To introduce the concept of a mathematical structure called an algebraic group. To illustrate group concepts, we introduce cyclic and dihedral groups.

Size: px
Start display at page:

Download "To introduce the concept of a mathematical structure called an algebraic group. To illustrate group concepts, we introduce cyclic and dihedral groups."

Transcription

1 1 Section 6. Introduction to the Algebraic Group Purpose of Section: To introduce the concept of a mathematical structure called an algebraic group. To illustrate group concepts, we introduce cyclic and dihedral groups. Introduction The theory of groups is an area of mathematics which is concerned with underlying relationships of things, and arguably the most powerful tool ever created for illuminating structure, both mathematical and physical. The word group was first used by the French genius Evariste Galois in 1830, who wrote his seminal paper on the unsolvability of the 5 th order polynomial equation, the night before he was killed in a stupid duel at the age of 0. Other early contributors to the development of group theory were Joseph Louis Lagrange, ( ), Niels Abel ( ), Augustin-Louis Cauchy ( ), Arthur Cayley ( , Camille Jordan ( ), Ludwig Sylow ( ) and Marius Sophus Lie ( ). Now, merely more than a century later, group theory has resulted in an amazing unification of areas of mathematics, including algebra and geometry, long thought to be separate and unrelated. It is often said that whenever groups make an appearance in a subject, simplicity is created from chaos. Group theory has played (and is playing) a crucial role for both chemists and physicists to penetrate the deep underlying relationships in our amazing world. Binary Operation and Groups A binary operation on a set A is a rule, which assigns to each pair of elements of A a unique element of A. Thus, a binary operation is simply a function f : A A A. Two common binary operations familiar to the reader are +, i which assign the sum a + b R and product aib R to a pair ( a, b) R R of real numbers. We now give a formal definition of a group.

2 Definition: An algebraic group g G (or simply group) is a set of elements with a binary operation, say, satisfying the closure property as well as the following properties: a, b G a * b G Associative: is associative, that is, for every a, b, c G, we have ( a b) c = a ( b c). Identity: G has a unique identity 1 e. That is, for any element a G we have a * e = e* a = a Inverse: Every element a G has a unique inverse. That is, for a G 1 there exists an element a 1 1 G that satisfies a * a = a * a = e. We often denote a group G with operation " " G. by {, } Often it happens that a b = b a for all a, b G. When this happens the group is called a commutative (or Abelian) group. We often denote the group operation a b as ab, or maybe something more suggestive like if the group operation is addition or closely resembles addition. A group is called finite if it contains a finite number of elements and the number of elements in the group is called the order of the group. If the order of a group G is n, we denote this by writing G = n If the group is not of finite order we say it is of infinite order. In Plain English Associative: The associative property tells us when we combine three elements a, b, c G (keeping them in the same order), the result is unchanged regardless of which two elements are combined first. There are examples of algebraic groups where the operation is not associative, the cross product of vectors in vector analysis is an example of a non-associative operation, as well as the difference between two numbers 3, but by far the majority of binary operations in mathematics are associative. 1 It is not necessary to state that the identity is unique since it can be proven there is only one identity. In a more lengthy treatment of groups, we would define the existence of an identity and then prove it is unique, but here we assume uniqueness to shorten the discussion. Again, it can be proven that the inverse is unique so it is not really necessary to assume uniqueness of an inverse in the definition. 3 ( a b) c a ( b c).

3 3 Identity: The identity element of a group depends on the binary relation and is the unique element e G that leaves every element a G unchanged when combined with e. In the group of the integers Z with the binary operation + (addition), the identity is 0 since a + 0 = 0 + a = a for every integer a. If the binary operation is (multiplication), then the identity e is 1 since a 1 = 1 a = a for all a in the group. 1 a Inverse: The inverse of an element a depends (of course) on the element a, but also on the identity e and is an element such that when 1 1 combined with a yields the identity; that is aa = a a = e. For example, the inverse of an integer a with group operation addition + is its negative a since a + a = a + a =. ( ) ( ) 0 Table 1: Properties of Binary Operations Operation Associative Commutative Identity Inverse on P( A) Yes Yes Yes No on P( A) Yes Yes Yes Yes gcd on N Yes Yes No No + on R Yes Yes Yes Yes on R No No No No on Q Yes Yes Yes Yes min on R Yes Yes No No Example 1: Group Test Which of the following define a group on the set of integers? a) { Z,+} : Integers with the operation of addition. b) { Z, ( m + n) / } : Integers with operation of averaging two integers. c) { Z, } Integers with operation of taking the difference of two integers. Solution a) { Z,+} : We leave it to the reader to show {,+} b) { Z, ( m + n) / } Z is a group. See Problem 5. : Taking the average of two integers, say and 3, is not an integer, hence the averaging operation is not closed in Z. Hence Z with the averaging operation is not a group. There is no need to check the other properties required of a group.

4 4 c) { Z, } The integers Z with the difference operation is not a group since subtraction is not associative, i.e. m ( n p) ( m n) p. However, it does have an identity, 0 since 0 m = m 0 = m. Also, every integer has an m m = 0, m m = 0. Nevertheless, failure of the inverse, itself, i.e. ( ) associative property says it is not a group. Abstraction Abstraction reveals connections between different areas of mathematics since the process of abstraction allows one to see essential ideas and see the forest and not just the trees. This broad viewpoint can result in making new discoveries in one area of mathematics as a result of knowledge in other areas. A disadvantage might be that highly abstract mathematics is more difficult to master and tends to isolate mathematics from the outside world. Cayley Table The binary operation of a group can be illustrated by means of a Cayley 4 table as drawn in Table, which shows the products gig j of elements g i and g j of a group. It is much like the addition or multiplication tables the reader studied as a child, except a Cayley table can record any binary operation. A Cayley table is an example of a latin square, meaning that every element of the group occurs once and exactly once in every row and column. We examine the Cayley table to learn about the inner workings of a group. g1 g1 = e g g 3 g j = e e g 1 g g j g g g g g 3 gg j g g g 1 g g g g 3 gig j i i i Cayley Table for a Group Figure Example Below we illustrate the only groups of order and 3. Show they are both commutative. Find the inverse of each element in the group. Show that each group is associative. Convince yourself that the only groups of these orders are the ones given. Keep in mind every row and column of the multiplication i 4 Arthur Cayley ( ) was an English mathematician

5 5 table includes every element of the group exactly once. We leave this fun for the reader. Order * e a e e a a a e Example 3: The set G { a, b, c, d} illustrated by the Cayley table Order 3 * e a b e e a b a a b e b b e a = and binary operation * define a group a) Is there an identity element? If so, find it. b) Find the inverse of each element. c) Is the binary operation commutative? a * b* c = a * b * c d) Is ( ) ( ) Solution a) The identity is a since ab = ba = b, ac = ca = c, ad = da = d b) a = a, b = d, c = c, d = b c) yes, the multiplication table is symmetric around the main diagonal a * b* c = a * d = d and a * b * c = b* c = d. In general, there is d) yes, ( ) ( ) no quick way to verify the associative property like there is the commutative property. You have to check ALL possible arrangements to verify associativity. On the other hand, if one instance where associativity fails, then the binary operation * is not associative.

6 6 Example 4: : Klein 4-Group G = e, a, b, c described by the multiplication table in Show that the set { } Figure 1 forms a group. This group is called the Klein 5 4-group group, which is the symmetry group of a (non square) rectangle 6 studied in Section 6.1. There are exactly two distinct groups of order four, the Klein 4-group and the cyclic group Z 4 which we will study shortly. e a b c e e a b c a a e c b b b c e a c c b a e Multiplication Table for the Klein four-group Figure 1 Proof: First observe all products of elements of G belong to G since the table consists only of elements of G. The hardest requirement to check is r s t = r s t, where r, s can be associativity, which requires we check ( ) ( ) 3 any of the elements e, a, b, c, which means we have 4 = 48 equations to check since each or the r, s, t in the associative formula can take on one of four values e, a, b, c. The computations can be simplified by observing the group is commutative (i.e. r s = s r for all r, s G ), which implies ( r r) r = r ( r r) so we have associativity when r = s = t. Other shortcuts tricks can be used (as well as computer algebra systems) to shorten the list of elements you must check. In this example, we observe that the group operation is simply the composition of functions and we can resort to the fact that composition of functions is associative. Finding the group identity e is easy since multiplying any member of the group by e, either on the left of right, does not change the member. You can see that the first row and column of the table are the same as the group elements themselves. 5 Felix Klein ( ) was a German geometer and one of the major mathematicians of the 19 th century. 6 Try interpreting the elements, e a of the Klein group are 0 and 180 degree rotations of a rectangle, and c, d the horizontal and vertical flips of the rectangle..

7 7 1 r Finally, to find the inverse of an element r simply follow along the 1 row labeled " r " until you get to the group identity e, then the inverse r is the column label above e. You could also do the same thing by going down the column labeled " r " until reaching e, then the row label at the left of e is 1 r. In the Klein four-group each element 1, a, b, c is its own inverse since the identity e lies along the diagonal of the multiplication table. Familiar Groups You are familiar with more groups that you probably realize. Table shows just a few algebraic groups you might have seen in earlier studies. Group Elements Operation Identity Inverse Z n Z addition 0 n + Q n m / n m, n > 0 Z k { 0,1,,..., n 1} R { 0} GL x nonzero real number a, b R R ( ) (, R) general linear group SL (, R) special linear group a b c d ad bc 0 a b c d ad bc = 1 multiplication 1 n / m multiplication mod n Abelian yes yes 0 n k yes multiplication 1 1/ x yes vector addition matrix multiplication matrix multiplication Common Groups Table ( 0,0 ) ( a, b) yes d b ad bc ad bc c a ad bc ad bc d c b a Notational Note: Repeated multiplication of an element g of a group by itself n result in powers of an element and are denoted by g, n = 1,,.... When n = 0 we define 0 g as the identity 0 g = e. Cyclic Groups (Modular ( Arithmetic) The most common and most simple of all groups are the cyclic groups, which are well-known to every child who has learned to keep time. no no

8 8 Definition: A finite cyclic group ( Z ) n, of order n is a group that contains an element g Z called the generator of the group, such that n 3 n 1 {,,,,..., } g e g g g g = Z n. where powers of g are simply repeated multiplications 7,,... 3 g = g g g = g g of g ; that is For example, the three rotational symmetries {,, } e R R of an equilateral triangle form a cyclic group Z 3 with generator g = R10 (note that R = R, R = e ) Cyclic groups also describe modular (or clock) arithmetic, which is the type of arithmetic we carry out when keeping time on 4-hour timepiece, where the numbers wrap around after 4 hours. However, unless you are in the military, your clock lists hours from 0 to 11 so telling time is done, as mod(1) = 3 A.M.. they say, modulo 1, as in 5 hours after 9 P.M. is ( ) This leads us to the cyclic group Z 1 with elements Z 1 = { 0,1,,3, 4,5, 6, 7,8,9,10,11} and the group operation on Z 1 exactly what you do when you keep time, that is a b = ( a + b)mod1 where mod 1 refers to computing a b by computing ( a + b) then taking its remainder after dividing by 1. (We denote the group operation by to remind us it is addition, only reduced modulo 1.) For example = mod1 which, related to a 1-hour clock, translates into A.M is 5 ( ) hours after 9 P.M. The hours of the clock Z 1 and keeping time using this binary operation defines an Abelian group of order 1, called a cyclic group, whose Cayley table is shown in Table 3. 7 We use the word multiplication here, but keep in mind the group operation can mean any binary operation, even addition.

9 Cayley table for the cyclic group of 1 elements Table 3 Figure 3 shows various clocks that give rise to different cyclic groups. The cyclic group Z 6 consists of elements Z6 = { 0, r1, r, r3, r4, r5 } were as always e is the identity map and r j is rotation of the clock by 60 j degrees, j = 1,...,5. This set, along with the binary operation of doing one operation after another (function composition) forms a group with the following Cayley table. In other to make the table read faster, we have replaced the angles of rotation r j by the time on the hour hand, 1,,3,, 11.

10 10 Z Cyclic Group Of Order Z 3 Cyclic Group of Order 3 Z 4 Cyclic Group of Order 4 Cyclic Groups Z n Z 6 Cyclic Group of Order Z 1 Cyclic of Order 1 See Table 1 Cyclic Groups Figure 3

11 11 Example 5 (Symmetries of a Square) Figure 4 shows the eight symmetries of a square, called the octic group. and ar 70. a) Is the octic group commutative? Hint: Compare products R70a b) There are several subsets of the eight symmetries that form a group in their own right. These are called subgroups of the octic group. Can you find all ten of them? Solution a) The reader can check but R70a ar70. Hence, the octic group is not commutative. b) The 10 subgroups of the octic group are { } { } { } { } { } { } { } { } e, e, v, e, h, e, d, e, a, e, R, e, R, v, h, e, R. d, a, D The reader can visualize these symmetries and make Cayley tables for them. (See Problem 16.) These subgroups form a partially ordered set and can be put in the Hasse diagram shown in Figure 5.

12 1 Motion Symbol First and Final Positions No motion e Rotate 90 Counterclockwise R 90 Rotate 180 Counterclockwise R 180 Rotate 70 Counterclockwise R 70 Horizontal flip h Vertical flip v Anti-diagonal flip a Diagonal flip d Symmetries of a Square Figure 4

13 13 Hasse Diagram for the Subgroups of the Octic Group D 4 Figure 5 Symmetry Groups of n -gons: Dihedral Groups In Section 6.1 we saw how symmetries of a figure in the plane, which is a rigid motion which leaves the figure unchanged, can create new symmetries by following one symmetry after another. In this way, one creates an arithmetic of symmetries, where the composition of symmetries plays the role of multiplication, and the system contains identity elements, inverses and all the goodies on an arithmetic system. In other words, a group of symmetries, called the symmetry group of the figure. Every figure no matter how non symmetric has at least one symmetric group, namely the group consisting only of the identity or do nothing symmetry. The more symmetric a figure the more elements in its symmetry group. In Section 6.1 we saw that the (non square) rectangle had four symmetries; namely rotations

14 14 of 0 and 180 degrees, and a horizontal and vertical flip about the midlines, which constitute the Klein 4-group. On the other hand, the more symmetric square has 8 symmetries in its symmetry group. (Can you find them?) Some figures have both rotational and flip symmetries. A polygon is called regular if all its sides have the same length and all its angles are equal. An equilateral triangle is a regular 3-gon, a square is a regular 4-gon, a pentagon is a regular 5-gon and so on. The symmetry group of a regular n -gon, which has n rotational and n flip symmetries, for a total of n symmetries, is called the dihedral group of the n -gon and denoted by D n Can you find the 10 symmetries of the dihedral group D 5 of the pentagon drawn in Figure 5? Find the Symmetric Group D 5 Figure 5 We are getting ahead of ourselves, but in addition to the complete dihedral group of 10 symmetries of a pentagon, there is also a smaller group of five rotation symmetries, called a subgroup, which is a group in its own right, of the larger group of 10 symmetries as well as a subgroup of the 5 flip symmetries. Figure 6 shows commercial figures whose symmetry groups are the dihedral groups D1 D5. symmetries D 1 one rotation (0 degrees), one vertical flip 4 symmetries D two rotations, two flips (horizontal and vertical axes)

15 15 D 3 three rotations, three flips 6 symmetries D 4 four rotations, four flips 8 symmetries D 5 five rotations, five flips 10 symmetries Figure with Dihedral Symmetry Groups Figure 6

16 16 Problems 1. Do the following sets with given binary operations form a group? If it does, give the identity element and the inverse of each element. If it does not form a group, say why. a) All even numbers, addition 1,1, multiplication b) { } c) All positive real numbers, multiplication d) All nonzero real numbers, division e) All real matrices, matrix addition f) the four numbers 1, 1, i, i where i = 1 is the unit complex number, the binary operation is ordinary multiplication. g) { a, b, c } with operation defined by the Cayley table a b c a b c a b a b c c c a b. (Finish the Group) Complete the following Cayley table for a group of order three. e a b e e a b a a b b 3. (Finish the Group) Complete the following Cayley table for a group of order four without looking at the Cayley tables of the Klein 4- group or the cyclic group of order 4 in the text. e a b c e e a b c a a b b c c 4. (Verification of a Group) Do the nonzero integers with the operation of multiplication form a group?

17 17 (Group You are Well Familiar) Show that {,+} 5. (Group You are Well Familiar) integers with the operation of addition) is a group. Z is a group (i.e. the 6. (Property of a Group) Verify that for all elements a, b in a group the identity ( ) ab = b a holds. Hint: Show that ( ab)( a b ) = e. General Properties of Groups 7. Show that a group has exactly one identity. 8. Show that every element in a group has no more than one inverse. 9. Show that in a group the identity ( ) (( ab c) d ) ( ab)( cd ) = holds. 10. Let a, b, c are members of a set with a binary operation on the set. If a = b, then the multiplication rule says ca = cb. a) Show that in a group we can cancel the c. b) Given an example of elements in the set Z 6 = { 0,1,,3, 4,5} with addition modulo 6, where cancellation does not hold. 11. (Cyclic Group) The cyclic group of order 6 describes the rotational symmetries of a regular hexagon. Its Cayley table is shown in Table 4. For notational simplicity the group elements are the number of degrees required to map the hexagon back onto itself Cayley Table for Z 6 Table 4 a) What is the inverse of each element of the group? b) The order of an element of a group is defined as the (smallest) number of repeated operations on itself that results in the group identity. What is the order of each element of the group?

18 18 c) Show that by taking repeated operations with itself of the element 40 the set of elements obtained is itself a group. Construct the Cayley table of this group. 1. (Affine Group) The set G of all transformations from the plane to the p lane of the form x = ax + by + c y = dx + ey + f where a, b, c, d, e, and f are real numbers satisfying ad bc 0, is a group if we define the group operation of performing one operation after the other, this forms a group called the affine group. a) What is the identity of the affine group? That is, what are the values of,,,,, a b c d e f? b) What is the inverse of( 0,0 )? 13.. (Mod 5 Multiplication) Create the multiplication table for the integers mod 5 arithmetic. Show this defines 0,1,, 3, 4 where multiplication defined as ( ) a group. 14. (Mod 4 Multiplication) Create the multiplication table for the integers mod 4 and show that this does not define a 0,1,,3 for modular arithmetic ( ) group. In other that the numbers 0,1,,..., n 1 forms a group under mod ( n) multiplication, it must be true that n is a prime number. 15. (Subgroups of the Dihedral Group D 4 ) A subgroup of a group is a subset of the elements of a group which itself a group using the group operation of the larger group. The Hasse diagram for the subgroups of the symmetries of a square (i.e. the dihedral group ) is shown in Figure 6. The letter " F " represents the identity element, the other letters represent rotations and reflections of F. Interpret the rotations in the subgroups and make a Cayley table for them. 16. (Subgroups of the Octic Group) Make a Cayley table for the subgroups of the octic group. a) { e, v } b) { e, h } c) { e, d }

19 19 d) { e, a } e) { e, R 180} f) { e, R180, v, h } g) { e, R, d, a } 180 Hasse diagram for the Subgroups of D 4 Figure (Isomorphic Groups) Sometime two groups appear different but are really the same group. The two groups in Figures 7, called Group A and Group B look different, but are really the same, or what are called isomorphic groups. Convince yourself the two groups are the same by making the substitution, or isomorphism that sends Group A into Group B., 0 i, 1 a, b, 3 c

20 Group A i a b c i i a b c a a b c i b b c i a c c i a b Group B 18. (Harder to See Isomorphic Groups) Show that Group C and Group D are isomorphic (the same group) by making the substitution, 0 1, 1, 4, 3 3 in Group C, and then interchanging the 3 rd and 4 th columns, followed by the 3 rd and 4 th rows of the resulting table Group C Group D

S on n elements. A good way to think about permutations is the following. Consider the A = 1,2,3, 4 whose elements we permute with the P =

S on n elements. A good way to think about permutations is the following. Consider the A = 1,2,3, 4 whose elements we permute with the P = Section 6. 1 Section 6. Groups of Permutations: : The Symmetric Group Purpose of Section: To introduce the idea of a permutation and show how the set of all permutations of a set of n elements, equipped

More information

So let us begin our quest to find the holy grail of real analysis.

So let us begin our quest to find the holy grail of real analysis. 1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

More information

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

More information

Elements of Abstract Group Theory

Elements of Abstract Group Theory Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for

More information

Group Theory (MA343): Lecture Notes Semester I Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics, NUI Galway

Group Theory (MA343): Lecture Notes Semester I Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics, NUI Galway Group Theory (MA343): Lecture Notes Semester I 2013-2014 Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics, NUI Galway November 21, 2013 Contents 1 What is a group? 2 1.1 Examples...........................................

More information

4. MATRICES Matrices

4. MATRICES Matrices 4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

Geometric Transformations

Geometric Transformations Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

Diagonal, Symmetric and Triangular Matrices

Diagonal, Symmetric and Triangular Matrices Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

More information

1 Symmetries of regular polyhedra

1 Symmetries of regular polyhedra 1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an

More information

Solving a System of Equations

Solving a System of Equations 11 Solving a System of Equations 11-1 Introduction The previous chapter has shown how to solve an algebraic equation with one variable. However, sometimes there is more than one unknown that must be determined

More information

SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison

SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections

More information

4. FIRST STEPS IN THE THEORY 4.1. A

4. FIRST STEPS IN THE THEORY 4.1. A 4. FIRST STEPS IN THE THEORY 4.1. A Catalogue of All Groups: The Impossible Dream The fundamental problem of group theory is to systematically explore the landscape and to chart what lies out there. We

More information

MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction. MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

More information

Abstract Algebra Cheat Sheet

Abstract Algebra Cheat Sheet Abstract Algebra Cheat Sheet 16 December 2002 By Brendan Kidwell, based on Dr. Ward Heilman s notes for his Abstract Algebra class. Notes: Where applicable, page numbers are listed in parentheses at the

More information

5.3 The Cross Product in R 3

5.3 The Cross Product in R 3 53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

More information

Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices. A Biswas, IT, BESU SHIBPUR

Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices. A Biswas, IT, BESU SHIBPUR Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices A Biswas, IT, BESU SHIBPUR McGraw-Hill The McGraw-Hill Companies, Inc., 2000 Set of Integers The set of integers, denoted by Z,

More information

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

More information

Chapter 7. Permutation Groups

Chapter 7. Permutation Groups Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014

L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 Unfortunately, no one can be told what the Matrix is. You have to see it for yourself. -- Morpheus Primary concepts:

More information

Introduction to Matrix Algebra I

Introduction to Matrix Algebra I Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model

More information

Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2

Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2 Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the

More information

Notes on Algebraic Structures. Peter J. Cameron

Notes on Algebraic Structures. Peter J. Cameron Notes on Algebraic Structures Peter J. Cameron ii Preface These are the notes of the second-year course Algebraic Structures I at Queen Mary, University of London, as I taught it in the second semester

More information

Properties of Real Numbers

Properties of Real Numbers 16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should

More information

1. Determine all real numbers a, b, c, d that satisfy the following system of equations.

1. Determine all real numbers a, b, c, d that satisfy the following system of equations. altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c

More information

15.062 Data Mining: Algorithms and Applications Matrix Math Review

15.062 Data Mining: Algorithms and Applications Matrix Math Review .6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

More information

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m) Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

More information

8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

More information

Section 1.1. Introduction to R n

Section 1.1. Introduction to R n The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

More information

PYTHAGOREAN TRIPLES KEITH CONRAD

PYTHAGOREAN TRIPLES KEITH CONRAD PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

More information

Cryptography and Network Security. Prof. D. Mukhopadhyay. Department of Computer Science and Engineering. Indian Institute of Technology, Kharagpur

Cryptography and Network Security. Prof. D. Mukhopadhyay. Department of Computer Science and Engineering. Indian Institute of Technology, Kharagpur Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Module No. # 01 Lecture No. # 12 Block Cipher Standards

More information

Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter 4 described a mathematical system

Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter 4 described a mathematical system CHAPTER Number Theory FIGURE FIGURE FIGURE Plus hours Plus hours Plus hours + = + = + = FIGURE. Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter described a mathematical

More information

6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)

6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu) 6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,

More information

1 Introduction to Matrices

1 Introduction to Matrices 1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

More information

3 Some Integer Functions

3 Some Integer Functions 3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple

More information

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers

More information

We know a formula for and some properties of the determinant. Now we see how the determinant can be used.

We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we

More information

Let s just do some examples to get the feel of congruence arithmetic.

Let s just do some examples to get the feel of congruence arithmetic. Basic Congruence Arithmetic Let s just do some examples to get the feel of congruence arithmetic. Arithmetic Mod 7 Just write the multiplication table. 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 2 0

More information

(Q, ), (R, ), (C, ), where the star means without 0, (Q +, ), (R +, ), where the plus-sign means just positive numbers, and (U, ),

(Q, ), (R, ), (C, ), where the star means without 0, (Q +, ), (R +, ), where the plus-sign means just positive numbers, and (U, ), 2 Examples of Groups 21 Some infinite abelian groups It is easy to see that the following are infinite abelian groups: Z, +), Q, +), R, +), C, +), where R is the set of real numbers and C is the set of

More information

We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P. Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

More information

3. Applications of Number Theory

3. Applications of Number Theory 3. APPLICATIONS OF NUMBER THEORY 163 3. Applications of Number Theory 3.1. Representation of Integers. Theorem 3.1.1. Given an integer b > 1, every positive integer n can be expresses uniquely as n = a

More information

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair

More information

Just the Factors, Ma am

Just the Factors, Ma am 1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive

More information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

More information

POWER SETS AND RELATIONS

POWER SETS AND RELATIONS POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

More information

The Inverse of a Matrix

The Inverse of a Matrix The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square

More information

Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli

Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Prompt: A mentor teacher and student teacher are discussing

More information

. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9 Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

More information

Matrix Algebra and Applications

Matrix Algebra and Applications Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2 - Matrices and Matrix Algebra Reading 1 Chapters

More information

5.1 Commutative rings; Integral Domains

5.1 Commutative rings; Integral Domains 5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following

More information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

This assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the

More information

Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.

Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that

More information

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

More information

Quotient Rings and Field Extensions

Quotient Rings and Field Extensions Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

More information

6.3 Conditional Probability and Independence

6.3 Conditional Probability and Independence 222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

More information

1 Solving LPs: The Simplex Algorithm of George Dantzig

1 Solving LPs: The Simplex Algorithm of George Dantzig Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

More information

Solving Systems of Linear Equations. Substitution

Solving Systems of Linear Equations. Substitution Solving Systems of Linear Equations There are two basic methods we will use to solve systems of linear equations: Substitution Elimination We will describe each for a system of two equations in two unknowns,

More information

Solution to Homework 2

Solution to Homework 2 Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if

More information

The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23

The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23 (copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional

More information

CONTENTS 1. Peter Kahn. Spring 2007

CONTENTS 1. Peter Kahn. Spring 2007 CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................

More information

Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

More information

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form 1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

More information

9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.

9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11. 9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role

More information

Permutation Groups. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles April 2, 2003

Permutation Groups. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles April 2, 2003 Permutation Groups Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles April 2, 2003 Abstract This paper describes permutations (rearrangements of objects): how to combine them, and how

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

More information

SUM OF TWO SQUARES JAHNAVI BHASKAR

SUM OF TWO SQUARES JAHNAVI BHASKAR SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted

More information

Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2-D

Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2-D Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2-D Welcome everybody. We continue the discussion on 2D

More information

The Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006

The Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006 The Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006 joshua.zucker@stanfordalumni.org [A few words about MathCounts and its web site http://mathcounts.org at some point.] Number theory

More information

Number Sense and Operations

Number Sense and Operations Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents

More information

THREE DIMENSIONAL GEOMETRY

THREE DIMENSIONAL GEOMETRY Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

More information

Review of Fundamental Mathematics

Review of Fundamental Mathematics Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

More information

2. THE x-y PLANE 7 C7

2. THE x-y PLANE 7 C7 2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

More information

Permutation Groups. Rubik s Cube

Permutation Groups. Rubik s Cube Permutation Groups and Rubik s Cube Tom Davis tomrdavis@earthlink.net May 6, 2000 Abstract In this paper we ll discuss permutations (rearrangements of objects), how to combine them, and how to construct

More information

GROUPS ACTING ON A SET

GROUPS ACTING ON A SET GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

More information

Introduction to Modern Algebra

Introduction to Modern Algebra Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write

More information

Math Workshop October 2010 Fractions and Repeating Decimals

Math Workshop October 2010 Fractions and Repeating Decimals Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

Factoring Patterns in the Gaussian Plane

Factoring Patterns in the Gaussian Plane Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood

More information

Lecture L3 - Vectors, Matrices and Coordinate Transformations

Lecture L3 - Vectors, Matrices and Coordinate Transformations S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

More information

some algebra prelim solutions

some algebra prelim solutions some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no

More information

4.4 Clock Arithmetic and Modular Systems

4.4 Clock Arithmetic and Modular Systems 4.4 Clock Arithmetic and Modular Systems A mathematical system has 3 major properies. 1. It is a set of elements 2. It has one or more operations to combine these elements (ie. Multiplication, addition)

More information

4. How many integers between 2004 and 4002 are perfect squares?

4. How many integers between 2004 and 4002 are perfect squares? 5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started

More information

SECTION 8.3: THE INVERSE OF A SQUARE MATRIX

SECTION 8.3: THE INVERSE OF A SQUARE MATRIX (Section 8.3: The Inverse of a Square Matrix) 8.47 SECTION 8.3: THE INVERSE OF A SQUARE MATRIX PART A: (REVIEW) THE INVERSE OF A REAL NUMBER If a is a nonzero real number, then aa 1 = a 1 a = 1. a 1, or

More information

Answer Key for California State Standards: Algebra I

Answer Key for California State Standards: Algebra I Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

More information

SECTION 9-1 Matrices: Basic Operations

SECTION 9-1 Matrices: Basic Operations 9 Matrices and Determinants In this chapter we discuss matrices in more detail. In the first three sections we define and study some algebraic operations on matrices, including addition, multiplication,

More information

26 Ideals and Quotient Rings

26 Ideals and Quotient Rings Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 26 Ideals and Quotient Rings In this section we develop some theory of rings that parallels the theory of groups discussed

More information

2. If C is the midpoint of AB and B is the midpoint of AE, can you say that the measure of AC is 1/4 the measure of AE?

2. If C is the midpoint of AB and B is the midpoint of AE, can you say that the measure of AC is 1/4 the measure of AE? MATH 206 - Midterm Exam 2 Practice Exam Solutions 1. Show two rays in the same plane that intersect at more than one point. Rays AB and BA intersect at all points from A to B. 2. If C is the midpoint of

More information

ABEL S THEOREM IN PROBLEMS AND SOLUTIONS

ABEL S THEOREM IN PROBLEMS AND SOLUTIONS TeAM YYePG Digitally signed by TeAM YYePG DN: cn=team YYePG, c=us, o=team YYePG, ou=team YYePG, email=yyepg@msn.com Reason: I attest to the accuracy and integrity of this document Date: 2005.01.23 16:28:19

More information

Fractions and Decimals

Fractions and Decimals Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first

More information

Number Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may

Number Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition

More information

Date: Period: Symmetry

Date: Period: Symmetry Name: Date: Period: Symmetry 1) Line Symmetry: A line of symmetry not only cuts a figure in, it creates a mirror image. In order to determine if a figure has line symmetry, a figure can be divided into

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

More information

MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

More information

Group Theory: Basic Concepts

Group Theory: Basic Concepts Group Theory: Basic Concepts Robert B. Griffiths Version of 9 Feb. 2009 References: EDM = Encyclopedic Dictionary of Mathematics, 2d English edition (MIT, 1987) HNG = T. W. Hungerford: Algebra (Springer-Verlag,

More information

Group Theory. Contents

Group Theory. Contents Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

More information