1. 4.4 Clock Arithmetic and Modular Systems

2. 12-hour Clock System  Based on an ordinary clock face  12 replaced with a zero  Minute hand is left off

3. The clock system is FINITE  Also known as CLOSED  You will only get back a clock number no matter what operation you do to it

4. Addition in the clock system  Add by moving the hour had clockwise  Clock arithmetic only uses whole numbers

5. Example 1  6 + 3

6. Example 2  10 + 7

7. Example 3  11 + 4

8. Let’s make a table for clock addition!

9. Closure Property of Clock Addition Defined  If a, b are any clock #s, then a+b is also in the set under addition.

10. Commutative Property of Clock Addition  If a, b are any clock numbers, then a+b = b+a

11. Identity Property of Clock Addition  When an element and the identity are combined, the original element is returned  Ex: a + i = a a is returned, therefore i is the identity element.

12. Subtraction in Clock Arithmetic  Subtraction is possible by going counter clockwise  We will also use the additive inverse

13. Example 4!  5 - 7

14. Additive Inverse  An element combined with its additive inverse will return the identity  In our number system:

15. Determine 4’s additive inverse in clock arithmetic:  What number combined with 4 will return the identity?

16. Additive Inverse Property of Clock Addition  Every element of the system has an additive inverse  Table:

17. Subtraction of Clock Numbers  If a,b are clock numbers, then the difference, a-b is defined as: a + (-b): where -b is defined as the inverse of b.

18. Example 5!  5 – 7  5 + (-7)  5 + 5 = 10