Elementary algebra abdFrom Wikipedia, the free encyclopedia

Contents

1 Additive identity 11.1 Elementary examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4.1 The additive identity is unique in a group . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4.2 The additive identity annihilates ring elements . . . . . . . . . . . . . . . . . . . . . . . . 21.4.3 The additive and multiplicative identities are dierent in a non-trivial ring . . . . . . . . . . 2

1.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Additive inverse 42.1 Common examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Relation to subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Algebraic expression 73.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 In roots of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3.2 Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.4 Algebraic vs. other mathematical expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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3.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Algebraic fraction 104.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Rational fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Irrational fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Algebraic operation 125.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 Arithmetic vs algebraic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3 Properties of arithmetic and algebraic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Associative property 146.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 Generalized associative law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.4.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.4.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.5 Non-associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.5.1 Nonassociativity of oating point calculation . . . . . . . . . . . . . . . . . . . . . . . . . 196.5.2 Notation for non-associative operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7 Brahmaguptas identity 227.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.2 Application to Pells equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8 BrahmaguptaFibonacci identity 248.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.2 Related identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.3 Relation to complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.4 Interpretation via norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.5 Application to Pells equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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8.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9 Carlyle circle 279.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.2 Dening property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.3 Construction of regular polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

9.3.1 Regular pentagon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.3.2 Regular heptadecagon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.3.3 Regular 257-gon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.3.4 Regular 65537-gon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

9.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10 Change of variables 3210.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.2 Formal introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

10.3.1 Coordinate transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.3.2 Dierentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.3.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.3.4 Dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.3.5 Scaling and shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.3.6 Momentum vs. velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.3.7 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

10.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

11 Commutative property 3711.1 Common uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.2 Mathematical denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

11.3.1 Commutative operations in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.3.2 Commutative operations in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.3.3 Noncommutative operations in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . 3911.3.4 Noncommutative operations in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 40

11.4 History and etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.5 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

11.5.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.5.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

11.6 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.7 Mathematical structures and commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.8 Related properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

11.8.1 Associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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11.8.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.9 Non-commuting operators in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

11.12.1 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.12.2 Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.12.3 Online resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

12 Completing the square 4612.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

12.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4712.1.2 Basic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4712.1.3 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.1.4 Non-monic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.1.5 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

12.2 Relation to the graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.3 Solving quadratic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

12.3.1 Irrational and complex roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5012.3.2 Non-monic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

12.4 Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.4.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.4.2 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.4.3 Idempotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

12.5 Geometric perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5312.6 A variation on the technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

12.6.1 Example: the sum of a positive number and its reciprocal . . . . . . . . . . . . . . . . . . 5412.6.2 Example: factoring a simple quartic polynomial . . . . . . . . . . . . . . . . . . . . . . . 54

12.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5412.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

13 Constant term 5613.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

14 Cube root 5814.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

14.1.1 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5914.1.2 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

14.2 Impossibility of compass-and-straightedge construction . . . . . . . . . . . . . . . . . . . . . . . . 6114.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6214.4 Appearance in solutions of third and fourth degree equations . . . . . . . . . . . . . . . . . . . . . 6214.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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14.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6314.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6314.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

15 Cubic function 6415.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6515.2 Critical points of a cubic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6815.3 Roots of a cubic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

15.3.1 The nature of the roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6815.3.2 General formula for roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7015.3.3 Reduction to a depressed cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7115.3.4 Cardanos method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7115.3.5 Vietas substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7315.3.6 Lagranges method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7415.3.7 Trigonometric (and hyperbolic) method . . . . . . . . . . . . . . . . . . . . . . . . . . . 7515.3.8 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7615.3.9 Geometric interpretation of the roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

15.4 Collinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8015.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

16 Dierence of two squares 8516.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8516.2 Geometrical demonstrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8516.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

16.3.1 Factorisation of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8716.3.2 Complex number case: sum of two squares . . . . . . . . . . . . . . . . . . . . . . . . . . 8716.3.3 Rationalising denominators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8716.3.4 Mental arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8816.3.5 Dierence of two perfect squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

16.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8916.4.1 Dierence of two nth powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

16.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8916.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9016.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9016.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

17 Distributive property 9117.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

vi CONTENTS

17.2 Meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9117.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

17.3.1 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9217.3.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9317.3.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

17.4 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9317.4.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9317.4.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

17.5 Distributivity and rounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9417.6 Distributivity in rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9417.7 Generalizations of distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

17.7.1 Notions of antidistributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9517.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9517.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9617.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

18 Elementary algebra 9718.1 Algebraic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

18.1.1 Alternative notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9918.2 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

18.2.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9918.2.2 Evaluating expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9918.2.3 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10018.2.4 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

18.3 Solving algebraic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10218.3.1 Linear equations with one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.3.2 Linear equations with two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.3.3 Quadratic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10518.3.4 Exponential and logarithmic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10718.3.5 Radical equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10818.3.6 System of linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10918.3.7 Other types of systems of linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . 111

18.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11318.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11418.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

19 Proofs involving the addition of natural numbers 11619.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11619.2 Proof of associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11619.3 Proof of identity element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11619.4 Proof of commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11719.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

CONTENTS vii

19.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11719.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 118

19.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11819.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12019.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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 fromelementary mathematics, but additive identities occur in other mathematical structures where addition is dened, suchas 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 denitionLet N be a set which is closed under the operation of addition, denoted +. An additive identity for N is any elemente 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 belowfor proof).

A ring or eld is a group under the operation of addition and thus these also have a unique additive identity 0.This is dened to be dierent from the multiplicative identity 1 if the ring (or eld) 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 Mmn(R) of m by n matrices over a ring R, the additive identity is denoted 0 and is the m by nmatrix whose entries consist entirely of the identity element 0 in R. For example, in the 2 by 2 matrices overthe integers M2(Z) the additive identity is

0 =

0 00 0

1

2 CHAPTER 1. ADDITIVE IDENTITY

In the quaternions, 0 is the additive identity. In the ring of functions from R to R, the function mapping every number to 0 is the additive identity. In the additive group of vectors in Rn, the origin or zero vector is the additive identity.

1.4 Proofs

1.4.1 The additive identity is unique in a groupLet (G, +) be a group and let 0 and 0' in G both denote additive identities, so for any g in G,

0 + g = g = g + 0 and 0' + g = g = g + 0'

It follows from the above that

(0') = (0') + 0 = 0' + (0) = (0)

1.4.2 The additive identity annihilates ring elementsIn a system with a multiplication operation that distributes over addition, the additive identity is a multiplicativeabsorbing element, meaning that for any s in S, s0 = 0. This can be seen because:

s 0 = s (0 + 0) = s 0 + s 0) s 0 = s 0 s 0) s 0 = 0

1.4.3 The additive and multiplicative identities are dierent in a non-trivial ringLet R be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, or 0 = 1. Let r beany element of R. Then

r = r 1 = r 0 = 0

proving that R is trivial, that is, R = {0}. The contrapositive, that if R is non-trivial then 0 is not equal to 1, is thereforeshown.

1.5 See also 0 (number) Additive inverse Identity element Multiplicative identity

1.6 References David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley (3d ed.): 2003, ISBN 0-471-43334-9.

1.7. EXTERNAL LINKS 3

1.7 External links uniqueness of additive identity in a ring at PlanetMath.org. Margherita Barile, Additive Identity, MathWorld.

Chapter 2

Additive inverse

In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This numberis also known as the opposite (number),[1] sign change, and negation.[2] For a real number, it reverses its sign: theopposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additiveinverse of itself.The additive inverse of a is denoted by unary minus: a (see the discussion below). For example, the additive inverseof 7 is 7, because 7 + (7) = 0, and the additive inverse of 0.3 is 0.3, because 0.3 + 0.3 = 0 .The additive inverse is dened as its inverse element under the binary operation of addition (see the discussion below),which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, doubleadditive inverse has no eect: (x) = x.

2.1 Common examplesFor a number and, generally, in any ring, the additive inverse can be calculated using multiplication by 1; that is, n= 1 n . Examples of rings of numbers are integers, rational numbers, real numbers, and complex number.

2.1.1 Relation to subtractionAdditive inverse is closely related to subtraction, which can be viewed as an addition of the opposite:

a b = a + (b).

Conversely, additive inverse can be thought of as subtraction from zero:

a = 0 a.

Hence, unary minus sign notation can be seen as a shorthand for subtraction with 0 symbol omitted, although in acorrect typography there should be no space after unary "".

2.1.2 Other propertiesIn addition to the identities listed above, negation has the following algebraic properties:

(a + b) = (a) + (b)a (b) = a + b(a) b = a (b) = (a b)(a) (b) = a b

notably, (a)2 = a2

4

2.2. FORMAL DEFINITION 5

0

+i

i

1 +1

These complex numbers, two of eight values of 81, are mutually opposite

2.2 Formal denitionThe notation + is usually reserved for commutative binary operations, i.e. such that x + y = y + x, for all x, y . If suchan operation admits an identity element o (such that x + o ( = o + x ) = x for all x), then this element is unique ( o =o + o = o ). For a given x , if there exists x such that x + x ( = x + x ) = o , then x is called an additive inverse of x.If + is associative (( x + y ) + z = x + ( y + z ) for all x, y, z), then an additive inverse is unique

x = x + o = x + (x + x) = (x + x ) + x = o + x = x

For example, since addition of real numbers is associative, each real number has a unique additive inverse.

2.3 Other examplesAll the following examples are in fact abelian groups:

complex numbers: (a + bi) = (a) + (b)i. On the complex plane, this operation rotates a complex number180 degrees around the origin (see the image above).

6 CHAPTER 2. ADDITIVE INVERSE

addition of real- and complex-valued functions: here, the additive inverse of a function f is the function fdened by (f )(x) = f (x) , for all x, such that f + (f ) = o , the zero function ( o(x) = 0 for all x ).

more generally, what precedes applies to all functions with values in an abelian group ('zero' meaning then theidentity element of this group):

sequences, matrices and nets are also special kinds of functions. In a vector space the additive inverse v is often called the opposite vector of v; it has the samemagnitude as theoriginal and opposite direction. Additive inversion corresponds to scalar multiplication by 1. For Euclideanspace, it is point reection in the origin. Vectors in exactly opposite directions (multiplied to negative numbers)are sometimes referred to as antiparallel.

vector space-valued functions (not necessarily linear), In modular arithmetic, themodular additive inverse of x is also dened: it is the number a such that a + x 0 (mod n). This additive inverse always exists. For example, the inverse of 3 modulo 11 is 8 because it is thesolution to 3 + x 0 (mod 11).

2.4 Non-examplesNatural numbers, cardinal numbers, and ordinal numbers, do not have additive inverses within their respective sets.Thus, for example, we can say that natural numbers do have additive inverses, but because these additive inverses arenot themselves natural numbers, the set of natural numbers is not closed under taking additive inverses.

2.5 See also Absolute value (related through the identity | x | = | x | ) Multiplicative inverse Additive identity Involution (mathematics) Reection symmetry

2.6 Footnotes[1] Tussy, Alan; Gustafson, R. (2012), Elementary Algebra (5th ed.), Cengage Learning, p. 40, ISBN 9781133710790.

[2] The term negation bears a reference to negative numbers, which can be misleading, because the additive inverse of anegative number is positive.

2.7 References Margherita Barile, Additive Inverse, MathWorld.

Chapter 3

Algebraic expression

Rational expression redirects here. For the notion in formal languages, see regular expression.

In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraicoperations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational num-ber).[1] For example, 3x2 2xy+ c is an algebraic expression. Since taking the square root is the same as raising tothe power 12 ,

r1 x21 + x2

is also an algebraic expression. By contrast, transcendental numbers like and e are not algebraic.A rational expression is an expression that may be rewritten to a rational fraction by using the properties of thearithmetic operations (commutative properties and associative properties of addition and multiplication, distributiveproperty and rules for the operations on the fractions). In other words, a rational expression is an expression whichmaybe constructed from the variables and the constants by using only the four operations of arithmetic. Thus, 3x22xy+cy31is a rational expression, whereas

q1x21+x2 is not.

A rational equation is an equation in which two rational fractions (or rational expressions) of the form P (x)Q(x) areset equal to each other. These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undened, so that a solution causing formal division by zero is rejected.

3.1 Terminology

Algebra has its own terminology to describe parts of an expression:

1 Exponent (power), 2 coecient, 3 term, 4 operator, 5 constant, x; y - variables

7

8 CHAPTER 3. ALGEBRAIC EXPRESSION

3.2 In roots of polynomialsThe roots of a polynomial expression of degree n, or equivalently the solutions of a polynomial equation, can alwaysbe written as algebraic expressions if n < 5 (see quadratic formula, cubic function, and quartic equation). Such asolution of an equation is called an algebraic solution. But the Abel-Runi theorem states that algebraic solutions donot exist for all such equations (just for some of them) if n 5.

3.3 Conventions

3.3.1 VariablesBy convention, letters at the beginning of the alphabet (e.g. a; b; c ) are typically used to represent constants, andthose toward the end of the alphabet (e.g. x; y and z ) are used to represent variables.[2] They are usually written initalics.[3]

3.3.2 ExponentsBy convention, terms with the highest power (exponent), are written on the left, for example, x2 is written to the leftof x . When a coecient is one, it is usually omitted (e.g. 1x2 is written x2 ).[4] Likewise when the exponent (power)is one, (e.g. 3x1 is written 3x ),[5] and, when the exponent is zero, the result is always 1 (e.g. 3x0 is written 3 , sincex0 is always 1 ).[6]

3.4 Algebraic vs. other mathematical expressionsThe table below summarizes how algebraic expressions compare with several other types of mathematical expressions.A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient ofpolynomials, such as x2 + 4x + 4. An irrational algebraic expression is one that is not rational, such as x + 4.

3.5 See also Algebraic equation

Linear_equation#Algebraic_equations Algebraic function Analytical expression Arithmetic expression Closed-form expression Expression (mathematics) Polynomial Term (logic)

3.6 Notes[1] Morris, Christopher G. (1992). Academic Press dictionary of science and technology. p. 74.

[2] William L. Hosch (editor), The Britannica Guide to Algebra and Trigonometry, Britannica Educational Publishing, TheRosen Publishing Group, 2010, ISBN 1615302190, 9781615302192, page 71

3.7. REFERENCES 9

[3] James E. Gentle, Numerical Linear Algebra for Applications in Statistics, Publisher: Springer, 1998, ISBN 0387985425,9780387985428, 221 pages, [James E. Gentle page 183]

[4] DavidAlanHerzog, Teach Yourself Visually Algebra, Publisher JohnWiley&Sons, 2008, ISBN0470185597, 9780470185599,304 pages, page 72

[5] JohnC. Peterson, TechnicalMathematicsWith Calculus, Publisher Cengage Learning, 2003, ISBN0766861899, 9780766861893,1613 pages, page 31

[6] JeromeE.Kaufmann, Karen L. Schwitters,Algebra for College Students, Publisher Cengage Learning, 2010, ISBN0538733543,9780538733540, 803 pages, page 222

3.7 References James, Robert Clarke; James, Glenn (1992). Mathematics dictionary. p. 8.

3.8 External links Weisstein, Eric W., Algebraic Expression, MathWorld.

Chapter 4

Algebraic fraction

In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Twoexamples of algebraic fractions are 3xx2+2x3 and

px+2

x23 . Algebraic fractions are subject to the same laws as arithmeticfractions.A rational fraction is an algebraic fraction whose numerator and denominator are both polynomials. Thus 3xx2+2x3is a rational fraction, but not

px+2

x23 ; because the numerator contains a square root function.

4.1 TerminologyIn the algebraic fraction ab , the dividend a is called the numerator and the divisor b is called the denominator. Thenumerator and denominator are called the terms of the algebraic fraction.A complex fraction is a fraction whose numerator or denominator, or both, contains a fraction. A simple fractioncontains no fraction either in its numerator or its denominator. A fraction is in lowest terms if the only factor commonto the numerator and the denominator is 1.An expression which is not in fractional form is an integral expression. An integral expression can always be writtenin fractional form by giving it the denominator 1. A mixed expression is the algebraic sum of one or more integralexpressions and one or more fractional terms.

4.2 Rational fractionsSee also: Rational function

If the expressions a and b are polynomials, the algebraic fraction is called a rational algebraic fraction[1] or simplyrational fraction.[2][3] Rational fractions are also known as rational expressions. A rational fraction f(x)g(x) is calledproper if deg f(x) < deg g(x) , and improper otherwise. For example, the rational fraction 2xx21 is proper, and therational fractions x3+x2+1x25x+6 and x

2x+15x2+3 are improper. Any improper rational fraction can be expressed as the sum of

a polynomial (possibly constant) and a proper rational fraction. In the rst example of an improper fraction one has

x3 + x2 + 1

x2 5x+ 6 = (x+ 6) +24x 35

x2 5x+ 6 ;

where the second term is a proper rational fraction. The sum of two proper rational fractions is a proper rationalfraction as well. The reverse process of expressing a proper rational fraction as the sum of two or more fractions iscalled resolving it into partial fractions. For example,

2x

x2 1 =1

x 1 +1

x+ 1:

10

4.3. IRRATIONAL FRACTIONS 11

Here, the two terms on the right are called partial fractions.

4.3 Irrational fractionsAn irrational fraction is one that contains the variable under a fractional exponent.[4] An example of an irrationalfraction is

x12 13a

x13 x 12

:

The process of transforming an irrational fraction to a rational fraction is known as rationalization. Every irrationalfraction in which the radicals are monomials may be rationalized by nding the least common multiple of the indicesof the roots, and substituting the variable for another variable with the least common multiple as exponent. In theexample given, the least common multiple is 6, hence we can substitute x = z6 to obtain

z3 13az2 z3 :

4.4 Notes[1] Bansi Lal (2006). Topics in Integral Calculus. p. 53.

[2] rnest Borisovich Vinberg (2003). A course in algebra. p. 131.

[3] Parmanand Gupta. Comprehensive Mathematics XII. p. 739.

[4] Washington McCartney (1844). The principles of the dierential and integral calculus; and their application to geometry.p. 203.

4.5 ReferencesBrink, Raymond W. (1951). IV. Fractions. College Algebra.

Chapter 5

Algebraic operation

Algebraic operations in the solution to the quadratic equation. The radical sign, denoting a square root, is equivalent toexponentiation to the power of . The sign represents the equation written with either a + and with a - sign.

In mathematics, an algebraic operation is any one of the operations addition, subtraction, multiplication, division,raising to an integer power, and taking roots (fractional power). Algebraic operations are performed on an algebraicvariable, term or expression,[1] and work in the same way as arithmetic operations.[2]

5.1 Notation

Multiplication symbols are usually omitted, and implied when there is no operator between two variables or terms,or when a coecient is used. For example, 3 x2 is written as 3x2, and 2 x y is written as 2xy.[3] Sometimesmultiplication symbols are replaced with either a dot, or center-dot, so that x y is written as either x . y or x y.Plain text, programming languages, and calculators also use a single asterisk to represent the multiplication symbol,[4]and it must be explicitly used, for example, 3x is written as 3 * x.Rather than using the obelus symbol, , division is usual represented with a vinculum, a horizontal line, e.g. 3/x + 1.In plain text and programming languages a slash (also called a solidus) is used, e.g. 3 / (x + 1).Exponents are usually formatted using superscripts, e.g. x2. In plain text, and in the TeX mark-up language, the caretsymbol, ^, represents exponents, so x2 is written as x ^ 2.[5][6] In programming languages such as Ada,[7] Fortran,[8]Perl,[9] Python[10] and Ruby,[11] a double asterisk is used, so x2 is written as x ** 2.The plus-minus sign, , is used as a shorthand notation for two expressions written as one, representing one expressionwith a plus sign, the other with a minus sign. For example y = x 1 represents the two equations y = x + 1 and y = x 1. Sometimes it is used for denoting positive-or-negative term such as x.

12

5.2. ARITHMETIC VS ALGEBRAIC OPERATIONS 13

5.2 Arithmetic vs algebraic operationsAlgebraic operations work in the same way as arithmetic operations, as can be seen in the table below.Note: the use of the letters a and b is arbitrary, and the examples would be equally valid if we had used x and y .

5.3 Properties of arithmetic and algebraic operations

5.4 References[1] William Smyth, Elementary algebra: for schools and academies, Publisher Bailey and Noyes, 1864, "Algebraic Operations"

[2] Horatio Nelson Robinson, New elementary algebra: containing the rudiments of science for schools and academies, Ivison,Phinney, Blakeman, & Co., 1866, page 7

[3] Sin Kwai Meng, Chip Wai Lung, Ng Song Beng, Algebraic notation, in Mathematics Matters Secondary 1 Express Text-book, Publisher Panpac Education Pte Ltd, ISBN 9812738827, 9789812738820, page 68

[4] William P. Berlingho, Fernando Q. Gouva,Math through the Ages: A Gentle History for Teachers and Others, PublisherMAA, 2004, ISBN 0883857367, 9780883857366, page 75

[5] Ramesh Bangia, Dictionary of Information Technology, Publisher Laxmi Publications, Ltd., 2010, ISBN 9380298153,9789380298153, page 212

[6] George Grtzer, First Steps in LaTeX, Publisher Springer, 1999, ISBN 0817641327, 9780817641320, page 17

[7] S. Tucker Taft, Robert A. Du, Randall L. Brukardt, Erhard Ploedereder, Pascal Leroy, Ada 2005 Reference Manual,Volume 4348 of Lecture Notes in Computer Science, Publisher Springer, 2007, ISBN 3540693351, 9783540693352,page 13

[8] C. Xavier, Fortran 77AndNumericalMethods, PublisherNewAge International, 1994, ISBN812240670X, 9788122406702,page 20

[9] Randal Schwartz, brian foy, Tom Phoenix, Learning Perl, Publisher O'Reilly Media, Inc., 2011, ISBN 1449313140,9781449313142, page 24

[10] MatthewA. Telles, Python Power!: The Comprehensive Guide, Publisher Course Technology PTR, 2008, ISBN1598631586,9781598631586, page 46

[11] Kevin C. Baird,Ruby by Example: Concepts and Code, PublisherNo Starch Press, 2007, ISBN1593271484, 9781593271480,page 72

[12] Ron Larson, Robert Hostetler, Bruce H. Edwards, Algebra And Trigonometry: A Graphing Approach, Publisher: CengageLearning, 2007, ISBN 061885195X, 9780618851959, 1114 pages, page 7

5.5 See also Elementary algebra Order of operations

Chapter 6

Associative property

This article is about associativity in mathematics. For associativity in the central processing unit memory cache, seeCPU cache. For associativity in programming languages, see operator associativity.Associative and non-associative redirect here. For associative and non-associative learning, see Learning#Types.

In mathematics, the associative property[1] is a property of some binary operations. In propositional logic, associa-tivity is a valid rule of replacement for expressions in logical proofs.Within an expression containing two or more occurrences in a row of the same associative operator, the order inwhich the operations are performed does not matter as long as the sequence of the operands is not changed. That is,rearranging the parentheses in such an expression will not change its value. Consider the following equations:

(2 + 3) + 4 = 2 + (3 + 4) = 9

2 (3 4) = (2 3) 4 = 24:Even though the parentheses were rearranged, the values of the expressions were not altered. Since this holds truewhen performing addition and multiplication on any real numbers, it can be said that addition and multiplication ofreal numbers are associative operations.Associativity is not to be confused with commutativity, which addresses whether a b = b a.Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups andcategories) explicitly require their binary operations to be associative.However, many important and interesting operations are non-associative; some examples include subtraction, exponentiationand the vector cross product. In contrast to the theoretical counterpart, the addition of oating point numbers in com-puter science is not associative, and is an important source of rounding error.

6.1 DenitionFormally, a binary operation on a set S is called associative if it satises the associative law:

(x y) z = x (y z) for all x, y, z in S.

Here, is used to replace the symbol of the operation, which may be any symbol, and even the absence of symbollike for the multiplication.

(xy)z = x(yz) = xyz for all x, y, z in S.

The associative law can also be expressed in functional notation thus: f(f(x, y), z) = f(x, f(y, z)).

14

6.2. GENERALIZED ASSOCIATIVE LAW 15

A binary operation on the set S is associative when this diagram commutes. That is, when the two paths from SSS to S composeto the same function from SSS to S.

6.2 Generalized associative lawIf a binary operation is associative, repeated application of the operation produces the same result regardless how validpairs of parenthesis are inserted in the expression.[2] This is called the generalized associative law. For instance, aproduct of four elements may be written in ve possible ways:

1. ((ab)c)d

2. (ab)(cd)

3. (a(bc))d

4. a((bc)d)

5. a(b(cd))

If the product operation is associative, the generalized associative law says that all these formulas will yield the sameresult, making the parenthesis unnecessary. Thus the product can be written unambiguously as

abcd.

As the number of elements increases, the number of possible ways to insert parentheses grows quickly, but theyremain unnecessary for disambiguation.

6.3 ExamplesSome examples of associative operations include the following.

The concatenation of the three strings hello, " ", world can be computed by concatenating the rst twostrings (giving hello ") and appending the third string (world), or by joining the second and third string(giving " world) and concatenating the rst string (hello) with the result. The two methods produce thesame result; string concatenation is associative (but not commutative).

In arithmetic, addition and multiplication of real numbers are associative; i.e.,

16 CHAPTER 6. ASSOCIATIVE PROPERTY

(((ab)c)d)e

((ab)c)(de)

((ab)(cd))e

((a(bc))d)e

(ab)(c(de))

(a(bc))(de)

(ab)((cd)e)

(a(b(cd)))e

a(b(c(de)))

a((bc)(de))

a(b((cd)e))

a(((bc)d)e)

a((b(cd))e)

(a((bc)d))e

In the absence of the associative property, ve factors a, b, c, d, e result in a Tamari lattice of order four, possibly dierent products.

(x+ y) + z = x+ (y + z) = x+ y + z(x y)z = x(y z) = x y z

for all x; y; z 2 R:

Because of associativity, the grouping parentheses can be omitted without ambiguity.

6.3. EXAMPLES 17

(x + z+ y)

x + z)+ (y=

The addition of real numbers is associative.

Addition and multiplication of complex numbers and quaternions are associative. Addition of octonions is alsoassociative, but multiplication of octonions is non-associative.

The greatest common divisor and least common multiple functions act associatively.

gcd(gcd(x; y); z) = gcd(x; gcd(y; z)) = gcd(x; y; z)lcm(lcm(x; y); z) = lcm(x; lcm(y; z)) = lcm(x; y; z)

for all x; y; z 2 Z:

Taking the intersection or the union of sets:

(A \B) \ C = A \ (B \ C) = A \B \ C(A [B) [ C = A [ (B [ C) = A [B [ C

for all sets A;B;C:

IfM is some set and S denotes the set of all functions fromM toM, then the operation of functional compositionon S is associative:

(f g) h = f (g h) = f g h for all f; g; h 2 S:

Slightly more generally, given four sets M, N, P and Q, with h: M to N, g: N to P, and f: P to Q, then

(f g) h = f (g h) = f g h

as before. In short, composition of maps is always associative.

Consider a set with three elements, A, B, and C. The following operation:

is associative. Thus, for example, A(BC)=(AB)C = A. This operation is not commutative.

Because matrices represent linear transformation functions, with matrix multiplication representing functionalcomposition, one can immediately conclude that matrix multiplication is associative.

18 CHAPTER 6. ASSOCIATIVE PROPERTY

6.4 Propositional logic

6.4.1 Rule of replacementIn standard truth-functional propositional logic, association,[3][4] or associativity[5] are two valid rules of replacement.The rules allow one to move parentheses in logical expressions in logical proofs. The rules are:

(P _ (Q _R)), ((P _Q) _R)

and

(P ^ (Q ^R)), ((P ^Q) ^R);

where ", " is a metalogical symbol representing can be replaced in a proof with.

6.4.2 Truth functional connectivesAssociativity is a property of some logical connectives of truth-functional propositional logic. The following logicalequivalences demonstrate that associativity is a property of particular connectives. The following are truth-functionaltautologies.Associativity of disjunction:

(P _ (Q _R))$ ((P _Q) _R)

((P _Q) _R)$ (P _ (Q _R))Associativity of conjunction:

((P ^Q) ^R)$ (P ^ (Q ^R))

(P ^ (Q ^R))$ ((P ^Q) ^R)Associativity of equivalence:

((P $ Q)$ R)$ (P $ (Q$ R))

(P $ (Q$ R))$ ((P $ Q)$ R)

6.5 Non-associativityA binary operation on a set S that does not satisfy the associative law is called non-associative. Symbolically,

(x y) z 6= x (y z) for some x; y; z 2 S:

For such an operation the order of evaluation does matter. For example:

Subtraction

(5 3) 2 6= 5 (3 2)

6.5. NON-ASSOCIATIVITY 19

Division

(4/2)/2 6= 4/(2/2)

Exponentiation

2(12) 6= (21)2

Also note that innite sums are not generally associative, for example:

(1 1) + (1 1) + (1 1) + (1 1) + (1 1) + (1 1) + : : : = 0

whereas

1 + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) + (1 + : : : = 1

The study of non-associative structures arises from reasons somewhat dierent from the mainstream of classicalalgebra. One area within non-associative algebra that has grown very large is that of Lie algebras. There the associativelaw is replaced by the Jacobi identity. Lie algebras abstract the essential nature of innitesimal transformations, andhave become ubiquitous in mathematics.There are other specic types of non-associative structures that have been studied in depth; these tend to come fromsome specic applications or areas such as combinatorial mathematics. Other examples are Quasigroup, Quasield,Non-associative ring, Non-associative algebra and Commutative non-associative magmas.

6.5.1 Nonassociativity of oating point calculation

In mathematics, addition and multiplication of real numbers is associative. By contrast, in computer science, theaddition and multiplication of oating point numbers is not associative, as rounding errors are introduced whendissimilar-sized values are joined together.[6]

To illustrate this, consider a oating point representation with a 4-bit mantissa:(1.000220 + 1.000220) + 1.000224 = 1.000221 + 1.000224 = 1.0012241.000220 + (1.000220 + 1.000224) = 1.000220 + 1.000224 = 1.000224

Even though most computers compute with a 24 or 53 bits of mantissa,[7] this is an important source of roundingerror, and approaches such as the Kahan Summation Algorithm are ways to minimise the errors. It can be especiallyproblematic in parallel computing.[8] [9]

6.5.2 Notation for non-associative operations

Main article: Operator associativity

In general, parentheses must be used to indicate the order of evaluation if a non-associative operation appears morethan once in an expression. However, mathematicians agree on a particular order of evaluation for several commonnon-associative operations. This is simply a notational convention to avoid parentheses.A left-associative operation is a non-associative operation that is conventionally evaluated from left to right, i.e.,

x y z = (x y) zw x y z = ((w x) y) zetc.

9=; for all w; x; y; z 2 Swhile a right-associative operation is conventionally evaluated from right to left:

20 CHAPTER 6. ASSOCIATIVE PROPERTY

x y z = x (y z)w x y z = w (x (y z))etc.

9=; for all w; x; y; z 2 SBoth left-associative and right-associative operations occur. Left-associative operations include the following:

Subtraction and division of real numbers:

x y z = (x y) z for all x; y; z 2 R;x/y/z = (x/y)/z for all x; y; z 2 R with y 6= 0; z 6= 0:

Function application:

(f x y) = ((f x) y)

This notation can be motivated by the currying isomorphism.

Right-associative operations include the following:

Exponentiation of real numbers:

xyz

= x(yz):

The reason exponentiation is right-associative is that a repeated left-associative exponentiation operationwould be less useful. Multiple appearances could (and would) be rewritten with multiplication:

(xy)z = x(yz):

Function denition

Z! Z! Z = Z! (Z! Z)x 7! y 7! x y = x 7! (y 7! x y)

Using right-associative notation for these operations can be motivated by the Curry-Howard correspon-dence and by the currying isomorphism.

Non-associative operations for which no conventional evaluation order is dened include the following.

Taking the Cross product of three vectors:

~a (~b ~c) 6= (~a~b) ~c for some ~a;~b;~c 2 R3

Taking the pairwise average of real numbers:

(x+ y)/2 + z

26= x+ (y + z)/2

2for all x; y; z 2 R with x 6= z:

Taking the relative complement of sets (AnB)nC is not the same as An(BnC) . (Compare material nonim-plication in logic.)

6.6. SEE ALSO 21

6.6 See also Lights associativity test A semigroup is a set with a closed associative binary operation. Commutativity and distributivity are two other frequently discussed properties of binary operations. Power associativity, alternativity and N-ary associativity are weak forms of associativity.

6.7 References[1] Thomas W. Hungerford (1974). Algebra (1st ed.). Springer. p. 24. ISBN 0387905189. Denition 1.1 (i) a(bc) = (ab)c

for all a, b, c in G.

[2] Durbin, John R. (1992). Modern Algebra: an Introduction (3rd ed.). New York: Wiley. p. 78. ISBN 0-471-51001-7. Ifa1; a2; : : : ; an (n 2) are elements of a set with an associative operation, then the product a1a2 : : : an is unambiguous;this is, the same element will be obtained regardless of how parentheses are inserted in the product

[3] Moore and Parker

[4] Copi and Cohen

[5] Hurley

[6] Knuth, Donald, The Art of Computer Programming, Volume 3, section 4.2.2

[7] IEEEComputer Society (August 29, 2008). IEEE Standard for Floating-Point Arithmetic. IEEE. doi:10.1109/IEEESTD.2008.4610935.ISBN 978-0-7381-5753-5. IEEE Std 754-2008.

[8] Villa, Oreste; Chavarra-mir, Daniel; Gurumoorthi, Vidhya; Mrquez, Andrs; Krishnamoorthy, Sriram, Eects of Floating-Point non-Associativity on Numerical Computations on Massively Multithreaded Systems (PDF), retrieved 2014-04-08

[9] Goldberg, David, What Every Computer Scientist ShouldKnowAbout Floating Point Arithmetic (PDF),ACMComputingSurveys 23 (1): 548, doi:10.1145/103162.103163, retrieved 2014-04-08

Chapter 7

Brahmaguptas identity

In algebra, Brahmaguptas identity says that the product of two numbers of the form a2 + nb2 is itself a numberof that form. In other words, the set of such numbers is closed under multiplication. Specically:

a2 + nb2

c2 + nd2

= (ac nbd)2 + n (ad+ bc)2 (1)= (ac+ nbd)

2+ n (ad bc)2 ; (2)

Both (1) and (2) can be veried by expanding each side of the equation. Also, (2) can be obtained from (1), or (1)from (2), by changing b to b.This identity holds in both the ring of integers and the ring of rational numbers, andmore generally in any commutativering.

7.1 HistoryThe identity is a generalization of the so-called Fibonacci identity (where n=1) which is actually found in Diophantus'Arithmetica (III, 19). That identity was rediscovered by Brahmagupta (598668), an Indian mathematician andastronomer, who generalized it and used it in his study ofwhat is now called Pells equation. HisBrahmasphutasiddhantawas translated from Sanskrit into Arabic by Mohammad al-Fazari, and was subsequently translated into Latin in1126.[1] The identity later appeared in Fibonacci's Book of Squares in 1225.

7.2 Application to Pells equationIn its original context, Brahmagupta applied his discovery to the solution of what was later called Pells equation,namely x2 Ny2 = 1. Using the identity in the form

(x21 Ny21)(x22 Ny22) = (x1x2 +Ny1y2)2 N(x1y2 + x2y1)2;

he was able to compose triples (x1, y1, k1) and (x2, y2, k2) that were solutions of x2 Ny2 = k, to generate the newtriple

(x1x2 +Ny1y2 ; x1y2 + x2y1 ; k1k2):

Not only did this give a way to generate innitely many solutions to x2 Ny2 = 1 starting with one solution, but also,by dividing such a composition by k1k2, integer or nearly integer solutions could often be obtained. The generalmethod for solving the Pell equation given by Bhaskara II in 1150, namely the chakravala (cyclic) method, was alsobased on this identity.[2]

22

7.3. SEE ALSO 23

7.3 See also Brahmagupta matrix BrahmaguptaFibonacci identity Indian mathematics List of Indian mathematicians

7.4 References[1] George G. Joseph (2000). The Crest of the Peacock, p. 306. Princeton University Press. ISBN 0-691-00659-8.

[2] John Stillwell (2002), Mathematics and its history (2 ed.), Springer, pp. 7276, ISBN 978-0-387-95336-6

7.5 External links Brahmaguptas identity at PlanetMath Brahmagupta Identity on MathWorld A Collection of Algebraic Identities

Chapter 8

BrahmaguptaFibonacci identity

In algebra, the BrahmaguptaFibonacci identity or simply Fibonaccis identity (and in fact due to Diophantus ofAlexandria) says that the product of two sums each of two squares is itself a sum of two squares. In other words, theset of all sums of two squares is closed under multiplication. Specically:

a2 + b2

c2 + d2

= (ac bd)2 + (ad+ bc)2 (1)= (ac+ bd)

2+ (ad bc)2 : (2)

For example,

(12 + 42)(22 + 72) = 262 + 152 = 302 + 12:

The identity is a special case (n = 2) of Lagranges identity, and is rst found in Diophantus. Brahmagupta provedand used a more general identity (the Brahmagupta identity), equivalent to

a2 + nb2

c2 + nd2

= (ac nbd)2 + n (ad+ bc)2 (3)= (ac+ nbd)

2+ n (ad bc)2 ; (4)

showing that the set of all numbers of the form x2 + y2 is closed under multiplication.Both (1) and (2) can be veried by expanding each side of the equation. Also, (2) can be obtained from (1), or (1)from (2), by changing b to b.This identity holds in both the ring of integers and the ring of rational numbers, andmore generally in any commutativering.In the integer case this identity nds applications in number theory for example when used in conjunction with oneof Fermats theorems it proves that the product of a square and any number of primes of the form 4n + 1 is also asum of two squares.

8.1 History

The identity is actually rst found in Diophantus'Arithmetica (III, 19), of the third century A.D. It was rediscovered byBrahmagupta (598668), an Indian mathematician and astronomer, who generalized it (to the Brahmagupta identity)and used it in his study of what is now called Pells equation. His Brahmasphutasiddhanta was translated fromSanskrit into Arabic by Mohammad al-Fazari, and was subsequently translated into Latin in 1126.[1] The identitylater appeared in Fibonacci's Book of Squares in 1225.

24

8.2. RELATED IDENTITIES 25

8.2 Related identitiesAnalogous identities are Eulers four-square related to quaternions, andDegens eight-square derived from the octonionswhich has connections to Bott periodicity. There is also Psters sixteen-square identity, though it is no longer bilinear.

8.3 Relation to complex numbersIf a, b, c, and d are real numbers, this identity is equivalent to the multiplication property for absolute values ofcomplex numbers namely that:

ja+ bijjc+ dij = j(a+ bi)(c+ di)jsince

ja+ bijjc+ dij = j(ac bd) + i(ad+ bc)j;by squaring both sides

ja+ bij2jc+ dij2 = j(ac bd) + i(ad+ bc)j2;and by the denition of absolute value,

(a2 + b2)(c2 + d2) = (ac bd)2 + (ad+ bc)2:

8.4 Interpretation via normsIn the case that the variables a, b, c, and d are rational numbers, the identity may be interpreted as the statement thatthe norm in the eld Q(i) is multiplicative. That is, we have

N(a+ bi) = a2 + b2 and N(c+ di) = c2 + d2;

and also

N((a+ bi)(c+ di)) = N((ac bd) + i(ad+ bc)) = (ac bd)2 + (ad+ bc)2:Therefore the identity is saying that

N((a+ bi)(c+ di)) = N(a+ bi) N(c+ di):

8.5 Application to Pells equationIn its original context, Brahmagupta applied his discovery (the Brahmagupta identity) to the solution of Pells equation,namely x2 Ny2 = 1. Using the identity in the more general form

(x21 Ny21)(x22 Ny22) = (x1x2 +Ny1y2)2 N(x1y2 + x2y1)2;

26 CHAPTER 8. BRAHMAGUPTAFIBONACCI IDENTITY

he was able to compose triples (x1, y1, k1) and (x2, y2, k2) that were solutions of x2 Ny2 = k, to generate the newtriple

(x1x2 +Ny1y2 ; x1y2 + x2y1 ; k1k2):

Not only did this give a way to generate innitely many solutions to x2 Ny2 = 1 starting with one solution, but also,by dividing such a composition by k1k2, integer or nearly integer solutions could often be obtained. The generalmethod for solving the Pell equation given by Bhaskara II in 1150, namely the chakravala (cyclic) method, was alsobased on this identity.[2]

8.6 See also Brahmagupta matrix Indian mathematics List of Indian mathematicians Eulers four-square identity

8.7 References[1] George G. Joseph (2000). The Crest of the Peacock, p. 306. Princeton University Press. ISBN 0-691-00659-8.

[2] John Stillwell (2002), Mathematics and its history (2 ed.), Springer, pp. 7276, ISBN 978-0-387-95336-6

8.8 External links Brahmaguptas identity at PlanetMath Brahmagupta Identity on MathWorld A Collection of Algebraic Identities

Chapter 9

Carlyle circle

In mathematics, a Carlyle circle is a certain circle in a coordinate plane associated with a quadratic equation. Thecircle has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersectionsof the circle with the horizontal axis.[1] The idea of using such a circle to solve a quadratic equation is attributedto Thomas Carlyle (17951881).[2] Carlyle circles have been used to develop ruler-and-compass constructions ofregular polygons.

9.1 Denition

Carlyle circle of the quadratic equation x2 sx + p = 0.

Given the quadratic equation

x2 sx + p = 0

27

28 CHAPTER 9. CARLYLE CIRCLE

the circle in the coordinate plane having the line segment joining the points A(0, 1) and B(s, p) as a diameter is calledthe Carlyle circle of the quadratic equation.

9.2 Dening propertyThe dening property of the Carlyle circle can be established thus: the equation of the circle having the line segmentAB as diameter is

x(x s) + (y 1)(y p) = 0.

The abscissas of the points where the circle intersects the x-axis are the roots of the equation (obtained by setting y= 0 in the equation of the circle)

x2 sx + p = 0.

9.3 Construction of regular polygons

9.3.1 Regular pentagonThe problem of constructing a regular pentagon is equivalent to the problem of constructing the roots of the equation

z5 1 = 0.

One root of this equation is z0 = 1 which corresponds to the point P0(1, 0). Removing the factor corresponding tothis root, the other roots turn out to be roots of the equation

z4 + z3 + z2 + z + 1 = 0.

These roots can be represented in the form , 2, 3, 4 where = exp(2i/5). Let these correspond to the pointsP1, P2, P3, P4. Letting

p1 = + 4, p2 = 2 + 3

we have

p1 + p2 = 1, p1p2 = 1. (These can be quickly shown to be true by direct substitution into the quarticabove and noting that 6 = , and 7 = 2.)

So p1 and p2 are the roots of the quadratic equation

x2 + x 1 = 0.

The Carlyle circle associated with this quadratic has a diameter with endpoints at (0, 1) and (1, 1) and center at(1/2, 0). Carlyle circles are used to construct p1 and p2. From the denitions of p1 and p2 it also follows that

p1 = 2 cos (2/5), p2 = 2 cos (4/5).

These are then used to construct the points P1, P2, P3, P4.This detailed procedure involving Carlyle circles for the construction of regular pentagons is given below.[2]

1. Draw a circle in which to inscribe the pentagon and mark the center point O.

9.3. CONSTRUCTION OF REGULAR POLYGONS 29

Construction of regular pentagon using Carlyle circles

2. Draw a horizontal line through the center of the circle. Mark one intersection with the circle as point B.

3. Construct a vertical line through the center. Mark one intersection with the circle as point A.

4. Construct the point M as the midpoint of O and B.

5. Draw a circle centered at M through the point A. Mark its intersection with the horizontal line (inside theoriginal circle) as the pointW and its intersection outside the circle as the point V.

6. Draw a circle of radius OA and centerW. It intersects the original circle at two of the vertices of the pentagon.

7. Draw a circle of radius OA and center V. It intersects the original circle at two of the vertices of the pentagon.

8. The fth vertex is the intersection of the horizontal axis with the original circle.

9.3.2 Regular heptadecagon

There is a similar method involving Carlyle circles to construct regular heptadecagons.[2] The attached gure illustratesthe procedure.

30 CHAPTER 9. CARLYLE CIRCLE

Construction of a regular heptadecagon using Carlyle circles

9.3.3 Regular 257-gonTo construct a regular 257-gon using Carlyle circles, as many as 24 Carlyle circles are to be constructed. One of theseis the circle to solve the quadratic equation x2 + x 64 = 0.[2]

9.3.4 Regular 65537-gonThere is a procedure involving Carlyle circles for the construction of a regular 65537-gon. However there are practicalproblems for the implementation of the procedure, as, for example, it requires the construction of the Carlyle circlefor the solution of the quadratic equation x2 + x 214 = 0.[2]

9.4 References[1] Weisstein, Eric W. Carlyle Circle. From MathWorldA Wolfram Web Resource. Retrieved 21 May 2013.

[2] DeTemple, DuaneW. (Feb 1991). Carlyle circles and Lemoine simplicity of polygon constructions (PDF). The AmericanMathematical Monthly 98 (2): 97208. doi:10.2307/2323939. Retrieved 6 November 2011.

9.4. REFERENCES 31

Construction of a regular 257-gon using Carlyle circles

Chapter 10

Change of variables

Substitution (algebra)" redirects here. It is not to be confused with substitution (logic).

In mathematics, the operation of substitution consists in replacing all the occurrences of a free variable appearing inan expression or a formula by a number or another expression. In other words, an expression involving free variablesmay be considered as dening a function, and substituting values to the variables in the expression is equivalent toapplying the function dened by the expression to these values.A change of variables is commonly a particular type of substitution, where the substituted values are expressionsthat depend on other variables. This is a standard technique used to reduce a dicult problem to a simpler one. Achange of coordinates is a common type of change of variables. However, if the expression in which the variablesare changed involves derivatives or integrals, the change of variable does not reduce to a substitution.A very simple example of a useful variable change can be seen in the problem of nding the roots of the sixth orderpolynomial:

x6 9x3 + 8 = 0:Sixth order polynomial equations are generally impossible to solve in terms of radicals (see AbelRuni theorem).This particular equation, however, may be written

(x3)2 9(x3) + 8 = 0(this is a simple case of a polynomial decomposition). Thus the equation may be simplied by dening a new variablex3 = u. Substituting x by 3pu into the polynomial gives

u2 9u+ 8 = 0;which is just a quadratic equation with solutions:

u = 1 and u = 8:

The solutions in terms of the original variable are obtained by substituting x3 back in for u:

x3 = 1 and x3 = 8:

Then, assuming that x is real,

x = (1)1/3 = 1 and x = (8)1/3 = 2:

32

10.1. SIMPLE EXAMPLE 33

10.1 Simple exampleConsider the system of equations

xy + x+ y = 71

x2y + xy2 = 880

where x and y are positive integers with x > y . (Source: 1991 AIME)Solving this normally is not terrible, but it may get a little tedious. However, we can rewrite the second equationas xy(x + y) = 880 . Making the substitution s = x + y; t = xy reduces the system to s + t = 71; st = 880:Solving this gives (s; t) = (16; 55) or (s; t) = (55; 16): Back-substituting the rst ordered pair gives us x + y =16; xy = 55 , which easily gives the solution (x; y) = (11; 5): Back-substituting the second ordered pair gives usx+ y = 55; xy = 16 , which gives no solutions. Hence the solution that solves the system is (x; y) = (11; 5) .

10.2 Formal introductionLet A , B be smooth manifolds and let : A ! B be a Cr -dieomorphism between them, that is: is a r timescontinuously dierentiable, bijective map from A to B with r times continuously dierentiable inverse from B to A. Here r may be any natural number (or zero),1 (smooth) or ! (analytic).The map is called a regular coordinate transformation or regular variable substitution, where regular refers to theCr -ness of . Usually one will write x = (y) to indicate the replacement of the variable x by the variable y bysubstituting the value of in y for every occurrence of x .

10.3 Other examples

10.3.1 Coordinate transformationSome systems can be more easily solved when switching to cylindrical coordinates. Consider for example the equation

U(x; y; z) := (x2 + y2)

s1 x

2

x2 + y2= 0:

This may be a potential energy function for some physical problem. If one does not immediately see a solution, onemight try the substitution

(x; y; z) = (r; ; z) given by (r; ; z) = (r cos(); r sin(); z) .

Note that if runs outside a 2 -length interval, for example, [0; 2] , the map is no longer bijective. Therefore should be limited to, for example (0;1] [0; 2) [1;1] . Notice how r = 0 is excluded, for is not bijectivein the origin ( can take any value, the point will be mapped to (0, 0, z)). Then, replacing all occurrences of theoriginal variables by the new expressions prescribed by and using the identity sin2 x+ cos2 x = 1 , we get

V (r; ; z) = r2r1 r

2 cos2 r2

= r2p1 cos2 = r2 jsin j

Now the solutions can be readily found: sin() = 0 , so = 0 or = . Applying the inverse of shows that thisis equivalent to y = 0 while x 6= 0 . Indeed we see that for y = 0 the function vanishes, except for the origin.Note that, had we allowed r = 0 , the origin would also have been a solution, though it is not a solution to the originalproblem. Here the bijectivity of is crucial. Note also that the function is always positive (for x; y; z 2 R ), hencethe absolute values.

34 CHAPTER 10. CHANGE OF VARIABLES

10.3.2 DierentiationMain article: Chain rule

The chain rule is used to simplify complicated dierentiation. For example, to calculate the derivative

d

dx

sin(x2)

the variable x may be changed by introducing x2 = u. Then, by the chain rule:

d

dx=

d

du

du

dx=

d

dx(u)

d

du=

d

dx

x2 ddu

= 2xd

du

so that

d

dx

sin(x2)

= 2x

d

du(sin(u)) = 2x cos(x2)

where in the very last step u has been replaced with x2.

10.3.3 IntegrationMain article: Integration by substitution

Dicult integrals may often be evaluated by changing variables; this is enabled by the substitution rule and is analogousto the use of the chain rule above. Dicult integrals may also be solved by simplifying the integral using a changeof variables given by the corresponding Jacobian matrix and determinant. Using the Jacobian determinant and thecorresponding change of variable that it gives is the basis of coordinate systems such as polar, cylindrical, and sphericalcoordinate systems.

10.3.4 Dierential equationsVariable changes for dierentiation and integration are taught in elementary calculus and the steps are rarely carriedout in full.The very broad use of variable changes is apparent when considering dierential equations, where the independentvariables may be changed using the chain rule or the dependent variables are changed resulting in some dierentiationto be carried out. Exotic changes, such as the mingling of dependent and independent variables in point and contacttransformations, can be very complicated but allow much freedom.Very often, a general form for a change is substituted into a problem and parameters picked along the way to bestsimplify the problem.

10.3.5 Scaling and shiftingProbably the simplest change is the scaling and shifting of variables, that is replacing them with new variables thatare stretched and moved by constant amounts. This is very common in practical applications to get physicalparameters out of problems. For an nth order derivative, the change simply results in

dny

dxn=

yscalexnscale

dny^

dx^n

where

10.3. OTHER EXAMPLES 35

x = x^xscale + xshift

y = y^yscale + yshift:

This may be shown readily through the chain rule and linearity of dierentiation. This change is very common inpractical applications to get physical parameters out of problems, for example, the boundary value problem

d2u

dy2=

dp

dx; u(0) = u(L) = 0

describes parallel uid ow between at solid walls separated by a distance ; is the viscosity and dp/dx the pressuregradient, both constants. By scaling the variables the problem becomes

d2u^

dy^2= 1 ; u^(0) = u^(1) = 0

where

y = y^L and u = u^L2

dp

dx:

Scaling is useful for many reasons. It simplies analysis both by reducing the number of parameters and by simplymaking the problem neater. Proper scaling may normalize variables, that is make them have a sensible unitless rangesuch as 0 to 1. Finally, if a problem mandates numeric solution, the fewer the parameters the fewer the number ofcomputations.

10.3.6 Momentum vs. velocityConsider a system of equations

m _v = @H@x

m _x =@H

@v

for a given functionH(x; v) . The mass can be eliminated by the (trivial) substitution (p) = 1/m v . Clearly thisis a bijective map from R to R . Under the substitution v = (p) the system becomes

_p = @H@x

_x =@H

@p

10.3.7 Lagrangian mechanicsMain article: Lagrangian mechanics

Given a force eld (t; x; v) , Newton's equations of motion are

mx = (t; x; v)

36 CHAPTER 10. CHANGE OF VARIABLES

Lagrange examined how these equations of motion change under an arbitrary substitution of variables x = (t; y) ,v = @(t;y)@t +

@(t;y)@y w .

He found that the equations

@L

@y=

ddt@L

@w

are equivalent to Newtons equations for the function L = T V , where T is the kinetic, and V the potential energy.In fact, when the substitution is chosen well (exploiting for example symmetries and constraints of the system) theseequations are much easier to solve than Newtons equations in Cartesian coordinates.

10.4 See also Change of variables (PDE) Substitution property of equality Instantiation of universals

Chapter 11

Commutative property

For other uses, see Commute (disambiguation).In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

=

==

This image illustrates that addition is commutative.

It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar asthe name of the property that says 3 + 4 = 4 + 3 or 2 5 = 5 2, the property can also be used in more advancedsettings. The name is needed because there are operations, such as division and subtraction that do not have it (forexample, 3 5 5 3), such operations are not commutative, or noncommutative operations. The idea that simpleoperations, such as multiplication and addition of numbers, are commutative was for many years implicitly assumedand the property was not named until the 19th century when mathematics started to become formalized.

11.1 Common uses

The commutative property (or commutative law) is a property generally associatedwith binary operations and functions.If the commutative property holds for a pair of elements under a certain binary operation then the two elements aresaid to commute under that operation.

37

38 CHAPTER 11. COMMUTATIVE PROPERTY

11.2 Mathematical denitionsFurther information: Symmetric function

The term commutative is used in several related senses.[1][2]

1. A binary operation on a set S is called commutative if:x y = y x for all x; y 2 SAn operation that does not satisfy the above property is called non-commutative.

2. One says that x commutes with y under if:x y = y x

3. A binary function f : AA! B is called commutative if:f(x; y) = f(y; x) for all x; y 2 A

11.3 Examples

11.3.1 Commutative operations in everyday life Putting on socks resembles a commutative operation, since which sock is put on rst is unimportant. Eitherway, the result (having both socks on), is the same.

The commutativity of addition is observed when paying for an item with cash. Regardless of the order the billsare handed over in, they always give the same total.

11.3.2 Commutative operations in mathematicsTwo well-known examples of commutative binary operations:[1]

The addition of real numbers is commutative, since

y + z = z + y for all y; z 2 RFor example 4 + 5 = 5 + 4, since both expressions equal 9.

The multiplication of real numbers is commutative, since

yz = zy for all y; z 2 RFor example, 3 5 = 5 3, since both expressions equal 15.

Some binary truth functions are also commutative, since the truth tables for the functions are the same whenone changes the order of the operands.

For example, the logical biconditional function p q is equivalent to q p. This function is also writtenas p IFF q, or as p q, or as Epq.The last form is an example of the most concise notation in the article on truth functions, which lists thesixteen possible binary truth functions of which eight are commutative: Vpq = Vqp; Apq (OR) = Aqp;Dpq (NAND) = Dqp; Epq (IFF) = Eqp; Jpq = Jqp; Kpq (AND) = Kqp; Xpq (NOR) = Xqp; Opq = Oqp.

Further examples of commutative binary operations include addition and multiplication of complex numbers,addition and scalar multiplication of vectors, and intersection and union of sets.

11.3. EXAMPLES 39

b

b

aaa+

b

The addition of vectors is commutative, because ~a+~b = ~b+ ~a .

11.3.3 Noncommutative operations in everyday life

Concatenation, the act of joining character strings together, is a noncommutative operation. For example

EA+ T = EAT 6= TEA = T + EA

Washing and drying clothes resembles a noncommutative operation; washing and then drying produces amarkedly dierent result to drying and then washing.

Rotating a book 90 around a vertical axis then 90 around a horizontal axis produces a dierent orientationthan when the rotations are performed in the opposite order.

The twists of the Rubiks Cube are noncommutative. This can be studied using group theory.

Also thought processes are noncommutative: A person asked a question (A) and then a question (B) may givedierent answers to each question than a person asked rst (B) and then (A), because asking a question maychange the persons state of mind.

40 CHAPTER 11. COMMUTATIVE PROPERTY

11.3.4 Noncommutative operations in mathematics

Some noncommutative binary operations:[3]

Subtraction is noncommutative, since 0 1 6= 1 0 Division is noncommutative, since 1/2 6= 2/1 Some truth functions are noncommutative, since the truth tables for the functions are dierent when one changesthe order of the operands.

For example, the truth tables for f (A,B) = A B (A AND NOT B) and f (B,A) = B A are

Matrix multiplication is noncommutative since0 20 1

=

1 10 1

0 10 1

6=0 10 1

1 10 1

=

0 10 1

The vector product (or cross product) of two vectors in three dimensions is anti-commutative, i.e., b a = (a b).

11.4 History and etymology

The rst known use of the term was in a French Journal published in 1814

Records of the implicit use of the commutative property go back to ancient times. The Egyptians used the commuta-tive property of multiplication to simplify computing products.[4][5] Euclid is known to have assumed the commutativeproperty of multiplication in his book Elements.[6] Formal uses of the commutative property arose in the late 18th andearly 19th centuries, when mathematicians began to work on a theory of functions. Today the commutative propertyis a well known and basic property used in most branches of mathematics.The rst recorded use of the term commutative was in a memoir by Franois Servois in 1814,[7][8] which used theword commutatives when describing functions that have what is now called the commutative property. The word is acombination of the French word commuter meaning to substitute or switch and the sux -ative meaning tendingto so the word literally means tending to substitute or switch. The term then appeared in English in PhilosophicalTransactions of the Royal Society in 1844.[7]

11.5. PROPOSITIONAL LOGIC 41

11.5 Propositional logic

11.5.1 Rule of replacement

In truth-functional propositional logic, commutation,[9][10] or commutativity[11] refer to two valid rules of replacement.The rules allow one to transpose propositional variables within logical expressions in logical proofs. The rules are:

(P _Q), (Q _ P )

and

(P ^Q), (Q ^ P )

where ", " is a metalogical symbol representing can be replaced in a proof with.

11.5.2 Truth functional connectives

Commutativity is a property of some logical connectives of truth functional propositional logic. The following logicalequivalences demonstrate that commutativity is a property of particular connectives. The following are truth-functionaltautologies.

Commutativity of conjunction (P ^Q)$ (Q ^ P )

Commutativity of disjunction (P _Q)$ (Q _ P )

Commutativity of implication (also called the law of permutation(P ! (Q! R))$ (Q! (P ! R))

Commutativity of equivalence (also called the complete commutative law of equivalence)(P $ Q)$ (Q$ P )

11.6 Set theoryIn group and set theory, many algebraic structures are called commutative when certain operands satisfy the com-mutative property. In higher branches of mathematics, such as analysis and linear algebra the commutativity ofwell-known operations (such as addition and multiplication on real and complex numbers) is often used (or implicitlyassumed) in proofs.[12][13][14]

11.7 Mathematical structures and commutativity

A commutative semigroup is a set endowed with a total, associative and commutative operation.

If the operation additionally has an identity element, we have a commutative monoid

An abelian group, or commutative group is a group whose group operation is commutative.[13]

Acommutative ring is a ringwhosemultiplication is commutative. (Addition in a ring is always commutative.)[15]

In a eld both addition and multiplication are commutative.[16]

42 CHAPTER 11. COMMUTATIVE PROPERTY

11.8 Related properties

11.8.1 Associativity

Main article: Associative property

The associative property is closely related to the commutative property. The associative property of an expressioncontaining two or more occurrences of the same operator states that the order operations are performed in does notaect the nal result, as long as the order of terms doesn't change. In contrast, the commutative property states thatthe order of the terms does not aect the nal result.Most commutative operations encountered in practice are also associative. However, commutativity does not implyassociativity. A counterexample is the function

f(x; y) =x+ y

2;

which is clearly commutative (interchanging x and y does not aect the result), but it is not associative (since, forexample, f(4; f(0;+4)) = 1 but f(f(4; 0);+4) = +1 ). More such examples may be found in Commutativenon-associative magmas.

11.8.2 Symmetry

-10-8-6-4-2 0 2 4 6 8 10

Graph showing the symmetry of the addition function

Main article: Symmetry in mathematics

11.9. NON-COMMUTING OPERATORS IN QUANTUM MECHANICS 43

Some forms of symmetry can be directly linked to commutativity. When a commutative operator is written as abinary function then the resulting function is symmetric across the line y = x. As an example, if we let a function frepresent addition (a commutative operation) so that f(x,y) = x + y then f is a symmetric function, which can be seenin the image on the right.For relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, thenaRb, bRa .

11.9 Non-commuting operators in quantum mechanicsMain article: Canonical commutation relation

In quantum mechanics as formulated by Schrdinger, physical variables are represented by linear operators such as x(meaning multiply by x), and ddx . These two operators do not commute as may be seen by considering the eect oftheir compositions x ddx and ddxx (also called products of operators) on a one-dimensional wave function (x) :

xd

dx = x 0 6= d

dxx