### UPC Number (Universal Product Code) (10/19/2001)

Prepared by:

Joseph Malkevitch

Mathematics and Computing Department

York College (CUNY)

Jamaica, NY 11451

Email: malkevitch@york.cuny.edu (for additions, suggestions, and corrections)

The Universal Product Code, UPC number was developed to help retailers track and inventory
the sale of their products and speed customers through checkout lines. The code went
into common usage about 1973. In America, there are a variety of UPC systems in use that vary from one environment to another. Three major variants are:

1. The use of a 12 digit code, broken up into 4 fields. The first field, one digit
long, identifies some aspect of the item, for example, does it have variable weight,
is it a price reduction coupon, etc. The second field, 5 digits long identifies the
manufacture of the product, the third field, 5 digits long identifies the item number
of the particular manufacturer, and the last field, one digit long, is a check digit.

2. The use of a 8 digit code, broken into 3 fields, which is designed for small and
round objects. This is known as the Version E code and is surprisingly complex. There
are various procedures for reducing the 11 digit code (without the check digit) to
this 7 digit code (without the check digit).

3. The use of of a 13 digit code for books where the first digit is 9 and then there
are two block of 6 digits are displayed. This system is used to code the ISBN number
of the book together with a check digit.

For the basic system mentioned in item 1 above, the check digit in position 12 is
chosen so that 3 times the sum of the digits in the odd positions added to the numbers
in the even positions is a multiple of 10.

Reference:

Gallian, J., Modular arithmetic in the marketplace, American Mathematical Monthly
95 (1988) 548-551.

Malkevitch, J. et al, For All Practical Purposes, Fifth Edition, W.H. Freeman, New
York, 2000.

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