Numbers in Our Pockets
GCNU 1025
Numbers Save the Day
Numbers in our pockets
• A fundamental use of numbers: counting
Numbers in our pockets
• How about banknotes?
Numbers in our pockets
• ID number for identifying/tracking banknotes
Numbers in our pockets
• ID number for identifying/tracking banknotes
1. Numbers for identification
An ID (identification) number is a string of numbers
that is used for identification. It enables us to recognize
the object or person attach to that number.
Examples:
• Postal codes
• Phone numbers
• Country codes
• Numbers in cheques
Postal Codes
• A post code (or post code, postcode, ZIP code) is a
series of letters and/or numbers appended to a
postal address (or a delivery route).
• Different countries with different systems.
• Postal codes allow us to recognize the geographical
area to which the code refers.
• ZIP code in the address of Harvard University: MA 02138
• Some more specific ZIP codes have 4 extra digits (“ZIP+4”)
ZIP codes of US
Chinese postal codes
• 6 digits (e.g. 518053)
• 4 levels: 51-8-0-53
–
–
–
–
First 2 digits: province/municipality/autonomous region
3rd digit: postal zone
4th digit: district
Last 2 digits: local post office
Phone numbers for identification
Hong Kong telephone numbers
• Which are fixed land line numbers? How can
you tell?
A. 2363 2767
B. 3411 2801
C. 5589 4120
D. 6583 2042
E. 7729 1502
F. 8320 1529
G. 9258 4981
Hong Kong telephone numbers
• Number of a church: 2363 2767
• 363 xxxx: before 01/01/1995
• Where is the church? Kowloon? Hong Kong
Island? New Territories? Outlying Islands?
How can you tell?
Hong Kong telephone numbers
• Where is the church? Kowloon? Hong Kong
Island? New Territories? Outlying Islands?
How can you tell?
• 363 2767: a number in Kowloon
• 1970s: (3)63 2767
– First “3”: area code for Kowloon
– 632767 was the actual number
– No need to dial the area code if you are in
Kowloon
Hong Kong telephone numbers
• The leading digits of a Hong Kong telephone
number are partially related to the location of
the telephone line
Country codes
• How can a friend of yours from the US call you? Can
he/she just dial your number?
• Need to dial 852 before the number (and probably
also the number provided by the service provider)
Numbers for identification
• Why can’t a Hong Kong number be 99920842?
• Can the country code of China be 85?
• Example: 8526543210
– Interpretation #1: 852-6543210 (Hong Kong)
– Interpretation #2: 85-26543210 (not Hong Kong)
• Prefix property:
– No phone number (country code) can be the prefix of
another phone number (country code)
Numbers in Cheques
• Cheque number, routing/transit number for issuing
bank/branch, account number
Numbers in Our Pockets
GCNU 1025
Numbers Save the Day
2. Numbers for error-checking
Some identification numbers include error-checking
features to enable us to check their validity.
Examples:
• HKID card numbers
• Credit card numbers
Warning: designed to detect errors rather than
to prevent fraud.
HKID card numbers
• Some basics about HKID card numbers
• Check digits of HKID card numbers
Validity of Y123456(9)
• Rule: every valid HKID number has a special property
• Step 1 (of checking validity):
• Step 2 (of checking validity):
• Step 3 (of checking validity): is this “check sum” 286
divisible by 11? YES! The number is valid!
Classwork 1: validity of HKID number
Have a try checking your own HKID….
A missing digit of a valid HKID number
• If Y123464(*) is valid, what is “*”?
• Step 1:
Y
1
2
3
4
6
4
*
x
8
7
6
5
4
3
2
1
=
200
7
12
15
16
18
8
*
• Step 2:
–
–
–
–
Check sum = 276 + * (which is known to be divisible by 11)
What should “*” be? Can “*” be 0-9?
If Y123464(*) is valid, “*” has to be 10!
Use “A” to represent 10 (recall the base 16 arithmetic)
HKID card check digit
• Weighted modulo 11 check: check digit
appended to make check sum divisible by 11
• 0-9 or A (representing 10)
• Using “A” for “10” is not uncommon in other
numeral systems (compare: base 16 numbers
such as “B1A”)
HKID card check digit - Review
• Weighted modulo 11 check: check digit
appended to make check sum divisible by 11
• 0-9 or A (representing 10)
• Using “A” for “10” is not uncommon in other
numeral systems (compare: base 16 numbers
such as “B1A”)
HKID card check digit - Review
http://www.cp.hk/articles/identity.asp Quote:
Incidentally, the last character in brackets ( ) is actually not part of the HKID.
It is a check digit which depends on the characters in the HKID, resulting between
0-9 or A. It is generated by a fairly simple formula, and is used to check for errors,
because if you get one character wrong in the ID, then the check digit will be
wrong. You can use our new online HKID check digit generator to calculate the
check digit of any HKID - try it and see! Hours of fun for all the family.
http://www.cp.hk/dbpub/idcheck.asp
Place value: base 10 system
• Every digit must be from 0-9 (e.g. 1985)
• For a base 10 number “1985”:
Digit
Value of the position
Actual value
1
1,000
(10x10x10)
1,000
9
100
(10x10)
900
8
10
(10)
80
5
1
5
Place value: base 16 system
• Base 16: every digit must be from 0-15
10
11
12
13
14
15
A
B
C
D
E
F
• For a base 16 number “3A24”:
Digit
Value of the position
Actual value
3
4096
(16x16x16)
12,288
A (10)
256
(16x16)
2,560
2
16
(16)
32
4
1
4
Other numeral systems
Numeral system
Digits from
Base 10
0-9
Base 16
0-15
Base 60
0-59
Base 12
0-11
Base 2
0-1
Base 8
0-7
Duodecimal system
Hexadecimal system
Sexagesimal system
Stems-and-branches
Numeral systems
Numeral system
Uses
Binary (base 2)
Computers
Octal (base 8)
Used to be popular in Central America
Duodecimal (base 12)
1 foot = 12 inches
Hexadecimal (base 16)
1 pound = 16 ounces
Sexagesimal (base 60)
1 minute = 60 seconds
Number for error-checking: remarks
• Not for fraudulence prevention
– Schemes known to public
– A counterfeit (i.e. a fake) could still have a valid
number
– Other schemes available for such purpose
Numbers in Our Pockets
GCNU 1025
Numbers Save the Day
Credit card check scheme
• Weighted modulo 10 check: check digit (i.e.
last digit) appended to make check sum
divisible by 10
• How to calculate the check sum? How are the
digits weighted?
Validity of 5424 7171 3312 0044
• Rule: every valid credit card number has a special
property
• Step 1 (of checking validity):
• Step 2 (of checking validity):
• Step 3 (of checking validity): Is this “check sum” 50
divisible by 10? YES! The number is valid!
Credit card check scheme
Luhn algorithm
http://en.wikipedia.org/wiki/Luhn_algorithm
See also “bank card numbers” http://en.wikipedia.org/wiki/Bank_card_number
Classwork 2: validity of credit card number
Why not try the credit cards in your own pocket?
Numbers in Our Pockets
GCNU 1025
Numbers Save the Day
3. Barcodes
A 1D barcode is a series of dark bars and light spaces that
represent characters/data. A scanning device is needed to
pass over the bars and spaces to read the data.
Examples:
• UPC barcodes (1D)
• QR codes (2D)
Error-correcting feature of QR codes
UPC (barcode) check scheme
• A (one-dimensional) barcode represents data by
varying the width and the spacing of parallel lines
• Primary example: Universal Product Code (UPC)
UPC (barcode): basic features
• 12-digit
• 1st digit: product category
Prefix
Category
0, 1, 6, 7 or 8
Ordinary products
2
Variable-weight products
3
Drugs
5 or 9
Coupons
• Last digit: check digit
UPC (barcode): basic features
• Two sides separated watersheds/long guard bars
• Total width of strips representing a number is 7
• Total width of black strips is odd (left side)
– Example: 8 (WBBWBBB)
UPC (barcode): basic features
• Bars on two sides are like inverses. Bars on the right
becomes spaces on the left and vice versa.
– Example: the left representation of 8 is (WBBWBBB) and
the right representation of 8 is (BWWBWWW)
Odd width for left and even for right:
enables scanners to detect orientation
(up-side-down?)
UPC (barcode) check scheme
• Weighted modulo 10 check: check digit appended to
make check sum divisible by 10
• How to calculate the check sum? How are the digits
weighted?
Validity of 9-87654-32109-8
• Rule: every valid UPC number has a special property
• Step 1 (of checking validity):
9
8
7
6
5
4
3
2
1
0
9
8
x
3
1
3
1
3
1
3
1
3
1
3
1
=
27
8
21
6
15
4
9
2
3
0
27
8
• Step 2 (of checking validity):
27+8+21+6+15+4+9+2+3+0+27+8=130
• Step 3 (of checking validity): Is this “check sum” 130
divisible by 10? YES! The number is valid!
Classwork: validity of UPC number
Two-dimensional barcodes
• Limited amount of data stored in 1D barcodes such as UPC
• 2D barcode: rectangles (of different sizes) instead of strips (of
different widths)
• Primary example: QR (quick response) codes
QR code: basic features
• Storage capacity depends on data type, dimension
and error-correction level
– 40 different dimensions
– 4 levels of error correction power
Error-correction level % of info that can be restored
Level L (Low)
~7%
Level M (Medium)
~15%
Level Q (Quartile)
~25%
Level H (High)
~30%
http://en.wikipedia.org/wiki/QR_code Play it!
QR Code: Online Generator
http://www.yj580.com/qr/
Numbers in Our Pockets
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