HomeSS 1MathematicsConcept of Modular Arithmetic and Cyclic Events – SS 1 Mathematics

Concept of Modular Arithmetic and Cyclic Events – SS 1 Mathematics

MATHEMATICS LESSON NOTE

CLASS: SS 1
TERM: First Term
WEEK: 4
TOPIC: Concept of Modular Arithmetic and Cyclic Events

Instructional Materials

  • Wall charts
  • Online resources
  • Pictures
  • Related audio-visual materials
  • Mathematics textbooks
  • A chart showing modular arithmetic
  • Samples of duty shifts
  • Menstrual chart

Reference Materials

  • Scheme of work
  • Online information
  • Textbooks
  • Workbooks
  • Education curriculum

Previous Knowledge

Students have previous knowledge of:

  • Conversion from Base Ten to other bases
  • Conversion from one base to another

Behavioral Objectives

By the end of the lesson, students should be able to:

  1. Define modular arithmetic.
  2. Perform basic operations (addition, subtraction, multiplication, and division) in modular arithmetic.
  3. Apply modular arithmetic to real-life situations.

CONTENT

SUB-TOPIC 1: Definition of Modular Arithmetic

Modular arithmetic is a system of arithmetic where numbers “wrap around” after reaching a certain value called the modulus. Unlike standard arithmetic, which follows an infinite sequence, modular arithmetic cycles through a set of numbers repeatedly.

For example, on a clock, after 12, the numbers reset to 1 instead of continuing to 13, 14, and so on. This is an example of modular arithmetic with a modulus of 12.

Interactive Questions and Answers

1. What is a modulus?

  • The modulus is the value at which numbers “wrap around” in modular arithmetic.

2. What is standard arithmetic?

  • Standard arithmetic follows a fixed number sequence without wrapping around.

3. How do numbers “wrap around” in modular arithmetic?

  • Numbers reset to zero or another starting point once they reach the modulus.

4. What is the difference between modular arithmetic and standard arithmetic?

  • Modular arithmetic repeats after reaching a modulus, while standard arithmetic continues infinitely.

5. What happens if two numbers in modular arithmetic are not relatively prime to the modulus?

  • Their results may collide and get reduced modulo the modulus.

SUB-TOPIC 2: Operations in Modular Arithmetic

Addition in Modular Arithmetic

Addition in modular arithmetic follows these steps:

  1. Add the given numbers.
  2. Divide the sum by the modulus.
  3. The remainder is the final answer.

Examples:

  1. 3+73 + 7 (mod 10) = 10. Since 10 mod 10 = 0, the answer is 0.
  2. 5+35 + 3 (mod 7) = 8. Since 8 mod 7 = 1, the answer is 1.
  3. 2+12 + 1 (mod 3) = 3. Since 3 mod 3 = 0, the answer is 0.
  4. 4+34 + 3 (mod 5) = 7. Since 7 mod 5 = 2, the answer is 2.
  5. 7+57 + 5 (mod 9) = 12. Since 12 mod 9 = 3, the answer is 3.

Subtraction in Modular Arithmetic

Subtraction in modular arithmetic follows similar steps:

  1. Subtract the numbers.
  2. If the result is negative, add the modulus.
  3. Find the remainder when divided by the modulus.

Examples:

  1. 7−37 – 3 (mod 10) = 4. Since 4 mod 10 = 4, the answer is 4.
  2. 5−35 – 3 (mod 7) = 2. Since 2 mod 7 = 2, the answer is 2.
  3. 2−12 – 1 (mod 3) = 1. Since 1 mod 3 = 1, the answer is 1.
  4. 4−34 – 3 (mod 5) = 1. Since 1 mod 5 = 1, the answer is 1.
  5. 7−57 – 5 (mod 9) = 2. Since 2 mod 9 = 2, the answer is 2.

Multiplication in Modular Arithmetic

For multiplication in modular arithmetic:

  1. Multiply the numbers.
  2. Find the remainder when divided by the modulus.

Examples:

  1. 3×73 × 7 (mod 10) = 21. Since 21 mod 10 = 1, the answer is 1.
  2. 5×35 × 3 (mod 7) = 15. Since 15 mod 7 = 1, the answer is 1.
  3. 2×12 × 1 (mod 3) = 2. Since 2 mod 3 = 2, the answer is 2.
  4. 4×34 × 3 (mod 5) = 12. Since 12 mod 5 = 2, the answer is 2.
  5. 7×57 × 5 (mod 9) = 35. Since 35 mod 9 = 8, the answer is 8.

Division in Modular Arithmetic

Division in modular arithmetic requires finding the modular inverse. The modular inverse of a number aa modulo mm is the number bb such that:
a×b≡1(modm)a × b ≡ 1 (mod m).

Examples:

  1. 7÷37 ÷ 3 (mod 10) = 1
  2. 5÷35 ÷ 3 (mod 7) = 2
  3. 2÷12 ÷ 1 (mod 3) = 2
  4. 4÷34 ÷ 3 (mod 5) = 1
  5. 7÷57 ÷ 5 (mod 9) = 4

SUB-TOPIC 3: Application of Modular Arithmetic in Daily Life

Modular arithmetic is widely used in:

  • Timekeeping: The 12-hour and 24-hour clocks operate on mod 12 and mod 24, respectively.
  • Cryptography: Used in encryption and security algorithms.
  • Banking Systems: Used for generating and verifying account numbers.
  • Scheduling: Helps in rotating work shifts and class timetables.
  • Menstrual Cycle Calculations: Used in predicting periods based on a cycle length.

Evaluation Questions

  1. Reduce 72 to its simplest form in:
    a. Modulo 3
    b. Modulo 4
    c. Modulo 5
    d. Modulo 6
    e. Modulo 7
  2. Find the following sums in modulo 5:
    a. 3 + 9
    b. 65 + 32
    c. 41 + 52
    d. 8 + 17
  3. Solve the following in modulo 4:
    a. 2×22 × 2
    b. 5×75 × 7
    c. 6×736 × 73

Conclusion

The teacher will:

  • Mark students’ work and provide corrections.
  • Summarize the lesson.
  • Explain the importance of modular arithmetic in daily life.

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