...William Allan Nelson
- Multiply the first number of the age by 5. (If the age is less than 10, example 7, consider it as 07. If it is greater than 100, example 109, then take 10 as the first digit, 9 as the second one.)
- Add 3 to the result.
- Double the answer.
- Add the second digit of the number with the result.
- Subtract 6 from it.
- You will get your age.
- Add 18 to your birth month.
- Multiply by 25.
- Subtract 333.
- Multiply by 8.
- Subtract 554.
- Divide by 2.
- Add your birth date.
- Multiply by 5.
- Add 692.
- Multiply by 20.
- Add only the last two digits of your birth year.
- Subtract 32940 to get your birthday!
A boy is half the age of his elder brother and the sum of the figures in the elder brother's age is half the sum of those in the younger brother's age. What are their respective ages?
How many months have 28 days?
1.1 Sets of Numbers and Interval Notation
1.2 Operations on Real Numbers
1.3 Simplifying Expressions
1.4 Linear Equations in One Variable
1.5 Applications of Linear Equations in One Variable
1.6 Literal Equations and Applications to Geometry
1.7 Linear Inequalities in One Variable
1.8 Properties of Integer Exponents and Scientific Notation
2.1 The Rectangular Coordinate System and Midpoint Formula
2.2 Linear Equations in Two Variables
2.3 Slope of a Line
2.4 Equations of a Line
2.5 Applications of Linear Equations and Graphing
3.1 Solving Systems of Linear Equations by Graphing
3.2 Solving Systems of Equations by Using the Substitution Method
3.3 Solving Systems of Equations by Using the Addition Method
3.4 Applications of Systems of Linear Equations in Two Variables
3.5 Systems of Linear Equations in Three Variables and Applications
3.6 Solving Systems of Linear Equations by Using Matrices
3.7 Determinants and Cramer�s Rule
4.1 Introduction to Relations
4.2 Introduction to Functions
4.3 Graphs of Functions
4.4 Variation
5.1 Addition and Subtraction of Polynomials and Polynomial Functions
5.2 Multiplication of Polynomials
5.3 Division of Polynomials Problem Recognition Exercises�Operations on Polynomials
5.4 Greatest Common Factor and Factoring by Grouping
5.5 Factoring Trinomials
5.6 Factoring Binomials
5.7 Additional Factoring Summary
5.8 Solving Equations by Using the Zero Product Rule
6.1 Rational Expressions and Rational Functions
6.2 Multiplication and Division of Rational Expressions
6.3 Addition and Subtraction of Rational Expressions
6.4 Complex Fractions Problem Recognition Exercises�Operations on Rational Expressions
6.5 Rational Equations
6.6 Applications of Rational Equations and Proportions
7.2 Rational Exponents
7.3 Simplifying Radical Expressions
7.4 Addition and Subtraction of Radicals
7.5 Multiplication of Radicals
7.6 Rationalization
7.7 Radical Equations
7.8 Complex Numbers
8.2 Quadratic Formula
8.3 Equations in Quadratic Form
8.4 Graphs of Quadratic Functions
8.5 Vertex of a Parabola and Applications
9.2 Polynomial and Rational Inequalities
9.3 Absolute Value Equations
9.4 Absolute Value Inequalities Problem Recognition Exercises�Equations and Inequalities
9.5 Linear Inequalities in Two Variables
| Please let us know what you think to our NEW WEBSITE! |
|---|
3.1.4. Definition. The set of all permutations of a set S is denoted by Sym(S). The set of all permutations of the set {1,2,...,n} is denoted by Sn.
3.1.5. Proposition. If S is any nonempty set, then Sym(S) is a group under the operation of composition of functions.
2.3.5. Theorem. Every permutation in Sn can be written as a product of disjoint cycles. The cycles that appear in the product are unique.
2.3.8 Proposition. If a permutation in Sn is written as a product of disjoint cycles, then its order is the least common multiple of the lengths of its cycles.
3.6.1. Definition. Any subgroup of the symmetric group Sym(S) on a set S is called a permutation group or group of permutations.
3.6.2. Theorem. (Cayley) Every group is isomorphic to a permutation group.
3.6.3. Definition. Let n > 2 be an integer. The group of rigid motions of a regular n-gon is called the nth dihedral group, denoted by Dn.
We can describe the nth dihedral group as
Dn= { ak, akb | 0 k < n },
subject to the relations o(a) = n, o(b) = 2, and ba = a-1b.
2.3.11. Theorem. If a permutation is written as a product of transpositions in two ways, then the number of transpositions is either even in both cases or odd in both cases.
2.3.12. Definition. A permutation is called even if it can be written as a product of an even number of transpositions, and odd if it can be written as a product of an odd number of transpositions.
3.6.4. Proposition. The set of all even permutations of Sn is a subgroup of Sn.
3.6.5. Definition. The set of all even permutations of Sn is called the alternating group on n elements, and will be denoted by An.
Other examples
Example 3.1.4. (Group of units modulo n) Let n be a positive integer. The set of units modulo n, denoted by Zn�, is an abelian group under multiplication of congruence classes. Its order is given by the value (n) of Euler's phi-function.
3.1.10. Definition. The set of all invertible n � n matrices with entries in R is called the general linear group of degree n over the real numbers, and is denoted by GLn(R).
3.1.11. Proposition. The set GLn(R) forms a group under matrix multiplication.
3.3.3. Definition. Let G1 and G2 be groups. The set of all ordered pairs (x1,x2) such that x1 G1 and x2 G2 is called the direct product of G1 and G2, denoted by G1 � G2.
3.3.4. Proposition. Let G1 and G2 be groups.
(a) The direct product G1 � G2 is a group under the multiplication defined for all
(a1,a2), (b1,b2) G1 � G2 by
(a1,a2) (b1,b2) = (a1b1,a2b2).
(b) If the elements a1 G1 and a2 G2 have orders n and m, respectively, then in
G1 � G2 the element (a1,a2) has order lcm[n,m].
3.3.5. Definition. Let F be a set with two binary operations + and � with respective identity elements 0 and 1, where 1 is distinct from 0. Then F is called a field if
(i) the set of all elements of F is an abelian group under +;
(ii) the set of all nonzero elements of F is an abelian group under �;
(iii) a � (b+c) = a � b + a � c for all a,b,c in F.
3.3.6. Definition. Let F be a field. The set of all invertible n � n matrices with entries in F is called the general linear group of degree n over F, and is denoted by GLn(F).
3.3.7. Proposition. Let F be a field. Then GLn(F) is a group under matrix multiplication.
3.4.5. Proposition. If m,n are positive integers such that gcd(m,n)=1, then
Zm � Zn Zmn.
Example. 3.3.7. (Quaternion group)
Consider the following set of invertible 2 � 2 matrices with entries in the field of complex numbers.
� , � , � , � .
If we let
1 = , i = , j = , k =
then we have the identities
i2 = j2 = k2 = -1;
ij = k, jk = i, ki = j;
ji = -k, kj = -i, ik = -j.
These elements form a nonabelian group Q of order 8 called the quaternion group, or group of quaternion units.
4.1.1. Definition. Let F be a set on which two binary operations are defined, called addition and multiplication, and denoted by + and � respectively. Then F is called a field with respect to these operations if the following properties hold:
(i) Closure: For all a,b F the sum a + b and the product a�b are uniquely defined and belong to F.
(ii) Associative laws: For all a,b,c F,
a+(b+c) = (a+b)+c and a�(b�c) = (a�b)�c.
(iii) Commutative laws: For all a,b F,
a+b = b+a and a�b = b�a.
(iv) Distributive laws: For all a,b,c F,
a�(b+c) = (a�b) + (a�c) and (a+b)�c = (a�c) + (b�c).
(v) Identity elements: The set F contains an additive identity element, denoted by 0, such that for all a F,
a+0 = a and 0+a = a.
The set F also contains a multiplicative identity element, denoted by 1 (and assumed to be different from 0) such that for all a F,
a�1 = a and 1�a = a.
(vi) Inverse elements: For each a F, the equations
a+x = 0 and x+a = 0
have a solution x F, called an additive inverse of a, and denoted by -a. For each nonzero element a F, the equations
a�x = 1 and x�a = 1
have a solution x F, called a multiplicative inverse of a, and denoted by a-1.
4.1.4. Definition. Let F be a field. If am, am-1 , . . . , a1, a0 F, then any expression of the form
amxm + am-1xm-1 + � � � + a1x + a0
is called a polynomial over F in the indeterminate x with coefficients am, am-1, . . . , a0. The set of all polynomials with coefficients in F is denoted by F[x].
If n is the largest nonnegative integer such that an 0, then we say that the polynomial
f(x) = anxn + � � � + a0
has degree n, written deg(f(x)) = n, and an is called the leading coefficient of f(x).
If the leading coefficient is 1, then f(x) is said to be monic.
Two polynomials are equal by definition if they have the same degree and all corresponding coefficients are equal. It is important to distinguish between the polynomial f(x) as an element of F[x] and the corresponding polynomial function from F into F defined by substituting elements of F in place of x. If f(x) = amxm + � � � + a0 and c F, then f(c) = amcm + � � � + a0. In fact, if F is a finite field, it is possible to have two different polynomials that define the same polynomial function. For example, let F be the field Z5 and consider the polynomials x5 -2x + 1 and 4x + 1. For any c Z5, by Fermat's theorem we have c5 c (mod 5), and so
c5 -2c + 1 -c + 1 4c + 1 (mod 5),
which shows that x5 -2x + 1 and 4x + 1 are identical, as functions.
For the polynomials
f(x) = amxm + am-1xm-1 + � � � + a1x + a0
and
g(x) = bnxn + bn-1xn-1 + � � � + b1x + b0,
the sum of f(x) and g(x) is defined by just adding corresponding coefficients. The product f(x)g(x) is defined to be
ambnxn+m + � � � + (a2b0 + a1b1 + a0b2)x2 + (a1b0 + a0b1)x + a0b0.
The coefficient ck of xk in f(x)g(x) can be described by the formula
ck = ai bk-i.
This definition of the product is consistent with what we would expect to obtain using a naive approach: Expand the product using the distributive law repeatedly (this amounts to multiplying each term be every other) and then collect similar terms.
4.1.5. Proposition. If f(x) and g(x) are nonzero polynomials in F[x], then f(x)g(x) is nonzero and
deg(f(x)g(x)) = deg(f(x)) + deg(g(x)).
4.1.6. Corollary. If f(x),g(x),h(x) F[x], and f(x) is not the zero polynomial, then
f(x)g(x) = f(x)h(x) implies g(x) = h(x).
4.1.7. Definition. Let f(x),g(x) F[x]. If f(x) = q(x)g(x) for some q(x) F[x], then we say that g(x) is a factor or divisor of f(x), and we write g(x) | f(x).
The set of all polynomials divisible by g(x) will be denoted by < g(x) >.
4.1.8. Lemma. For any element c F, and any positive integer k,
(x - c) | (xk - ck).