4.4 Clock Arithmetic 4.4 Clock Arithmetic and Modular Systemsand Modular Systems

12-hour Clock System12-hour Clock System

Based on an Based on an ordinary clock faceordinary clock face

12 replaced with a 12 replaced with a zerozero

Minute hand is left Minute hand is left offoff

The clock system is The clock system is FINITEFINITE

Also known as Also known as CLOSEDCLOSED

You will only get You will only get back a clock back a clock number no matter number no matter what operation you what operation you do to itdo to it

Addition in the clock systemAddition in the clock system

Add by moving the Add by moving the hour had clockwisehour had clockwise

Clock arithmetic Clock arithmetic only uses whole only uses whole numbers numbers

Example 1Example 1

6 + 36 + 3

Example 2Example 2

10 + 710 + 7

Example 3Example 3

11 + 411 + 4

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

Closure Property of Clock Addition Closure Property of Clock Addition DefinedDefined

If a, b are any clock #s, then a+b is If a, b are any clock #s, then a+b is also in the set under addition.also in the set under addition.

Commutative Property of Clock Commutative Property of Clock AdditionAddition

If a, b are any clock numbers, then If a, b are any clock numbers, then a+b = b+a a+b = b+a

Identity Property of Clock AdditionIdentity Property of Clock Addition

When an element and the identity When an element and the identity are combined, the original element is are combined, the original element is returnedreturned

Ex: a + i = aEx: a + i = a

a is returned, therefore i is a is returned, therefore i is the identity element.the identity element.

Subtraction in Clock ArithmeticSubtraction in Clock Arithmetic

Subtraction is Subtraction is possible by going possible by going counter clockwisecounter clockwise

We will also use We will also use the additive the additive inverseinverse

Example 4!Example 4!

5 - 75 - 7

Additive InverseAdditive Inverse

An element combined with its An element combined with its additive inverse will return the additive inverse will return the identityidentity

In our number system:In our number system:

Determine 4’s additive inverse in Determine 4’s additive inverse in clock arithmetic:clock arithmetic:

What number combined with 4 will What number combined with 4 will return the identity?return the identity?

Additive Inverse Property of Clock Additive Inverse Property of Clock AdditionAddition

Every element of the system has an Every element of the system has an additive inverse additive inverse

Table:Table:

Subtraction of Clock NumbersSubtraction of Clock Numbers

If a,b are clock numbers, then the If a,b are clock numbers, then the difference, a-b is defined as:difference, a-b is defined as:

a + (-b): where -b is defined as a + (-b): where -b is defined as the inverse of b.the inverse of b.

Example 5!Example 5!

5 – 75 – 7 5 + (-7)5 + (-7) 5 + 5 = 105 + 5 = 10