Numbers in Our Pockets

GCNU 1025Numbers 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 2767B. 3411 2801C. 5589 4120D. 6583 2042E. 7729 1502F. 8320 1529G. 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 1025Numbers 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

Y 1 2 3 4 6 4 *x 8 7 6 5 4 3 2 1= 200 7 12 15 16 18 8 *

• If Y123464(*) is valid, what is “*”?• Step 1:

• 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”)

http://www.cp.hk/dbpub/idcheck.asp

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.

HKID card check digit - Review

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

• For a base 16 number “3A24”:

10 11 12 13 14 15

A B C D E F

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 systemsNumeral 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 systemsNumeral 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 1025Numbers 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 algorithmhttp://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 1025Numbers 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

• Last digit: check digit

Prefix Category

0, 1, 6, 7 or 8 Ordinary products

2 Variable-weight products

3 Drugs

5 or 9 Coupons

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):

• 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!

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

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

http://en.wikipedia.org/wiki/QR_code Play it!

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%

QR Code: Online Generator

http://www.yj580.com/qr/

Numbers in Our Pockets

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