4.4 Clock Arithmetic and Modular Systems

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## 4.4 Clock Arithmetic and Modular Systems

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**12-hour Clock System**• Based on an ordinary clock face • 12 replaced with a zero • Minute hand is left off**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**Addition in the clock system**• Add by moving the hour had clockwise • Clock arithmetic only uses whole numbers**Example 1**• 6 + 3**Example 2**• 10 + 7**Example 3**• 11 + 4**Closure Property of Clock Addition Defined**• If a, b are any clock #s, then a+b is also in the set under addition.**Commutative Property of Clock Addition**• If a, b are any clock numbers, then a+b = b+a**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.**Subtraction in Clock Arithmetic**• Subtraction is possible by going counter clockwise • We will also use the additive inverse**Example 4!**• 5 - 7**Additive Inverse**• An element combined with its additive inverse will return the identity • In our number system:**Determine 4’s additive inverse in clock arithmetic:**• What number combined with 4 will return the identity?**Additive Inverse Property of Clock Addition**• Every element of the system has an additive inverse • Table:**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.**Example 5!**• 5 – 7 • 5 + (-7) • 5 + 5 = 10