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

© 2008 Pearson Addison-Wesley. All rights reserved 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

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

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

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

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

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

© 2008 Pearson Addison-Wesley. All rights reserved 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 Back 4' 3' 1'1' 2'

© 2008 Pearson Addison-Wesley. All rights reserved Symmetries - Rotational M rotate 90 ° N rotate 180 ° P rotate 270 ° Q original

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

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

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

© 2008 Pearson Addison-Wesley. All rights reserved 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.

© 2008 Pearson Addison-Wesley. All rights reserved 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

© 2008 Pearson Addison-Wesley. All rights reserved Arrangements of A*: B*: C*: D*: E*: F*:

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