Elementary algebra abde From Wikipedia, the free encyclopedia Contents

1 Additive identity 1 1.1 Elementary examples ...... 1 1.2 Formal definition ...... 1 1.3 Further examples ...... 1 1.4 Proofs ...... 2 1.4.1 The additive identity is unique in a group ...... 2 1.4.2 The additive identity annihilates ring elements ...... 2 1.4.3 The additive and multiplicative identities are different in a non-trivial ring ...... 2 1.5 See also ...... 2 1.6 References ...... 2 1.7 External links ...... 3

2 Additive inverse 4 2.1 Common examples ...... 4 2.1.1 Relation to subtraction ...... 4 2.1.2 Other properties ...... 4 2.2 Formal definition ...... 5 2.3 Other examples ...... 5 2.4 Non-examples ...... 6 2.5 See also ...... 6 2.6 Footnotes ...... 6 2.7 References ...... 6

3 Algebraic expression 7 3.1 Terminology ...... 7 3.2 In roots of polynomials ...... 8 3.3 Conventions ...... 8 3.3.1 Variables ...... 8 3.3.2 Exponents ...... 8 3.4 Algebraic vs. other mathematical expressions ...... 8 3.5 See also ...... 8 3.6 Notes ...... 8 3.7 References ...... 9

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3.8 External links ...... 9

4 Algebraic fraction 10 4.1 Terminology ...... 10 4.2 Rational fractions ...... 10 4.3 Irrational fractions ...... 11 4.4 Notes ...... 11 4.5 References ...... 11

5 Algebraic operation 12 5.1 Notation ...... 12 5.2 Arithmetic vs algebraic operations ...... 13 5.3 Properties of arithmetic and algebraic operations ...... 13 5.4 References ...... 13 5.5 See also ...... 13

6 Associative property 14 6.1 Definition ...... 14 6.2 Generalized associative law ...... 15 6.3 Examples ...... 15 6.4 Propositional logic ...... 18 6.4.1 Rule of replacement ...... 18 6.4.2 Truth functional connectives ...... 18 6.5 Non-associativity ...... 18 6.5.1 Nonassociativity of floating point calculation ...... 19 6.5.2 Notation for non-associative operations ...... 19 6.6 See also ...... 21 6.7 References ...... 21

7 Brahmagupta’s identity 22 7.1 History ...... 22 7.2 Application to Pell’s equation ...... 22 7.3 See also ...... 23 7.4 References ...... 23 7.5 External links ...... 23

8 Brahmagupta–Fibonacci identity 24 8.1 History ...... 24 8.2 Related identities ...... 25 8.3 Relation to complex numbers ...... 25 8.4 Interpretation via norms ...... 25 8.5 Application to Pell’s equation ...... 25 8.6 See also ...... 26 CONTENTS iii

8.7 References ...... 26 8.8 External links ...... 26

9 Carlyle circle 27 9.1 Definition ...... 27 9.2 Defining property ...... 28 9.3 Construction of regular polygons ...... 28 9.3.1 Regular pentagon ...... 28 9.3.2 Regular heptadecagon ...... 29 9.3.3 Regular 257-gon ...... 30 9.3.4 Regular 65537-gon ...... 30 9.4 References ...... 30

10 Change of variables 32 10.1 Simple example ...... 33 10.2 Formal introduction ...... 33 10.3 Other examples ...... 33 10.3.1 Coordinate transformation ...... 33 10.3.2 Differentiation ...... 34 10.3.3 Integration ...... 34 10.3.4 Differential equations ...... 34 10.3.5 Scaling and shifting ...... 34 10.3.6 Momentum vs. velocity ...... 35 10.3.7 Lagrangian mechanics ...... 35 10.4 See also ...... 36

11 Commutative property 37 11.1 Common uses ...... 37 11.2 Mathematical definitions ...... 38 11.3 Examples ...... 38 11.3.1 Commutative operations in everyday life ...... 38 11.3.2 Commutative operations in mathematics ...... 38 11.3.3 Noncommutative operations in everyday life ...... 39 11.3.4 Noncommutative operations in mathematics ...... 40 11.4 History and etymology ...... 40 11.5 Propositional logic ...... 41 11.5.1 Rule of replacement ...... 41 11.5.2 Truth functional connectives ...... 41 11.6 Set theory ...... 41 11.7 Mathematical structures and commutativity ...... 41 11.8 Related properties ...... 42 11.8.1 Associativity ...... 42 iv CONTENTS

11.8.2 Symmetry ...... 42 11.9 Non-commuting operators in quantum mechanics ...... 43 11.10See also ...... 43 11.11Notes ...... 44 11.12References ...... 44 11.12.1 Books ...... 44 11.12.2 Articles ...... 45 11.12.3 Online resources ...... 45

12 Completing the square 46 12.1 Overview ...... 47 12.1.1 Background ...... 47 12.1.2 Basic example ...... 47 12.1.3 General description ...... 48 12.1.4 Non-monic case ...... 48 12.1.5 Formula ...... 48 12.2 Relation to the graph ...... 49 12.3 Solving quadratic equations ...... 50 12.3.1 Irrational and complex roots ...... 50 12.3.2 Non-monic case ...... 51 12.4 Other applications ...... 51 12.4.1 Integration ...... 51 12.4.2 Complex numbers ...... 52 12.4.3 Idempotent matrix ...... 53 12.5 Geometric perspective ...... 53 12.6 A variation on the technique ...... 53 12.6.1 Example: the sum of a positive number and its reciprocal ...... 54 12.6.2 Example: factoring a simple quartic polynomial ...... 54 12.7 References ...... 54 12.8 External links ...... 54

13 Constant term 56 13.1 See also ...... 57

14 Cube root 58 14.1 Formal definition ...... 59 14.1.1 Real numbers ...... 59 14.1.2 Complex numbers ...... 60 14.2 Impossibility of compass-and-straightedge construction ...... 61 14.3 Numerical methods ...... 62 14.4 Appearance in solutions of third and fourth degree equations ...... 62 14.5 History ...... 63 CONTENTS v

14.6 See also ...... 63 14.7 References ...... 63 14.8 External links ...... 63

15 Cubic function 64 15.1 History ...... 65 15.2 Critical points of a cubic function ...... 68 15.3 Roots of a cubic function ...... 68 15.3.1 The nature of the roots ...... 68 15.3.2 General formula for roots ...... 70 15.3.3 Reduction to a depressed cubic ...... 71 15.3.4 Cardano’s method ...... 71 15.3.5 Vieta’s substitution ...... 73 15.3.6 Lagrange’s method ...... 74 15.3.7 Trigonometric (and hyperbolic) method ...... 75 15.3.8 Factorization ...... 76 15.3.9 Geometric interpretation of the roots ...... 77 15.4 Collinearities ...... 78 15.5 Applications ...... 78 15.6 See also ...... 78 15.7 Notes ...... 78 15.8 References ...... 80 15.9 External links ...... 81

16 Difference of two squares 85 16.1 Proof ...... 85 16.2 Geometrical demonstrations ...... 85 16.3 Uses ...... 87 16.3.1 Factorisation of polynomials ...... 87 16.3.2 Complex number case: sum of two squares ...... 87 16.3.3 Rationalising denominators ...... 87 16.3.4 Mental arithmetic ...... 88 16.3.5 Difference of two perfect squares ...... 88 16.4 Generalizations ...... 89 16.4.1 Difference of two nth powers ...... 89 16.5 See also ...... 89 16.6 Notes ...... 90 16.7 References ...... 90 16.8 External links ...... 90

17 Distributive property 91 17.1 Definition ...... 91 vi CONTENTS

17.2 Meaning ...... 91 17.3 Examples ...... 92 17.3.1 Real numbers ...... 92 17.3.2 Matrices ...... 93 17.3.3 Other examples ...... 93 17.4 Propositional logic ...... 93 17.4.1 Rule of replacement ...... 93 17.4.2 Truth functional connectives ...... 94 17.5 Distributivity and rounding ...... 94 17.6 Distributivity in rings ...... 94 17.7 Generalizations of distributivity ...... 95 17.7.1 Notions of antidistributivity ...... 95 17.8 Notes ...... 95 17.9 References ...... 96 17.10External links ...... 96

18 Elementary algebra 97 18.1 Algebraic notation ...... 97 18.1.1 Alternative notation ...... 99 18.2 Concepts ...... 99 18.2.1 Variables ...... 99 18.2.2 Evaluating expressions ...... 99 18.2.3 Equations ...... 100 18.2.4 Substitution ...... 102 18.3 Solving algebraic equations ...... 102 18.3.1 Linear equations with one variable ...... 103 18.3.2 Linear equations with two variables ...... 103 18.3.3 Quadratic equations ...... 105 18.3.4 Exponential and logarithmic equations ...... 107 18.3.5 Radical equations ...... 108 18.3.6 System of linear equations ...... 109 18.3.7 Other types of systems of linear equations ...... 111 18.4 See also ...... 113 18.5 References ...... 114 18.6 External links ...... 115

19 Equating coefficients 116 19.1 Example in real fractions ...... 116 19.2 Example in nested radicals ...... 117 19.3 Example of testing for linear dependence of equations ...... 117 19.4 Example in complex numbers ...... 118 19.5 References ...... 118 CONTENTS vii

20 Equation 119 20.1 Introduction ...... 119 20.1.1 Parameters and unknowns ...... 119 20.1.2 Analogous illustration ...... 120 20.1.3 Identities ...... 121 20.2 Properties ...... 122 20.3 Algebra ...... 123 20.3.1 Polynomial equations ...... 123 20.3.2 Systems of linear equations ...... 123 20.4 Geometry ...... 125 20.4.1 Analytic geometry ...... 125 20.4.2 Cartesian equations ...... 125 20.4.3 Parametric equations ...... 126 20.5 Number theory ...... 127 20.5.1 Diophantine equations ...... 127 20.5.2 Algebraic and transcendental numbers ...... 127 20.5.3 Algebraic geometry ...... 127 20.6 Differential equations ...... 127 20.6.1 Ordinary differential equations ...... 128 20.6.2 Partial differential equations ...... 128 20.7 Types of equations ...... 128 20.8 See also ...... 129 20.9 References ...... 129 20.10External links ...... 130

21 Euler’s four-square identity 131 21.1 See also ...... 132 21.2 References ...... 132 21.3 External links ...... 132

22 Extraneous and missing solutions 133 22.1 Extraneous solutions: multiplication ...... 133 22.2 Extraneous solutions: rational ...... 134 22.3 Missing solutions: division ...... 134 22.4 Other operations ...... 135 22.5 See also ...... 135

23 Proofs involving the addition of natural numbers 136 23.1 Definitions ...... 136 23.2 Proof of associativity ...... 136 23.3 Proof of identity element ...... 136 23.4 Proof of commutativity ...... 137 viii CONTENTS

23.5 See also ...... 137 23.6 References ...... 137 23.7 Text and image sources, contributors, and licenses ...... 138 23.7.1 Text ...... 138 23.7.2 Images ...... 141 23.7.3 Content license ...... 143 Chapter 1

Additive identity

In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.

1.1 Elementary examples • The additive identity familiar from elementary mathematics is zero, denoted 0. For example, 5 + 0 = 5 = 0 + 5 • In the natural numbers N and all of its supersets (the integers Z, the rational numbers Q, the real numbers R, or the complex numbers C), the additive identity is 0. Thus for any one of these numbers n, n + 0 = n = 0 + n

1.2 Formal definition

Let N be a set which is closed under the operation of addition, denoted +. An additive identity for N is any element e such that for any element n in N,

e + n = n = n + e

Example: The formula is n + 0 = n = 0 + n.

1.3 Further examples • In a group the additive identity is the identity element of the group, is often denoted 0, and is unique (see below for proof). • A ring or field is a group under the operation of addition and thus these also have a unique additive identity 0. This is defined to be different from the multiplicative identity 1 if the ring (or field) has more than one element. If the additive identity and the multiplicative identity are the same, then the ring is trivial (proved below). • In the ring Mm×n(R) of m by n matrices over a ring R, the additive identity is denoted 0 and is the m by n matrix whose entries consist entirely of the identity element 0 in R. For example, in the 2 by 2 matrices over the integers M2(Z) the additive identity is 0 0 0= (0 0)

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