What is senary?

Senary is a numeric system using a base of six instead of ten.

Our normal base system is called decimal because it is based on the number ten. This is seen in the concept of place value: the number 2345 represents 2 sets of a thousand, 3 sets of a hundred, 4 sets of ten, and 5 ones. Each place value represents an additional power of ten: a ten is ten ones, a hundred is ten sets of ten, a thousand is ten sets of a hundred, et cetera. (For more information about base systems, see the article on Base at Wolfram MathWorld, or Positional Notation at Wikipedia)

But who decided that we should group things by tens? Why not another number? Senary is a system where things are grouped in sets of six. The number 2345 in senary would represent 2 sets of 216 (i.e. six sets of 36, or six cubed), 3 sets of 36 (i.e. six sets of 6, or six squared), 4 sets of 6, and 5 ones. Each place value represents an additional power of six. Thus, 2345 in senary would represent a total of 2(216) + 3(36) + 4(6) + 5 = 569 in decimal.

Since the sets in decimal and senary are of different sizes, 2345 in decimal and 2345 in senary represent two very different values. As a result, we need a way of distinguishing between a number in decimal and a number in senary. There are a couple of ways to do this. The most common way is to use a subscript 6 to designate that a number is written in a base of six (in senary), e.g. 2345₆. Recognizing that numbers in decimal and senary represent different values, it is helpful to make a table showing the corresponding numbers in decimal and senary. The corresponding numbers in the table below have the same value, but are represented differently because of the different base.

(To see other corresponding numbers in decimal and senary, visit the Base Conversion tool at the Math Is Fun website.)

It is interesting to see the difference in counting in senary as opposed to decimal. Note that the highest digit ever used is a 5. This is because anytime there is a group of six items, we form an additional larger group of those six items. In decimal, once we have ten items, we call it a group of ten. Once we have ten groups of ten, we call it a group of a hundred. Once we have ten groups of a hundred, we call it a thousand. And it continues. But what do we call groups of six? Or six groups of six? Or six groups of six groups of six? If we are to talk about numbers in senary, it is helpful to have terms for the various place values in senary. Instead of saying a number is "two-three-four-five in base six", or saying "2 sets of 216, 3 sets of 36, 4 sets of 6, and 5 ones," we designate terms to represent each place value or group of blocks. Thus, a group of six is called "sen", six groups of six is called a "hex", and six groups of six groups of six is called a "kilsen". The table below shows the name of several numbers.