### 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.

 Decimal Senary Decimal Senary Decimal Senary Decimal Senary 1 1 13 216 25 416 37 1016 2 2 14 226 26 426 38 1026 3 3 15 236 27 436 39 1036 4 4 16 246 28 446 40 1046 5 5 17 256 29 456 41 1056 6 106 18 306 30 506 42 1106 7 116 19 316 31 516 43 1116 8 126 20 326 32 526 44 1126 9 136 21 336 33 536 45 1136 10 146 22 346 34 546 46 1146 11 156 23 356 35 556 47 1156 12 206 24 406 36 1006 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.

 Number Name Number Name Number Name Number Name Number Name 1 one 216 twosen-one 416 foursen-one 1016 one hex one 1216 one hex twosen-one 2 two 226 twosen-two 426 foursen-two 1026 one hex two ... ... 3 three 236 twosen-three 436 foursen-three 1036 one hex three 2436 two hex foursen-three 4 four 246 twosen-four 446 foursen-four 1046 one hex four ... ... 5 five 256 twosen-five 456 foursen-five 1056 one hex five 5556 five hex fisen-five 106 sen 306 thrisen 506 fisen 1106 one 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