...Eleanor Doan
Algebra
From the 825 A.D. book, "ilm al-jabr w'al Maqa balah" (translated "The Science of Cancellation and Reduction") by the great Arabian Mathematician Mohammed Ibn Musa Khowarizimi. After years of bad pronunciation by Europeans, it came down as "aljabra" and eventually, "algebra".
If 3 cats can kill 3 rats in 3 minutes, how long will it take 100 cats to kill 100 rats?
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
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Additive Postulate of Inequality
If a < b, then a + c < b + c
Additive Postulate of Zero
a + 0 = a and 0 + a = a
Associate Postulate of Addition
(a + b) + c = a + (b + c)
Associative Postulate of Multiplication
(ab)c = a(bc)
Commutative Postulate of Addition
a + b = b + a
Commutative Postulate of Multiplication
ab = ba
Multiplication Postulate of One
a x 1 = a and 1 x a = a
Postulate of Additive Inverses
a + (-a) = 0 and (-a) + a = 0
Postulate of Comparison
Only one can be true: a < b, a = b, or b < a
Postulate of Multiplicative Inverses
a x (1/a) = 1 and (1/a) x a = 1
Reflexive Property of Equality
a = a
Symmetric Property of Equality
If a = b, then b = a
Transitive Property
If a = b and b = c, then a = c
Transitive Property of Inequality
If a < b and b < c, then a < c
Zero Product Property
If ab = 0, then a = 0 or b = 0
The set {..., -2, -1, 0, 1, 2, 3, ...} is called the set of integers, and will be denoted by Z.
1.1.1. Definition. An integer a is called a multiple of an integer b if a=bq for some integer q. In this case we also say that b is a divisor of a, and we use the notation b | a.
In the above case we can also say that b is a factor of a, or that a is divisible by b. If b is not a divisor of a, meaning that a bq for all q Z, then we write b a. The set of all multiples of an integer a will be denoted by
aZ = { m Z | m=aq for some q Z }.
1.1.2. Axiom. [Well-Ordering Principle] Every nonempty set of natural numbers contains a smallest element.
1.1.3 Theorem. [Division Algorithm] For any integers a and b, with b>0, there exist unique integers q (the quotient) and r (the remainder) such that a=bq+r, with 0 r
1.1.4. Theorem. Let I be a nonempty set of integers that is closed under addition and subtraction. Then I either consists of zero alone or else contains a smallest positive element, in which case I consists of all multiples of its smallest positive element.
1.1.5. Definition. A positive integer d is called the greatest common divisor of the nonzero integers a and b if
(i) d is a divisor of both a and b, and
(ii) any divisor of both a and b is also a divisor of d.
We will use the notation gcd(a,b), or simply (a,b), for the greatest common divisor of a and b.
1.1.6. Theorem. Any two nonzero integers a and b have a greatest common divisor, which can be expressed as the smallest positive linear combination of a and b. Moreover, an integer is a linear combination of a and b if and only if it is a multiple of their greatest common divisor.
The greatest common divisor of two numbers can be computed by using a procedure known as the Euclidean algorithm. First, note that if a 0 and b | a, then gcd(a,b) = |b|. The next observation provides the basis for the Euclidean algorithm. If a=bq+r, then (a,b)=(b,r). Thus given integers a>b>0, the Euclidean algorithm uses the division algorithm repeatedly to obtain
a = bq1 + r1, with 0 r1< b b = r1q2 + r2, with 0 r2< r1, etc.
Since r1 > r2 > . . . , the remainders get smaller and smaller, and after a finite number of steps we obtain a remainder rn+1 = 0. Thus gcd(a,b) = gcd(b,r1) = . . . = rn.
1.2.1. Definition. The nonzero integers a and b are said to be relatively prime if (a,b)=1.
1.2.2 Proposition. Let a,b be nonzero integers. Then (a,b)=1 if and only if there exist integers m,n such that
ma + nb = 1 .
1.2.3 Proposition. Let a,b,c be integers.
(a) If b | ac, then b | (a,b)c.
(b) If b | ac and (a,b)=1, then b | c.
(c) If b | a, c | a and (b,c)=1, then bc | a.
(d) (a,bc)=1 if and only if (a,b)=1 and (a,c)=1.
1.2.4. Definition. An integer p>1 is called a prime number if its only divisors are � 1 and � p. An integer a > 1 is called composite if it is not prime.
1.2.5. Lemma. [Euclid] An integer p>1 is prime if and only if it satisfies the following property: If p | ab for integers a and b, then either p | a or p | b.
1.2.6. Theorem. [Fundamental Theorem of Arithmetic] Any integer a>1 can be factored uniquely as a product of prime numbers, in the form
a = p1m1 p2m2 � � � pnmn
where p1 < p2 < � � � < pn and the exponents m1, m2 , . . . , mn are all positive.
1.2.7. Theorem. [Euclid] There exist infinitely many prime numbers.
1.2.8. Definition. A positive integer m is called the least common multiple of the nonzero integers a and b if
(i) m is a multiple of both a and b, and
(ii) any multiple of both a and b is also a multiple of m.
We will use the notation lcm[a,b] for the least common multiple of a and b.
1.2.9. Proposition. Let a and b be positive integers with prime factorizations
a = p1a1 p2a2 � � � pnan and b = p1b1 p2b2 � � � pnbn ,
where ai 0 and bi 0 for all i (allowing use of the same prime factors.)
For each i let di =min { ai, bi } and let mi =max { ai, bi }. Then we have the following factorizations:
(a) gcd(a,b) = p1d1 p2d2 � � � pndn
(b) lcm[a,b] = p1m1 p2m2 � � � pnmn
1.3.1. Definition. Let n be a positive integer. Integers a and b are said to be congruent modulo n if they have the same remainder when divided by n. This is denoted by writing a b (mod n).
1.3.2. Proposition. Let a,b, and n>0 be integers. Then a b (mod n) if and only if n | (a-b).
When working with congruence modulo n, the integer n is called the modulus. By the preceding proposition, a b (mod n) if and only if a-b=nq for some integer q. We can write this in the form a=b+nq, for some integer q. This observation gives a very useful method of replacing a congruence with an equation (over Z). On the other hand, Proposition 1.3.3 shows that any equation can be converted to a congruence modulo n by simply changing the = sign to . In doing so, any term congruent to 0 can simply be omitted. Thus the equation a=b+nq would be converted back to a b (mod n).
1.3.3 Proposition. Let n>0 be an integer. Then the following conditions hold for all integers a,b,c,d:
(a) If a c (mod n) and b d (mod n), then
then a b c d (mod n), and ab cd (mod n).
(b) If a+c a+d (mod n), then c d (mod n).
If ac ad (mod n) and (a,n)=1, then c d (mod n).
1.3.4. Proposition. Let a and n>1 be integers. There exists an integer b such that ab 1 (mod n) if and only if (a,n)=1.