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