Elementary algebra aei 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 i ii CONTENTS 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 ....................