Laws of algebraKinds of NumbersCoordinates on the Line and OrderExponentsCoordinates in the Plane and GraphsLinesParabolas and EllipsesSolving EquationsSystems of EquationsSymmetryCompleting the SquareFunctionsInverse FunctionsAverage Rate of ChangePolynomialsRational FunctionsExponentials and LogarithmsExponential Growth and DecayThe Natural LogarithmSequences and SeriesInfinite SeriesComplex NumbersDivision of PolynomialsThe Fundamental Theorem of AlgebraWhere to Look in the TextbookCalc PrepJWR & MWSNov 2, 2009Contents1 Laws of algebra 22 Kinds of Numbers 63 Coordinates on the Line and Order 84 Exponents 125 Coordinates in the Plane and Graphs 136 Lines 177 Parabolas and Ellipses 208 Solving Equations 239 Systems of Equations 2610 Symmetry 2711 Completing the Square 2812 Functions 3013 Inverse Functions 3314 Average Rate of Change 36115 Polynomials 3716 Rational Functions 4117 Exponentials and Logarithms 4318 Exponential Growth and Decay 4619 The Natural Logarithm 4720 Sequences and Series 5121 Infinite Series 5422 Complex Numbers 5623 Division of Polynomials 5824 The Fundamental Theorem of Algebra 58A Where to Look in the Textbook 611 Laws of algebra1.1. Terminology and Notation. In this section we review the notationsused in algebra. Some are peculiar to these notes. For example the notationA := B indicates that the equality holds by definition of the notations in-volved. (See for example Paragraph 1.2 which follows.) Two other notationswhich will become important when we solve equations are =⇒ and ⇐⇒ .The notation P =⇒ Q means that P implies Q i.e. “If P , then Q”. Forexample, x = 2 =⇒ x2= 4. (Note however that the converse statementx2= 4 =⇒ x = 2 is not always true since it might be that x = −2.) Thenotation P ⇐⇒ Q means P =⇒ Q and Q =⇒ P , i.e. “P if and only ifQ”. For example 3x − 6 = 0 ⇐⇒ x = 2. The notations =⇒ and ⇐⇒are explained more carefully in Paragraphs 8.2 and 8.3 below.1.2. Implicit Multiplication. In mathematics the absence of an operationsymbol usually indicates multiplication: ab mean a × b. Sometimes a dot2is used to indicate multiplication and in computer languages an asterisk isoften used.ab := a ·b := a ∗b := a ×b1.3. Order of operations. Parentheses are used to indicate the orderof doing the operations: in evaluating an expression with parentheses theinnermost matching pairs are evaluated first as in((1 + 2)2+ 5)2= (32+ 5)2= (9 + 5)2= 142= 196.There are conventions which allow us not to write the parentheses. Forexample, multiplication is done before additionab + c means (ab) + c and not a(b + c),and powers are done before multiplication:ab2c means a(b2)c and not (ab)2c.In the absence of other rules and parentheses, the left most operations aredone first.a − b − c means (a − b) − c and not a − (b − c).The long fraction line indicates that the division is done last:a + bcmeans (a + b)/c and not a + (b/c).In writing fractions the length of the fraction line indicates which fraction isevaluated first:abcmeans a/(b/c) and not (a/b)/c,abcmeans (a/b)/c and not a/(b/c).The length of the horizontal line in the radical sign indicates the order ofevaluation:√a + b meansp(a + b) and not (√a) + b.√a + b means (√a) + b and notp(a + b).31.4. The Laws of Algebra. There are four fundamental operations whichcan be performed on numbers.1. Addition. The sum of a and b is denoted a + b.2. Multiplication. The product of a and b is denoted ab.3. Reversing the sign. The negative of a is denoted −a.4. Inverting. The reciprocal of a (for a 6= 0) is denoted by a−1or by1a.These operations satisfy the following laws.Associative a + (b + c) = (a + b) + c a(bc) = (ab)cCommutative a + (b + c) = (a + b) + c a(bc) = (ab)cIdentity a + 0 = 0 + a = a a · 1 = 1 · a = aInverse a + (−a) = (−a) + a = 0 a · a−1= a−1· a = 1Distributive a(b + c) = ab + ac (a + b)c = ac + bcThe operations of subtraction and division are then defined bya − b := a + (−b) a ÷ b :=ab:= a · b−1= a ·1b.All the rules of calculation that you learned in elementary school follow fromthe above fundamental laws. In particular, the Commutative and AssociativeLaws say that you can add a bunch of numbers in any order and similarlyyou can multiply a bunch of numbers in any order. For example,(A+B)+(C +D) = (A+C)+(B +D), (A·B)·(C ·D) = (A·C)·(B ·D).1.5. Because both addition and multiplication satisfy the commutative, as-sociative, identity, and inverse laws, there are other analogies:4(i) −(−a) = a (a−1)−1= a(ii) −(a + b) = −a − b (ab)−1= a−1b−1(iii) −(a − b) = b − aab−1=ba(iv) (a − b) + (c − d) = (a + c) − (b + d)ab·cd=acbd(v) a − b = (a + c) − (b + c)ab=acbc(vi) (a − b) − (c − d) = (a − b) + (d − c)a/bc/d=ab·dcThese identities are proved in the Guided Exercises. (An identity is anequation which is true for all values of the variables which appear in it.)1.6. Here are some further identities which are proved using the distributivelaw.(i) a · 0 = 0 (ii) −a = (−1)a(iii) a(−b) = −ab (iv) (−a)(−b) = ab(v)ab+cd=ad + cbbd(vi) (a + b)(c + d) = ab + ad + bc + bd(vii) (a + b)2= a2+ 2ab + b2(viii) (a + b)(a − b) = a2− b2These are also proved in the Guided Exercises.1.7. The following Zero-Product Property will be used to solve equations.pq = 0 ⇐⇒ p = 0 or q = 0 (or both).Proof: If p = 0 (or q = 0) then pq = 0 by (i) in Paragraph 1.6 . Conversely,if p 6= 0 then q = p−1pq = p−10 = 0.5Definition 1.8. For a natural number n and any number a the nth powerof a isan:= a · a · a ···a| {z }n factorsThe zeroth power isa0:= 1and negative powers are defined bya−n:=1an.1.9. The following laws are easy to understand when m and n are integers.In Theorem 4.1 below we will learn that these laws also hold whenever aand b are positive real numbers and m and n are any real numbers, not justintegers.(i) aman= am+ne.g. a2a3= (aa)(aaa) = a5(ii) (am)n= amne.g. (a2)3= (aa)(aa)(aa) = a6(iii)aman= am−ne.g.a2a5= a−3=1a3(iv) (ab)m= ambme.g. (ab)2= (ab)(ab) = (aa)(bb) = a2b2(v)abm=ambme.g.ab2=ab·ab=aabb=a2b22 Kinds of Numbers2.1. We distinguish the following different kinds of numbers.• The natural numbers are 1, 2, 3 . . ..• The integers are . . . − 3, −2, −1, 0, 1, 2, 3 . . ..• The rational numbers are ratios of integers like 3/2, 14/99, −1/2.6• The real numbers are numbers which have an infinite decimal expan-sion like32= 1.5000 . . . ,1499=