The Mathematics
of Barcodes |
## Student Glossary**Binary number system**- Base two
number system as opposed to the decimal (base 10) system which is the standard
system. For example, 5 in base 10 is 101 in binary (1 four, no twos,
and 1 one).
**Complement Rule**- The sum of the probabilities
of an event happening or not happening is one [P(a) + P(not a) = 1].
For example, if the probability of rain is .4, the probability of not having
rain is .6.
**Congruence ()**- Symbol used to describe
numbers in modular arithmetic that are equivalent to each other.
These numbers have the same remainder when divided by the modulus.
In mod 10, for example, 23133.
**Counting Principle**- Method used to find
the number of ways of making several decisions in succession. To
find the number of ways, multiply the number of choices that can be made
in each decision.
**Modular Arithmetic**- An arithmetic system
based on the remainder obtained when dividing a given number by the base
number which is called a modulus.
**Permutation**- Any ordered arrangement
of a set of objects. In the coding permutation (1248), 1 is coded
as 2, 2 becomes 4, 4 is coded as 8, and 8 becomes 1.
**Probability**- The chance or likelihood
of an event happening. The theoretical probability of an event happening
is defined as the number of favorable outcomes divided by the total number
of outcomes. When rolling a die, for example, the chance of getting
an even number is 3/6 because 3 of the 6 numbers on a die are even.
## UPC Encoding Scheme- It codes only numbers from 0 to 9.
- A narrow dark bar indicates a binary one and a narrow white space a binary zero.
- A bar or space may be from 1 to 4 spaces wide.
- Each digit is coded by 2 bars and 2 spaces occupying a total of 7 spaces.
- Digits to the left of the center pattern are coded space-bar-space-bar.
- Digits to the right of the center pattern are coded bar-space-bar-space.
- Each digit has two binary codes. One is for the left hand of the center pattern and one for the right.
- Digits to the left of the center pattern are made up of an odd number (3 or 5) of binary ones (odd parity) and the digits on the right are made up of an even number (2 or 4) of binary ones (even parity).
- The binary number that is the used for each number is the binary code, not the binary representation for that number.
- The check digit is the twelfth digit encoded into the symbology but is not necessarily printed in numeral form.
- Each bar code consists of a right and left quiet zone, a right and left guard pattern, a center pattern and of course the twelve number codes.
- The quiet zone is 11 times the narrow bar width on the left and 7 times the narrow bar width on the right.
- The guard pattern for each side is bar-space-bar (101).
- The center pattern divides the 12 numbers in half and consists of space-bar-space-bar-space (01010).
- A complete UPC-A symbology is made up of 30 bars and 29 spaces.
- The complete symbology including quiet zones at 100% magnification is approximately 1.5 inches horizontally and 1.0 inches vertically.
Table 1 - UPC ENCODING
SCHEME
Bar Codes
from common household products
Area 10 Mathematics and Technology Professional Development Center Permission is granted to duplicate these materials for classroom use.
Last updated on 1/30/1999 |