# Dozen

A dozen (doz or dz) is a set (or group) of 12 items.

Groupings of twelve are thought to be one of the earliest primitive groupings (e.g. 12 months in a year), possibly due to the convenience of the number 12. 12 is the smallest number with four factors (not including 1 and itself): 2, 3, 4, and 6. This means that it can be easily divisible into halves, thirds, quarters, or sixths, which can be helpful for packaging or distributing different things.

Eggs, cookies, donuts, flowers, and disposable cups are just a few examples of items that are commonly packaged or sold by the dozen.

There are a few other measures that are related to the dozen. They include the baker's dozen, gross (twelve dozen), great gross (twelve gross), and small gross (ten dozen):

Dozen | 12 |

Baker's dozen | 13 |

Small gross | 120 (12 × 10) |

Gross | 144 (12^{2}) |

Great gross | 1728 (12^{3}) |

## Dozenal (duodecimal) system

The duodecimal system, like the decimal numeral system (the most widely used system in the world), is a positional numeral system. It is sometimes referred to as the base 12 system, in the same manner that the decimal system is referred to as base 10. The system originated in Mesopotamia along with other systems such as the sexagesimal system (base 60).

In the duodecimal system, each position represents a power of 12, where the value of a digit in a given position is that digit multiplied by the power of 12 represented by that position. For example, the numeral "10" in the duodecimal system does not represent the number "ten" as it would in the decimal system. In the duodecimal system, "10" represents the number 12 (in the decimal system), while 12 in duodecimal is 1 dozen and 2 units, or 14 in the decimal system. Regardless of the base being used in a positional numeral system, the same concepts apply. Briefly, expanding 10 and 12 in the duodecimal system works as follows:

10 = 1 × 12^{1} + 0 × 12^{0} = 12 + 0 = 12

12 = 1 × 12^{1} + 2 × 12^{0} = 12 + 2(1) = 14

As a comparison, below is the expansion of 10 and 12 in the decimal numeral system:

10 = 1 × 10^{1} + 0 × 10^{0} = 10 + 0 = 10

12 = 1 × 10^{1} + 2 × 10^{0} = 10 + 2(1) = 12

This page won't go into detail on converting between the duodecimal and decimal system, since it is not widely used, but it does have some advantages over the decimal system. One of these is that some of the more common fractions used in the decimal system have simpler, terminating decimal representations in the duodecimal system:

### Did you know?

One of the prevailing theories for the origin of the baker's dozen dates back to medieval England, where strict laws dictated the price of bread based on the amount of wheat used to make it. Bakers could be severely punished for selling undersized loaves, so rather than risk accidentally selling a loaf that was too small, bakers would include an extra 1 (sometimes even 2) loaves in the "dozen."