The Mathematics
of Barcodes |
Students are made aware of the UPC-A and IBM bar coding systems, and use modular arithmetic to understand the meaning of the code numbers.
## Introduction:You may have noticed that many things you buy come with a bar code. These activities will help you decipher such codes. What is clock or modular arithmetic? Look at the student copy for a definition. Put this definition in your own words. What does congruent mean? Again, look at the student copy and put the definition in your own words. [Go over the reference sheet with students.]
- To determine the validity of bar code numbers.
- To calculate the 12th digit (check number) for the UPC-A numbering system which is left off of many products.
- To discover the mathematics involved in the IBM bar coding system.
## Part A- 18 divided by 5 has remainder 3. 18 is congruent (mod 5) to 8, 13, 23 and many others.
- 26 divided by 11 has remainder 4. 26 is congruent (mod 11) to 15, 37, 48, and many others.
- 057627529316 is not a valid number. 3(0 + 7 + 2 + 5 + 9 + 1) + (5 + 6 + 7 + 2 + 3 + 6) = 3(24) + (29) = 72 + 29 = 101 º 1 (mod 10).
- All three 12 digit codes on the reference sheet are valid.
- Answers will vary.
## Part B
## Part C- Answers will vary.
- Not a valid IBM code number. For 345678921, which has an odd
number of digits, the congruence statement is
3 + 8 + 5 + 3 + 7 + 7 + 9 + 4 + 1 º 0 (mod 10). Since the sum of the left-hand side, 47, is not congruent to 0, the number 345678921 is not a valid IBM code number - C = 6. For 9776827468C, which has an odd number of digits,
the congruence statement is
9 + 5 + 7 + 3 + 8 + 4 + 7 + 8 + 6 + 7 + C º 0 (mod 10). This simplifies to 64 + C º 0. - C = 2. For 320053461C, which has an even number of digits,
the congruence statement would be
6 + 2 + 0 + 0 + 1 + 3 + 8 + 6 + 2 + C º 0 (mod 10). This simplifies to 28 + C º 0 (mod 10). - Answers will vary.
## DiscussionAfter Part C, #5 is completed, compile results from the entire class to find the percentage of valid UPC-A numbers chosen by randomly selecting 12 digits numbers.Area 10 Mathematics and Technology Professional Development Center Permission is granted to duplicate these materials for classroom use.
Last updated on 1/30/1999 |