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4.4 Clock Arithmetic and Modular Systems. 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.

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slide2

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

slide3

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.

slide4

12 Hour Clock Addition Table

When the same number is always found on the other side of the diagonal, the system is commutative.

slide5

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.

slide6

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.

slide7

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?

slide8

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

slide10

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.

slide11

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.

slide12

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

slide13

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.

slide14

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?

slide15

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.

slide19

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

slide20

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.

slide21

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, … }

slide22

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, … }