1. Chapter 4 Numeration and Mathematical Systems © 2008 Pearson Addison-Wesley. All rights reserved

2. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-2 Chapter 4: Numeration and Mathematical Systems 4.1 Historical Numeration Systems 4.2 Arithmetic in the Hindu-Arabic System 4.3 Conversion Between Number Bases 4.4 Clock Arithmetic and Modular Systems 4.5 Properties of Mathematical Systems 4.6Groups

3. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-3 Chapter 1 Section 4-6 Groups

4. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-4 Groups Symmetry Groups Permutation Groups

5. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-5 Group A mathematical system is called a group if, under its operation, it satisfies the closure, associative, identity, and inverse properties.

6. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-6 Example: Checking Group Properties Does the set {–1, 1} under the operation of multiplication form a group? Solution –11 1 1 1 All of the properties to be a group (closure, associative, identity, inverse) are satisfied as can be seen by the table.

7. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-7 Example: Checking Group Properties Does the set {–1, 1} under the operation of addition form a group? Solution +–11 –20 102 No, right away it can be seen that closure is not satisfied.

8. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-8 Symmetry Groups A group can be built on sets of objects other than numbers. Consider the group of symmetries of a square. Start with a square labeled below. Front 1 2 4 3 Back 4' 3' 1'1' 2'

9. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-9 Symmetries - Rotational 4 1 3 2 M rotate 90 ° 3 4 2 1 N rotate 180 ° 2 3 1 4 P rotate 270 ° 1 2 4 3 Q original

10. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-10 Symmetries - Flip 1 2 4 3 Flip about horizontal line through middle. 2' 1' 3' 4' 1 2 4 3 Flip about vertical line through middle. 4' 3' 1' 2' S R

11. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-11 Symmetries - Flip 1 2 4 3 Flip about diagonal line upper left to lower right. 3' 2' 4' 1' 1 2 4 3 4' 2' 3' V T Flip about diagonal line upper right to lower left.

12. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-12 Symmetries of the Square □MNPQRSTV MNPQMVTRS NPQMNSRVT PQMNPTVSR QMNPQRSTV RTSVRQNMP SVRTSNQPM TSVRTPMQN VRTSVMPNQ

13. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-13 Example: Verifying a Subgroup Form a mathematical system by using only the set {M, N, P, Q} from the group of symmetries of a square. Is this new system a subgroup? □MNPQ MNPQM NPQMN PQMNP QMNPQ Solution The operational table is given and the system is a group. The new group is a subgroup of the original group.

14. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-14 Permutation Groups A group comes from studying the arrangements, or permutations, of a list of numbers. The next slide shows the possible permutations of the numbers 1-2-3.

15. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-15 Arrangements of 1-2-3 A*: 1-2-3 2-3-1 B*: 1-2-3 2-1-3 C*: 1-2-3 1-2-3 D*: 1-2-3 1-3-2 E*: 1-2-3 3-1-2 F*: 1-2-3 3-2-1

16. © 2008 Pearson Addison-Wesley. All rights reserved 4-6-16 Example: Combining Arrangements Find D*E*. Solution 1-2-3 1-3-2 Rearrange according to D*. 3 E* replaces 1 with 3. 3 1 E* replaces 2 with 1. 3-2-1 E* replaces 3 with 2.