Symmetric senary

Symmetric senary is a tiny numeral system which is intended for people to use in their everyday life. It is especially useful for dividing things up in practical ways, and easily estimating things like tips and mileage without ever needing a calculator.

A numeral system is a way of writing numbers. There are many ways of writing numbers. For example, the first four natural numbers can be written as 1, 2, 3, 4 using the decimal system, or as •, ••, •••, •••• using the tally system, or as I, II, III, IV using the Roman system.

All three numeral systems represent the same numbers. However, one of these systems is far better than the others. There would be massive repercussions if the world were forced to revert to Roman numerals, and our situation would be absolutely dire under the tally system.

The choice of our numeral system is very important, but it is not generally given much thought.

In this article, I argue that the perfect numeral system for everyday life is positional, balanced, and written in base six. This gives us the perfect numeral system: symmetric senary.

Here are the seven digits:

3 2 1 0 1 2 3

The red, underlined digits represent negative numbers. For example, the digit 1 represents the decimal value -1. As a result, 3 - 1 = 2 in decimal is equivalent to 3 + 1 = 2 in symmetric senary. The negative digits provide the same functionality as subtraction and negation.

Symmetric senary is a positional numeral system: the value of a digit depends on its position in the number. For example, in the decimal number 11, the digit “1” can represent the value one or the value ten.

Symmetric senary is a positional numeral system based on the number six. The number 10 represents the value six (61 = 6). The number 100 represents six groups of six (62 = 36)—and so on.

The largest digit is 3, but you can continue counting all the way to six if you make use of the negative digits. For example, the next number after three is 4 = 6 - 2 = 10 + 2 = 12.

Here are the natural numbers from one to fifteen:

1 2 3 12 11 10 11 12 13 22 21 20 21 22 23

You can see that the “milestone” numbers like 10 dominate their local environment: all nearby numbers start with 10 plus a correction term. This makes it easy to estimate and round numbers.

Here is a calculator that can convert numbers from decimal to symmetric senary:

Symmetric senary

You can also use this calculator to convert fractions and decimals. For example:
1/2 = 0.3
1/3 = 0.2

Numbers in symmetric senary tend to have beautiful patterns. For example, every prime number greater than 3 ends in a 1 or a 1.


Here are the arithmetic tables in symmetric senary:

+ 1 2 3
1 2 3 12
2 3 12 11
3 12 11 10
+ 1 2 3
1 0 1 2
2 1 0 1
3 2 1 0
× 2 3
2 12 10
3 10 13

In addition, it’s useful to know the basic properties of numbers:

Multiplying by 1 negates a number—flipping the sign of every digit in the number. For example:


Numbers in symmetric senary can end in a positive or a negative digit. As a result, when adding columns of numbers, the trailing digits tend to cancel out. For example:

12 11 + 13
2 1 + 33

The number 3 is halfway between 0 and 10, so it can be written in two equivalent ways: 3 or 13. You can use these two forms interchangeably when solving addition problems. For example:

3 + 3
3 + 13



Many numbers have both positive and negative digits. When settling a debt, the debtor pays the positive digits, and receives the negative digits back in change. The number itself shows the most efficient way to make change, without having to rely on a cash register.

For example, if your bill comes to 30.11, the most efficient way to pay is to give the cashier three 10 bills and a coin worth 0.10. The cashier can then give you a coin worth 0.01 in exchange, and the debt is settled.

In each transaction, you tend to both give and receive. Because you’re often giving coins away, you tend to cycle through your minor currency quickly. It generally does not accumulate in your pockets or in an enormous coin jar at home.

At some point, you may actually run out of coins—and therefore be unable to continue paying in the most efficient manner. In that case, you’ll be forced to pay in larger denominations. For example, if you owe 0.10, you can give the cashier a 1 bill. The cashier will change your 1 bill into six 0.10 coins. The cashier will then take one of the coins to settle the bill, and give the rest to you. That means you’ll be stuck with five extra coins. It’s inefficient, but it’s no different from how transactions always work out in practice under the current decimal system.


Halves and thirds have simple decimal expansions. For example, 1/2 = 0.3 and 1/3 = 0.2. As a result, you can use the convenience of the decimal system to scale your recipes for more or fewer guests, without needing to work in improper fractions.

Graphic design

Graphic design often involves calculating simple, harmonious ratios. The ratios 2 and 3 are particularly powerful. When you choose your widths and lengths using symmetric senary, your graphic elements can be subdivided in many visually impactful ways. For example, if you ever need to divide a 100 px photograph into thirds, you’ll find that each square of the grid has a perfect 20 px width.

Lengths chosen according the decimal system don’t have such nice divisibility properties. For example, if you need to divide a 100 px photograph into thirds, each square of the grid will have a width of 33.3333… px. This will often cause one-pixel display errors in graphic design software. Web browsers can even catastrophically miswrap floated div elements for exactly this reason.


An extra digit

The most common everyday numbers tend to gain a digit when they are converted from decimal to symmetric senary. For example, the number 4 in decimal becomes 12 in symmetric senary. This makes it a little more difficult to remember a long number, such as an important phone number.

Other numeral systems, such as symmetric dozenal, have fewer digits than decimal. However, these systems pay for their short numbers through the increased difficulty of their arithmetic tables. In symmetric senary, arithmetic is very easy.