LEARNER’S MODULE:
Mathematics in the Modern World
Chapter 6: CODES
I. OBJECTIVES:
At the end of this chapter, students should be able to:
a. Using coding schemes to encode and decode different types of
information for identification, privacy and security purposes; and
b. Exemplify honesty and integrity when using codes for security
purposes.
II. OVERVIEW
The utility of mathematics goes beyond the mundane. Mathematics
enables the development of codes and ciphers that are useful to
individuals and to society.
Coding Scheme (Tabut, 2015) is a convention that associates each
character with character set with a unique bit pattern-a binary
representation of the integers from 0.
III. DISCUSSION
SAMPLE CODES AND CIPHERS
A. The Caesar’s Cipher
Nearly 2000 years ago, Julius Caesar was busy taking over the world,
invading countries the size of the Roman Empire. He needed a way of
communicating his battle plans and tactics to everyone on his side
without the enemy finding it. So, Caesar would write messages to his
generals in code. Instead of writing the letter ‘A’, he would write the
letter that comes three places further on in the alphabet, the letter ‘D’.
Instead of a ‘B’, he would write an ‘E’, instead of a ‘C’, he would write
an ‘F’ and so on. When he got to the end of the alphabet, however, he
would have to go right back to the beginning, so instead of an ‘X’, he
would write an ‘A’, instead of a ‘Y’, he’d write a ‘B’ and instead of ‘Z’,
he’d write a ‘C’
Complete the table to find out how Caesar would encode the following
message:
Caesar’s message A T T A C K A T D A W N
B U
C V
Coded message D
B. The Credit Card Cipher
Credit card encryption is a security measure used to reduce the
likelihood of a credit or debit card information being stolen. Credit card
encryption involves both the security of the card, the security of the
terminal where a card is scanned, and the security of the transmission of
the card’s information between the terminal and a back-end computer
system.
BRYAN S. AMBRE | Sir Amber
bryanamber029@gmail.com
0955-417-3898
LEARNER’S MODULE:
Mathematics in the Modern World
C. The Shifting Cipher
This cipher got its name thru its process – shifting the alphabet forward
three places, is the same as adding three to your starting number:
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
For example, encoding the letter ‘A’ is 0+3=3, which is a ‘D’, coding ‘I’ is 8+3=11,
which is ‘L’.
However, you do have to be careful when you get to the end of the
alphabet, because there is no letter 26, so you have to go back to number 0. In
math we call this ‘MOD 26’, instead of writing 26, we go back to 0.
Thus, EGG is ciphered as 7-9-9 or HJJ.
SUN is ciphered as 21-23-16 0r VXQ.
YACHT is ciphered as 27-3-5-10-22 (since 27 is higher than 26, 27 is translated
as 1)…B-D-F-K-W.
Have a go at coding your name by adding 3 to every letter. Then have a
go at coding your name by shifting the alphabet forward by more places by
adding greater numbers e.g. adding 5, then adding 10, then decoding is
subtraction, so if you’ve coded a message by adding 5, you will decode the
message by subtracting 5.
D. The Tripling Cipher
Let’s encipher the word CAT using tripling formula. The position 2, 0, and
19 are tripled to 6, 0, 57.
In mod 26, the 57 is reduced to 5, or the letter F (recall that we take away
all the 26s). so the formula for a cipher number c# for each plain number
p# is GAF.
What is GAF is given and we want the original term, how do we do it?
E. The ISBN Cipher
ISBN, as we know is the International Standard Book Number, which
is present at the back of every book.
BRYAN S. AMBRE | Sir Amber
bryanamber029@gmail.com
0955-417-3898
LEARNER’S MODULE:
Mathematics in the Modern World
In its 10 digits, the ISBN uniquely identifies the book as well as telling
you its country of origin and its publisher. But that’s not all this code does.
The ISBN has a little magic worked into it.
Say, I want to order a book and I know its ISBN. I type the number in,
but I’m in a hurry and I made a mistake. You might think an amazing
property: they can detect errors in themselves.
Let me show you how.
Here is a genuine ISBN, from one of my favorite books: 05-214-27061.
Underneath, I have multiplied the digits by its position in the ISBN
code. So – 0 gets multiplied by 1, 5 by 2, 2 by 3, and so on… You shall
get these digits: 0, 10, 6, 4, 20, 12, 49, 0, 54, 10
The sum of these number is 165.
Wait, I will give another two examples before revealing the secret… Here
the first line is the ISBN, the second is the series in which each number is multiplied
by its position, the third is the sum of these:
186-2307-369
1, 16, 18, 8, 15, 0, 49, 24, 54, 90
Sum: 275
0486-2566-42
0, 8, 24, 24, 10, 30, 42, 48, 36, 20
Sum: 242
Notice anything about all these sums that have been cooked out of the ISBN?
Have you spotted the pattern?
The calculation always gives a sum that’s divisible by 11. This is not an
amazing coincidence, but an example of cunning mathematical design.
It’s only the first nine digits that contain the information about the book. The
tenth digit is included just to make this total number extracted from ISBN divisible
by 11. You might have spotted that some books have an X instead of a number
as their tenth digit. Because these nine digits are already divisible by 11.
Look for some books, try the ISBN cipher.
F. The Binary Number System
The binary number system is a numbering system that represents
numeric values using two unique digits (0 and 1). Most computing
devices use binary numbering to represent electronic circuit voltage
state, (i.e., on/off switch), which considers 0 voltage input as off and 1
as on. This is also known as the base 2 number system, or the binary
numbering system.
Microcontrollers only use binary logic in computing. Compare this to
the decimal numbering system, which is comprised of 10 unique digits
(0-9). For example, a user input of 345 (in decimal) is 101011001 in binary
form.
BRYAN S. AMBRE | Sir Amber
bryanamber029@gmail.com
0955-417-3898
LEARNER’S MODULE:
Mathematics in the Modern World
Suppose we want to write the word “LOVE” in a binary code. The
process is to look for the binary codes of the letters L, O, V, and E. That
is,
LETTERS BINARY CODES
L 01001100
O 01001111
V 01010110
E 01000101
Can you do the lower case of the word “love”
LETTERS BINARY CODES
l 01101100
o 01101111
v 01110110
e 01100101
Thus, the word, “Chemistry” will have the binary codes,
LETTERS BINARY CODES
C 01000011
h 01101000
e 01100101
m 01101101
i 01101001
s 01110011
t 01110100
r 01110010
y 01111001
Operations on Binary Systems
BRYAN S. AMBRE | Sir Amber
bryanamber029@gmail.com
0955-417-3898
LEARNER’S MODULE:
Mathematics in the Modern World
a. Addition of Binary Numbers
To add binary numbers, let us remember the following “Magic
Table”:
0+0=0
1+0=1
0+1=1
1 + 1 = 10
1 + 1 + 1 = 11
Example:
1 1 1 1 1
1 1 0 1 0 1 1 = 1(26)+1(25)+0(24)+1(23)+0(22)+1(21)+1(20)=64+32+8+2+1=107
+ 1 1 1 0 0 1 = 1(25)+1(24)+1(23)+1(22)+0(21)+1(20)= 32+16+8+1 = 57
7 5 1
1 0 1 0 0 1 0 0 = 1(2 ) + 0 + 1(2 ) + 0 + 0 + 1(2 ) + 0 + 0 = 𝟏𝟐𝟖 + 𝟑𝟐 + 𝟒 = 𝟏𝟔𝟒
b. Subtraction of Binary Numbers
To subtract binary numbers, then let us use the following “Magic
Table”:
1–1=0
0–0=0
1–0=1
0 – 1 = 1 but carry is required 10 – 1 = 1
Example 1:
1 0 1 1 1 = 16 + 4 + 2 + 1 = 23
- 1 0 1 0 1 = 16 + 4 + 1 = 21
0 0 0 1 0 = =2
Example 2:
1 1 0 = 1 0 10 = (borrow 1 from 1 to make 0 to be 10) = 4 + 2 = 6
- 1 0 1 = 1 0 1 = =4+1=5
0 0 1 = =1
c. Multiplication of Binary Numbers
To multiply binary numbers, we will do the same process as
decimals.
Example
0 1 0 1 = 4+1= 5
x 1 0 1 1 = 8 + 2 + 1 = x 11
0 1 0 1 55
0 1 0 1
0 0 0 0
+ 0 1 0 1
0 1 1 0 1 1 1 = 32 + 16 + 4 + 2 + 1 = 55
BRYAN S. AMBRE | Sir Amber
bryanamber029@gmail.com
0955-417-3898
LEARNER’S MODULE:
Mathematics in the Modern World
d. Division of Binary Numbers
In dividing binary numbers, we will use the same process as in
decimals.
Example:
0 0 1 1 0 1
1 1 1 0 0 1 1 1
1 1 = (Borrow 1 from the next digit)
0 0 1 1
1 1
0 0 1 1
1 1
0 0
BRYAN S. AMBRE | Sir Amber
bryanamber029@gmail.com
0955-417-3898
