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)

Base 10 Blocks showing 2 thousand, 3 hundred, 4 tens, and 5 ones.
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.   
2345 in base six represented by base six blocks.

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.

 Decimal Senary  DecimalSenary   DecimalSenary   DecimalSenary 
 1  13216   25416   371016 
 2  14226   26426   381026 
 3  15236   27436   391036 
 4  16246   28446   40 1046 
 5  17256   29456   41 1056
 6106   18306   30506  42 1106
 7116   19316   31516  43 1116 
 8126   20326   32526  44 1126 
 9136   21336   33536  45 1136 
 10146  22346   34546  46 1146 
 11156  23356   35556  47 1156 
 12206   24406   361006  48 1206 

(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.

 NumberName  Number Name  NumberName  NumberName   NumberName 
one  216 twosen-one 416 foursen-one  1016  one hex one  1216 one hex twosen-one
 two 226 twosen-two  426 foursen-two  1026 one hex two   ......
 three 236 twosen-three  436 foursen-three  1036 one hex three  2436 two hex foursen-three
 four 246 twosen-four  446  foursen-four 1046 one hex four   ... ...
 five 256 twosen-five  456 foursen-five  1056one hex five   5556 five hex fisen-five
106  sen 306 thrisen  506 fisen  1106one hex sen   10006 one kilsen
116  seno 316 thrisen-one  516 fisen-one  1116 one hex seno   ... ...
126  sentoo 326 thrisen-two  526 fisen-two  1126 one hex sentoo   23456 two kilsen, three hex foursen-five
136  sentri 336 thrisen-three  536 fisen-three  1136 one hex sentri   ...... 
146 senfor 346 thrisen-four  546 fisen-four  1146 one hex senfor   1,000,0006 one milsen
156 senfi 356 thrisen-five  556 fisen-five  1156 one hex senfi   ... ...
206  twosen 406 foursen  1006 one hex  1206 

one hex twosen   1,000,000,0006 one gigsen