990080 remainder 11

10990080÷12 = 915840 remainder 0

915840÷12 = 76320 remainder 0

76320÷12 = 6360 remainder 0

6360÷12 = 530 remainder 0

530÷12 = 44 remainder 2

44÷12 = 3 remainder 8

3÷12 = 0 remainder 3

Remember that the digit eleven is represented by the symbol B in duodecimal, so reading the remainders from bottom to top we get 3820000B as the base-12 representation of 131880971_{dec}.

To check our work, we compute

3*12^7 + 8*12^6 + 2*12^5 + 0*12^4 + 0*12^3 + 0*12^2 + 0*12^1 + 11*12^0

= 131880971 in base-10,

which confirms that our base conversion is correct.

## Other Base Conversions

## Converting the Long Way

The remainder method is the most efficient way to convert from one base to another, but you can also find the dozenal representation of a number by expressing the decimal number as a a sum of multiples of powers of 12. For example, let's convert 204635_{dec} to a base-12 integer. We start by finding the logarithm of 204635 in base 12 and rounding it down to the nearest integer:

log_{12}(204635) = 4.921 → 4

This means that the highest power of 12 that divides into 204635 is 12^4 = 20736. The quotient of 204635 by 20736 gives us the first digit of the the base-12 representation:

204635÷20736 = **9** remainder 18011

Now we divide the remainder by the next highest power of 12, which is 12^3 = 1728. This gives us the second digit of the base-12 string:

18011÷1728 = 10 remainder 731 = **A** remainder 731

Continuing on with the procedures gives us

731÷144 = **5** remainder 11 = 5 remainder B

B÷12 = **0** remainder B

B÷1 = **B** remainder 0

Therefore the original number converted to base-12 is **9A50B**.

This is similar to the process of computing remainders demonstrated above, but instead of constantly dividing by 12, we divide by decreasing powers of 12 all the way down to 12^0 = 1. And instead of reading off the *remainders backward*, we read off the *quotients forward*.

## Other Representations of Base-12 Digits

Using A = 10 and B = 11 is the most common way to write digits in base-12, but there are other sets of symbols people use based on the words "ten" and "eleven." Here are a few

- English: T = 10, E = 11
- Latin: D = 10, U = 11
- Roman/English: X = 10, E = 11
- Greek: Δ = 10, Ε = 11

Some also use the rotated 2 symbol "ᘔ" for ten and the rotated 3 symbol "Ɛ" for eleven, because ᘔ looks like a "T" and Ɛ looks like an "E." These alternative symbols for ten and eleven are not so strange, considering that 6 and 9 are also 180-degree rotations of one another.

## Representations of Common Numbers in Base-12

In duodecimal, twelve is represented as 10, thirteen is represented as 11, 60 is represented as 50, and 144 is represented as 100.

In duodecimal you also avoid repeating decimal representations of common fractions such as 1.3, 2/3, 1/6, 5/6, 1/9, etc. In base-12, these are represented as

1/3 = 0.4

2/3 = 0.8

1/6 = 0.2

5/6 = 0.A

1/9 = 0.14

However, 1/5 becomes a repeating decimal in base-12, it's 0.24972497...

Irrational numbers are non-repeating decimals regardless of the integer base. For example, the dozenal representation of the mathematical constant pi is

π = 3.184809493B918664573A6211BB151551A05729290A7809A492..

## Adoption of the Dozenal System

Throughout the years, many people have advocated for the adoption of a dozenal system of counting, citing numerous advantages over the decimal system. For example, the number 12 is deeply involved in the way we measure time -- 5*12 seconds in a minute, 5*12 minutes in an hour, 2*12 hours in a day, 12 months in a year. Multiples of 2 and 3 are also integral in the Imperial system of weights and measures:

- 16 cups = 8 pints = 4 quarts = 1 gallon
- 36 inches = 3 feet = 1 yard
- 16 ounces = 1 pound
- 1 gross = 12 dozen = 144 units

However, the Imperial system also makes use of 5s, 7s, 11s, so one could use the same argument to advocate for a base-14 or base-15 system. For example

- 1 gallon = 3*7*11 cubic inches = 231 cubic inches
- 1 mile = 32*3*5*11 feet = 5280 feet
- 1 stone = 2*7 pounds = 14 pounds.
- 1 week = 7 days
- 1 lunar month = 4*7 days = 28 days

It is interesting to note that the Dozenal Societies of the UK and US prefer the term "dozenal" rather than "duodecimal" since the latter is a Latin construction derived from the base-10 system of counting.

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