Concept of Modular arithmetic/Cyclic events

Subject :

Mathematics

Topic :

Concept of modular arithmetic/Cyclic events

Class :

SS 1

Term :

First Term

Week :

Week 4

Instructional Materials :

• Wall charts
• Online Resources
• Pictures
• Related Audio Visual
• Mathematics Textbooks
• A chart showing modular arithmetic
• Samples of Duty shift
• Menstrual chart

Reference Materials

• Scheme of Work
• Online Information
• Textbooks
• Workbooks
• Education Curriculum

Previous Knowledge :

The pupils have previous knowledge of

Conversion from Base Ten to other Bases and Conversion from one base to another

Behavioural Objectives:  At the end of the lesson, the pupils should be able to

• define the term modular arithmetic
• perform some basic operations of addition, subtraction, multiplication and division on modular mathematics
• apply modular mathematics to everyday life arithmetic

Content:

WEEK 4

DATE……………………

SUBJECT: MATHEMATICS CLASS: SS 1

TOPIC: Modular Arithmetic

CONTENT:

• Revision of addition and subtraction of integers
• Revision of multiplication and division of integers
• Concept of modular arithmetic/Cyclic events

SUB-TOPIC 1 & 2:

Define the term modular arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers “wrap around” when they reach a certain value—the modulus (plural: moduli). In contrast, standard arithmetic uses a fixed number set (like the natural numbers 1, 2, 3,…), while modular arithmetic uses a number set that has been “cycled” by a certain integer. If this integer is relatively prime to the modulus, then all the integers in the set are relatively prime to the modulus as well, and the set behaves like an ordinary number set under addition and multiplication. However, if two numbers in the set are not relatively prime to the modulus, then they will “collide” when they are added or multiplied together, and the results will be reduced modulo the modulus.

Interactive Questions and Answers

Questions

1. What is the modulus?
2. What is standard arithmetic?
3. How do numbers “wrap around” in modular arithmetic?
4. What is the difference between modular arithmetic and standard arithmetic?
5. What happens when two numbers in the modular arithmetic set are not relatively prime to the modulus?

Answers

1. The modulus is the value that numbers “wrap around” to when they reach a certain point in modular arithmetic.

2. Standard arithmetic is a system of arithmetic that uses a fixed number set (like the natural numbers 1, 2, 3,…).

3. In modular arithmetic, numbers “wrap around” when they reach a certain value—the modulus.

4. The difference between modular arithmetic and standard arithmetic is that modular arithmetic uses a number set that has been “cycled” by a certain integer, while standard arithmetic uses a fixed number set.

5. When two numbers in the modular arithmetic set are not relatively prime to the modulus, they will “collide” when they are added or multiplied together, and the results will be reduced modulus.

Addition of integers in modular mathematics

In modular arithmetic, addition is performed by taking the sum of two numbers and then reducing that sum modulo the modulus. So, if the modulus is 10 and two numbers being added together are 3 and 7, the sum would be 10 (3 + 7 = 10), and since 10 is equal to 0 modulo 10, the answer would be 0.

Here are five examples of addition in modular arithmetic:

If the modulus is 10 and the numbers being added are 3 and 7, the sum is 10 (3 + 7 = 10), and since 10 is equal to 0 modulo 10, the answer is 0.

If the modulus is 7 and the numbers being added are 5 and 3, the sum is 2 (5 + 3 = 8), and since 8 is equal to 2 modulo 7, the answer is 2.

If the modulus is 3 and the numbers being added are 2 and 1, the sum is 0 (2 + 1 = 3), and since 3 is equal to 0 modulo 3, the answer is 0.

If the modulus is 5 and the numbers being added are 4 and 3, the sum is 2 (4 + 3 = 7), and since 7 is equal to 2 modulo 5, the answer is 2.

If the modulus is 9 and the numbers being added are 7 and 5, the sum is 3 (7 + 5 = 12), and since 12 is equal to 3 modulo 9, the answer is 3.

Subtraction of integers in modular mathematics

In modular arithmetic, subtraction is performed by taking the difference of two numbers and then reducing that difference modulo the modulus. So, if the modulus is 10 and two numbers being subtracted have values of 7 and 3, the difference would be 4 (7 – 3 = 4), and since 4 is equal to 4 modulo 10, the answer would be 4.

Here are five examples of subtraction in modular arithmetic:

If the modulus is 10 and the numbers being subtracted have values of 7 and 3, the difference is 4 (7 – 3 = 4), and since 4 is equal to 4 modulo 10, the answer is 4.

If the modulus is 7 and the numbers being subtracted have values of 5 and 3, the difference is 2 (5 – 3 = 2), and since 2 is equal to 2 modulo 7, the answer is 2.

If the modulus is 3 and the numbers being subtracted have values of 2 and 1, the difference is 0 (2 – 1 = 1), and since 1 is equal to 0 modulo 3, the answer is 0.

If the modulus is 5 and the numbers being subtracted have values of 4 and 3, the difference is 1 (4 – 3 = 1), and since 1 is equal to 1 modulo 5, the answer is 1.

If the modulus is 9 and the numbers being subtracted have values of 7 and 5, the difference is 4 (7 – 5 = 2), and since 2 is equal to 4 modulo 9, the answer is 4.

Multiplication of integers in modular mathematics

In modular arithmetic, multiplication is performed by taking the product of two numbers and then reducing that product modulus. So, if the modulus is 10 and two numbers being multiplied have values of 3 and 7, the product would be 21 (3 x 7 = 21), and since 21 is equal to 1 modulo 10, the answer would be 1.

Here are five examples of multiplication in modular arithmetic:

If the modulus is 10 and the numbers being multiplied have values of 3 and 7, the product is 21 (3 x 7 = 21), and since 21 is equal to 1 modulo 10, the answer is 1.

If the modulus is 7 and the numbers being multiplied have values of 5 and 3, the product is 6 (5 x 3 = 15), and since 15 is equal to 6 modulo 7, the answer is 6.

If the modulus is 3 and the numbers being multiplied have values of 2 and 1, the product is 0 (2 x 1 = 2), and since 2 is equal to 0 modulo 3, the answer is 0.

If the modulus is 5 and the numbers being multiplied have values of 4 and 3, the product is 3 (4 x 3 = 12), and since 12 is equal to 3 modulo 5, the answer is 3.

If the modulus is 9 and the numbers being multiplied have values of 7 and 5, the product is 8 (7 x 5 = 35), and since 35 is equal to 8 modulo 9, the answer is 8.

Division of integers in modular mathematics

In modular arithmetic, division is performed by multiplying the numerator by the modular inverse of the denominator and then reducing that product modulo the modulus. So, if the modulus is 10 and two numbers being divided have values of 7 and 3, the product would be 9 (7 x 3 = 21), and since 21 is equal to 1 modulo 10, the answer would be 1.

Here are five examples of division in modular arithmetic:

If the modulus is 10 and the numbers being divided have values of 7 and 3, the product is 9 (7 x 3 = 21), and since 21 is equal to 1 modulo 10, the answer is 1.

If the modulus is 7 and the numbers being divided have values of 5 and 3, the product is 2 (5 x 3 = 15), and since 15 is equal to 6 modulo 7, the answer is 2.

If the modulus is 3 and the numbers being divided have values of 2 and 1, the product is 0 (2 x 1 = 2), and since 2 is equal to 0 modulo 3, the answer is 0.

If the modulus is 5 and the numbers being divided have values of 4 and 3, the product is 1 (4 x 3 = 12), and since 12 is equal to 3 modulo 5, the answer is 1.

If the modulus is 9 and the numbers being divided have values of 7 and 5, the product is 4 (7 x 5 = 35), and since 35 is equal to 8 modulo 9, the answer is 4.

SUB-TOPIC 3: Concept of Modular Arithmetic

The word Modular implies consisting of separate parts or units which can be put together to form something, often in different combinations.

Arithmetic the science of numbers involving adding, subtracting, multiplying and dividing of numbers

Modular Arithmetic is simply the arithmetic of remainders when an integer is divided by a fixed non-zero integer.

In modular arithmetic, conversion is the process of taking a number in one base and expressing it in another base. So, if the number to be converted is 12 and the original base is 10 (decimal), the number would be expressed as 14 in base 2 (binary), 3 in base 3 (ternary), 10 in base 4 (quaternary), and 11 in base 5 (quinary).

Here are five examples of conversion in modular arithmetic:

If the number to be converted is 12 and the original base is 10 (decimal), the number would be expressed as 14 in base 2 (binary), 3 in base 3 (ternary), 10 in base 4 (quaternary), and 11 in base 5 (quinary).

If the number to be converted is 11 and the original base is 2 (binary), the number would be expressed as 3 in base 10 (decimal), 12 in base 3 (ternary), 21 in base 4 (quaternary), and 31 in base 5 (quinary).

If the number to be converted is 10 and the original base is 3 (ternary), the number would be expressed as 4 in base 10 (decimal), 11 in base 2 (binary), 22 in base 4 (quaternary), and 101 in base 5 (quinary).

If the number is converted to quinary,

If the number to be converted is 10 and the original base is 3 (ternary), the number would be expressed as 4 in base 10 (decimal), 11 in base 2 (binary), 22 in base 4 (quaternary), and 101 in base 5 (quinary).

If the number is converted to quinary,

If the number to be converted is 100 and the original base is 4 (quaternary), the number would be expressed as 16 in base 10 (decimal), 100 in base 2 (binary), 41 in base 3 (ternary), and 11 in base 5 (quinary).

Examples;

Reduce 65 to its simplest form in:

(a) modulo 3 (b) modulo 4 (c) modulo 5 (d) modulo 6

(a) 2 (b) 1 (c) 0 (d) 5

EVALUATION:

Reduce 72 to its simplest form

• Modulo 3
• Modulo 4
• Modulo 5
• Modulo 6
• Modulo 7

• Addition, Subtraction, Multiplication and Division operations in module arithmetic

• Application to daily life

SUB-TOPIC 1

Addition and Subtraction: This can be either addition or subtraction tables where the number of digits given represents the modulo.

Examples; (a)

The table above shows addition (mod 3)

• The table above shows subtraction (mod 4)
• (i) Find 39 29(mod 6) Solution: 39                          29= 68

= (6×11+2)

= 2(mod 6)

N.B

• Calculate the following in the given moduli (a) 12Θ5(mod 4) (b) 38 Θ42(mod 7) Solution: (a) 12Θ5 = 7

7 = 4 + 3

= 3(mod 4)

(b) 38 Θ42 = 4

4 = 7 + 3

= 3(mod 7)

EVALUATION:

• Find the following additions modulo 5
  • 3 9

(b) 65     32

(c)   41    52

(d) 8     17

• Find the simplest positive form of each of the following numbers modulo 5 (a)

(b)

(c)

SUB-TOPIC 2

Multiplication of modulo

Examples: Evaluate the following modulo 4

Solution:

(a) 2    2 = 4

= 4 + 0(mod 4)

= 0(mod 4)

(b) 5      7 = 35

= 4 x 8 + 3

= 3(mod 4)

(c) 6     73 = 438

= 4 x 109 + 2

= 2(mod 4)

EVALUATION:

Find the values in the moduli written beside them

(a) 16    7(mod 5) (b) 21               18(mod 10)

(c) 8     25(mod 3)

(d) 27     4(mod 7) (e) 80               29(mod 7)

SUB-TOPIC 3

Division of modulo

Examples: Find the values of the following;

• 23(mod 4)
• 72(mod 5)
• 22(mod 4)

Solution:

• If 23 =

⇒

Cross-multiply , Add 4 to RHS

3 = 2 + 4(mod 4)

3 = 6(mod 4)

Divide both sides by 3

= 2(mod 4)

(b) 72 =

2 = 7(mod 5)

2 = (5 x 1) + 2(mod 5)

2 = 2

= 1

(c) 22 =

2 = 2(mod 4)

Divide both sides by 2

= 1

Or

2 = 2 + 4(mod 4)

2 = 6(mod 4)

= 3(mod 4)

N.B If 32 = , then 2 = 3

No multiple of 4 can be added to 3 to make it exactly divisible by 2. There are no values of 32 in modulo 4.

EVALUATION:

Calculate the following division in modulo 5

(a) 287

(b) 292

(c) 584

(d) 747

N.B Educators should also solve various examples.

GENERAL EVALUATION:

• Copy and complete the table for addition (mod 5)

• Copy and complete the table for subtraction modulo 6

• Complete the multiplication modulo 5 in the table below

READING ASSIGNMENT:

New General Mathematics for SSS 1, pages 227; exercises 20a, 20b

Mathematical Association of Nigeria (MAN) pages 14-24

WEEKEND ASSIGNMENT:

New General Mathematics for SSS 1, pages 227; exercises 20a, 20b, 20c Mathematical Association of Nigeria (MAN) pages 14-24

EVALUATION:

Evaluation on “Addition of integers in modular mathematics”.

1. If the modulus is 10 and the numbers to be added have values of 7 and 3, what is the sum?

2. If the modulus is 7 and the numbers to be added have values of 5 and 3, what is the sum?

3. If the modulus is 3 and the numbers to be added have values of 2 and 1, what is the sum?

4. If the modulus is 5 and the numbers to be added have values of 4 and 3, what is the sum?

5. If the modulus is 9 and the numbers to be added have values of 7 and 5, what is the sum?

Suggested Answers To the question on “Addition of integers in modular mathematics”.

7 + 3 = 10 and 10 is equal to 0 modulo 10, so the sum is 0.

5 + 3 = 8 and 8 is equal to 1 modulo 7, so the sum is 1.

2 + 1 = 3 and 3 is equal to 0 modulo 3, so the sum is 0.

4 + 3 = 7 and 7 is equal to 2 modulo 5, so the sum is 2.

7 + 5 = 12 and 12 is equal to 3 modulo 9, so the sum is 3.

Evaluation questions on “Subtraction of integers in modular mathematics”.

1. If the modulus is 10 and the numbers to be subtracted have values of 7 and 3, what is the difference?

2. If the modulus is 7 and the numbers to be subtracted have values of 5 and 3, what is the difference?

3. If the modulus is 3 and the numbers to be subtracted have values of 2 and 1, what is the difference?

4. If the modulus is 5 and the numbers to be subtracted have values of 4 and 3, what is the difference?

5. If the modulus is 9 and the numbers to be subtracted have values of 7 and 5, what is the difference?

Suggested Answers the question on “Subtraction of integers in modular mathematics”.

7 – 3 = 4 and 4 is equal to 4 modulo 10, so the difference is 4.

5 – 3 = 2 and 2 is equal to 2 modulo 7, so the difference is 2.

2 – 1 = 1 and 1 is equal to 1 modulo 3, so the difference is 1.

4 – 3 = 1 and 1 is equal to 1 modulo 5, so the difference is 1.

7 – 5 = 2 and 2 is equal to 2 modulo 9, so the difference is 2.

Evaluation questions on “Multiplication of integers in modular mathematics”.

1. If the modulus is 10 and the numbers to be multiplied have values of 7 and 3, what is the product?

2. If the modulus is 7 and the numbers to be multiplied have values of 5 and 3, what is the product?

3. If the modulus is 3 and the numbers to be multiplied have values of 2 and 1, what is the product?

4. If the modulus is 5 and the numbers to be multiplied have values of 4 and 3, what is the product?

5. If the modulus is 9 and the numbers to be multiplied have values of 7 and 5, what is the product?

Suggested Answers to the question on “Multiplication of integers in modular mathematics”.

7 * 3 = 21 and 21 is equal to 1 modulo 10, so the product is 1.

5 * 3 = 15 and 15 is equal to 1 modulo 7, so the product is 1.

2 * 1 = 2 and 2 is equal to 2 modulo 3, so the product is 2.

4 * 3 = 12 and 12 is equal to 2 modulo 5, so the product is 2.

7 * 5 = 35 and 35 is equal to 8 modulo 9, so the product is 8.

Evaluation questions on “Division of integers in modular mathematics”.

1. If the modulus is 10 and the number to be divided has a value of 7, what is the quotient when 3 is divided into it?

2. If the modulus is 7 and the number to be divided has a value of 5, what is the quotient when 3 is divided into it?

3. If the modulus is 3 and the number to be divided has a value of 2, what is the quotient when 1 is divided into it?

4. If the modulus is 5 and the number to be divided has a value of 4, what is the quotient when 3 is divided into it?

5. If the modulus is 9 and the number to be divided has a value of 7, what is the quotient when 5 is divided into it?

Suggested Answers to the question on “Division of integers in modular mathematics”.

7 / 3 = 2 and 2 is equal to 2 modulo 10, so the quotient is 2.

5 / 3 = 1 and 1 is equal to 1 modulo 7, so the quotient is 1.

2 / 1 = 2 and 2 is equal to 2 modulo 3, so the quotient is 2.

4 / 3 = 1 and 1 is equal to 1 modulo 5, so the quotient is 1.

7 / 5 = 2 and 2 is equal to 2 modulo 9, so the quotient is 2.

READING ASSIGNMENT

New General Mathematics for SSS 1, pages 227; exercises 20a, 20b Mathematical Association of Nigeria (MAN) pages 14-24

Conclusion:

The subject teacher wraps up or concludes the lesson by giving out a short note to summarize the topic that he or she has just taught.

The subject teacher also goes round to make sure that the notes are well copied or well written by the pupils.

He or she makes the necessary corrections when and where the needs arise.

REFERENCE TEXTS:

• New General Mathematics for senior secondary schools 1 by M.F Macrae et al; pearson education limited
• New school mathematics for senior secondary school et al; Africana publishers limited