4.4 Clock Arithmetic and Modular Systems

1. 4.4 Clock Arithmetic and Modular Systems

2. A mathematical system has a set of elements, one or more operations for combining those elements, and one or more relations for comparing those elements. A clock can demonstrate a mathematical system. Add by moving in a clockwise direction 0 1 11 2 10 9 3 8 4 5 7 6

3. 12 Hour Clock Addition Table The set of elements that are in the clock system is: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} The elements in the addition table are the same as the elements in the system. We say that the clock system is closed for the operation of addition.

4. 12 Hour Clock Addition Table When the same number is always found on the other side of the diagonal, the system is commutative.

5. 12 Hour Clock Addition Table When the sum is the same when we add two of the three terms in a different order then the system is associative.

6. 12 Hour Clock Addition Table When zero is added to any number the sum is the original number. Zero is the additive identity for clock arithmetic.

7. 12 Hour Clock Addition Table When two numbers add to give a sum that is the additive identity (zero), then they are additive inverses. 5 is the additive inverse of 7 and 7 is the additive inverse of 5. What is the additive inverse of 3?

8. Subtraction with Clock Arithmetic If a and b are elements in clock arithmetic, then the difference, a – b = a + (– b ) 0 1 11 2 10 9 3 8 4 5 7 6

9. 12 Hour Clock Addition Table

10. 12 Hour Clock Addition Table Multiply, then divide the product by 12 (the number of elements in the system.) The product will be the remainder.

11. 12 Hour Clock Addition Table Multiply, then divide the product by 12 (the number of elements in the system.) The product will be the remainder.

12. Multiply, then divide the product by 12 (the number of elements in the system.) The product will be the remainder.

13. We can do this with a 9 hour clock. There will be 9 elements in the new system. Multiply, then divide the product by 9 (the number of elements in the system.) The product will be the remainder.

14. 9 Hour Clock Multiplication Table What is the identity element for multiplication? What is the inverse of 4? What is the inverse of 8? What is the inverse of 3? Is multiplication closed? Is multiplication commutative? Is multiplication associative?

15. Clock arithmetic is a modular system. The same operations can be performed in any modulus (the modulus is the number of elements in the system.) Multiply, then divide the product by 15 (the modulus.) The product will be the remainder.

16. Multiply, then divide the product by 11 (the modulus.) The product will be the remainder.

17. Add, then divide the product by 6 (the modulus.) The sum will be the remainder.

18. Subtract, then divide the difference by 21 (the modulus.) The difference will be the remainder.

19. Congruence in Modular Systems The integers a and b are congruent modulo m(where m is a natural number greater than 1 called the modulus) if and only if the difference a – b is divisible by m. 21 is divisible by 7 418 is divisible by 11

20. Congruence in Modular Systems The integers a and b are congruent modulo mif and only if the same remainder is obtained when a and b are divided by m.

21. Solving Modular Equations Check the possible solutions 0, 1, 2, 3, and 4 The solutions are x = {2, 2 + 5, 2 + 5 + 5, 2 + 5 + 5 +5, … } The solutions are x = {2, 7, 12, 17, … }

22. Solving Modular Equations Check the possible solutions 0, 1, 2, 3, 4, 5, and 6 The solutions are x = {3, 3 + 7, 3 + 7 + 7, 3+ 7 + 7 + 7, … } The solutions are x = {3, 10, 17, 24, … }