Download 3a - Math TAMU

Chapter 16 Identification Numbers
Identification Numbers
Modern identification numbers serve at least two functions:

The number should unambiguously identify the person or thing with which it is
associated .

The number should have a “self-checking” aspect.
Codes

A group of symbols that represent information.

Codes have been invented for storing, securing, and transmitting information.

Examples: Hieroglyphics, the Greek alphabet, Roman numerals, Morse code,
and the “genetic code” used to describe DNA
Check Digits
A digit included in an identification number for the purpose of error detection.

Mathematical calculations or schemes are used on the digits of the
identification number to assign the check digit.

Computers use the check digit to help detect typing errors during data entry to
prevent and detect fraud and to find other errors.
U.S. Postal Service money order with identification number 5025978014
and check digit 5.
Universal Product Code (UPC)

A bar code and identification
number that are used on most retail items.

By using weighted schemes in the
calculation of the check digit, the UPC
code can achieve greater error detection—
up to 100% of all single-digit errors and
most other types of errors.

Example: Consider the number 6 54954 20117 6 found on the bottom of
a bag of juice.

The first digit identifies a broad category of goods.

The next five digits identify the manufacturer.

The next five digits identify the product.

The last is a check digit.
Thus in general we have
Bank Identification Number
Here is what the string of numbers at the bottom of a check mean
0113 Bank’s Federal Reserve District,
office, and state or special collection
arrangement
0014
Bank’s ID number
2
Check digit ( error detection )
12345678 Customer's check account
0101
Customer's check number
International Standard Book Number (ISBN)

A ten-digit (or thirteen) identification number ( ISBN-10 or ISBN-13) used on
books throughout the world that contains a check digit for error detection.

This scheme can detect 100% of single errors and 100% of transposition errors.
A correctly coded ten-digit ISBN a1 a2 …a10 has the property that 10a1 +
9a2 + 8a3 + 7a4 + 6a5 + 5a6 + 4a7 + 3a8 + 2a9 + a10 is evenly divisible
by 11.
VIN System – Vehicle Identification Number

The manufacturer gives each automobile and
truck a unique VIN. A typical VIN has 17
alphanumeric characters that code information
such as country where the vehicle was built,
manufacturer, make, body style, engine type,
plant where the vehicle was built, model year,
model, type of restraint, a check digit, and a
production sequence number.
The Zip Code
The ZIP Code

Identification numbers sometimes encode geographic data.

In 1963, the U.S. Postal Service numbered every American post office with a
five-digit ZIP code—numbers begin with 0 at the points farthest east, and work
up to 9 at the points farthest west.

In 1983, the U.S. Postal Service added four digits to the ZIP code, ZIP + 4 code.
Bar Codes
ZIP Code Bar Code
The simplest bar code is the
Postnet code used by the U.S. Postal
Service and commonly found on business
reply forms.
UPC Bar Code
The UPC bar code was first used on grocery
items in 1973 and has since spread to most
retail products.
Encoding Personal Data

Encoding – Translating data into code.

Coding License Numbers
Some states assign driver’s license numbers with personal data encoded
into the number.
Personal data that may be encoded are name, sex, and date of birth.
Coding license numbers solely from personal data enables automobile
insurers, government entities, and law enforcement agencies to determine
the number from the personal data.

Social Security Numbers
There is no personal data encoded in the Social Security number.
The Social Security numbering scheme predates computers.
The only personal information that can be deduced from the Social
Security number is where the holder obtained it. The first three digits
show (the state usually) where the applicant applied, with a few
exceptions.
Mathematical Model
As we know in every identification number there is a check digit and
there are different mathematical schemes to do it. For example we have
that:
1) The US postal service the first ten digits of the 11-digit
identification number 63024383845 simply identify the money order
and the last number, 5, is the check digit which is the reminder of the
sum of the first ten digits divided by 9:
6 + 3 + 0 + 2 + 4 + 3 + 8 + 3 + 8 + 4 = 41
=> 41= 9*4 + 5
remainder
2) The scheme used on airlines tickets, UPS packages, and National
rental cars assigns the remainder after division by 7 of the number
itself as the check digit. Thus the identification number 540047 has
the check digit 4 because
540047 = 77149 * 7 + 4
remainder
3) The UPC scheme is more complicated. Given an identification
number of the form
a1 a2 a3 a 4 a5 a6 a7 a8 a9 a10 a11 a 12
the last digit is a check digit, chosen in such a way that the following
calculation
3a1 +a2+3a3+a4 +3a5+a6 +3a7+a8 +3a9+a10+3a11+a12
results in a number evenly divided by 10. Thus the UPC identification
number 0 38000 00127 7 has as a check digit 7 because
3∗0+3+3∗8+0+3∗0+0+3∗0+0+3∗1+2+3∗7=53 (+7=60 )
4) The US Banking system uses a variation of the UPC system. In a
routing number of the form
a1 a2 a3 a 4 a5 a6 a7 a8 a9
the last digit is the check digit, chosen as the last digit of the sum
7a1 +3a2 +9a3+7a4 +3a5 +9a6 +7a7 +3a8
Thus the routing number 071000013 has as a check digit 3 because
7∗0+3∗7+9∗1+7∗0+3∗0+9∗0+7∗0+3∗1=33
5) The Postnet code consists of ten digits
a1 a2 a3 a4 a5 a6 a7 a8 a9C
( ZIP + 4 + C ) where the last one, C, is a check digit chosen in such
a way that the following calculation
a1 +a 2+a3 +a 4 +a5 +a6 +a7 +a8 +a9 +C
results in a number evenly divided by 10. Thus the ZIP + 4 code
number 605159968 has a check digit the number 1 because
6 + 0 + 5 + 1 + 5 + 9 + 9 + 6 + 8 = 49 ( + 1 = 50 )
Example
1) Given the UPC code 0 69000 00341 6 for some product, after
passing through the scanner machine we obtain an error. What is
happening in this case ?
OBS
In the UPC code and US banking systems the digits are multiplied by
3 and 1 in the first case, and by 7, 3, and 9 in the second case . These
numbers are called the weights.
Types of error creating identification numbers.
1) Replacing one digit with a different digit ( single digit error )
Example: 32 rather 12
2) Transposing two adjacent digits ( adjacent transposition error )
Example: 21 rather than 12
3) Transposing a sequence of digits ( jump transposition error )
Example: 321 rather than 123 .
Now since we are going to deal with remainder in the calculation of
the check digits we will introduce some basic concepts from the
branch of mathematics called Number Theory ( modular arithmetic ).
Definition
Let a, b, and m be integers with m > 2. Then a is congruent with b
modulo m, written as
a ≡ b mod m
if an only if a – b is evenly divided by m .
Examples
1) 32 ≡ 2 mod 5
b/c (32 – 2) / 5 = 30 / 5 = 6 .
2) 29 ≡ 8 mod 3
b/c (29 – 8) / 3 = 21 / 3 = 7 .
3) 69 ≡ 5 mod 8
b/c (69 – 5) / 8 = 64 / 8 = 8 .
4) 33 ≢ 3 mod 4
b/c (33 - 3) / 4 = 30 / 4 ≠ not an integer .
Definition
Let x and y be two integers with y ≠ 0 we define x mod y as the
remainder of divide x by y.
Examples
1) 67 mod 5 = 2 because 67 = (13)(5) + 2
remainder
2) 39 mod 3 = 3 because 38 = (12)(3) + 3
remainder
3) 21 mod 4 = 1 because 21 = (5)(4) + 1
remainder
4) 58 mod 6 = 4 because 58 = (9)(6) + 4
remainder
5) A code a 1 a 2 a 3 a 4 a 5 uses the last digit ( a 5 ) as the check
digit. The check digit is found using the formula
a 1 + a 4 + 2( a 2 + a 3 ) mod 10
a) What is the check digit for the code 6728 ?
b) For the identification number 324x6. What is the value of x?
c) Assume that instead of the identification number 6728 we type
erroneously 7628 . Will the scheme detect the error ? How about
typing 6278 ?
6) A code a 1 a 2 a 3 a 4 uses the last last digit, ( a 4 ) as the check
digit. The check digit is found using the formula
a1+3a2+4a3
mod 10
The first digit is not readable so the code looks like x922. If the check
digit is known to be correct, what is the value of the missing digit ?
7) A code a 1 a 2 a 3 a 4 uses the last last digit ( a 4 ) as the check
digit. The check digit is found using the formula
9a 1 + 4 a 2 + 8 a 3
mod 10
Will this check digit find all transposition errors in the first and
second positions?
8)A code a 1 a 2 a 3 a 4 uses the last last digit ( a 4 ) as the check
digit. The check digit is found using the formula
2a 1 + 3 a 2 + 5 a 3
mod 9
Will this check digit find all single digit errors in the first position?
9) In Florida, the last three digits of the driver's license of a female
with birth month m and birth day b are 40( m – 1 ) + b + 500. For both
males and females, the fourth and fifth digits from the end give the
year of birth. Determine the last five digits of a Florida driver's license
number for a female born on July 18, 1942. Explain why a Florida
driver's license number that ends with the five numbers 99718 is not
valid ?
10) The last three digits of a man's ID number are the birthday of the
year based on each month of the year having 42 days. If the person is
a woman, 600 is added to the birth day.
a) What are the last three digits of a man's ID number if he was born
on August the 5th ?
b) What do you know about a person if the last thre digits of the
person's ID number are 618.
c) What do you know about a person if the last three digits of the ID
number are 502 ?
11) A code is given by a 1 a 2 a 3 a 4 where a 4 is the check digit.
The check digit is given by the formula
7a 1 + 2 a 2 + 5 a 3
mod 9
a) Determine the value of x in the code 2x45 given that the check digit
is valid.
b) Determine if the check digit will find all single digit errors in the
third position
c) Determine if the check digit will find all transposition
errors in the second and third position