Algebra to Calculus

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1  Polynomials
1.1  The Fundamental Theorem of Algebra	5
1.2  Polynomials For Sums of Square Roots	10
1.3  Gauss' Lemma Without Divisibility Arguments	16
1.4  Polynomials From Pascal's Triangle	18
1.5  Fundamental Theorem For Palindromic Polynomials	22
1.6  Reducing Quartics to Cubics	24
1.7  On the Solution of the Cubic	26
1.8  Intersections of Polynomials	29
1.9  Cramer's Paradox	32
1.10  A Unique Intersection Pattern for Plane Curves	35
1.11  The Resultant and Bezout's Theorem	37
1.12  Galois Groups	45
1.13  Determining the Galois Group of a Polynomial	53
2  Trigonometry
2.1  From Broken Chords To Trigonometric Identities	56
2.2  Tangents, Exponentials, and π	61
2.3  Machin’s Merit	65
2.4  Tangent To π	69
2.5  Radical Expression For cos(2π/7)	76
2.6  Quintisection of an Angle	77
2.7  Infinite Products and a Tangent Fan	79
2.8  The Twelve-Step Cycle of 4/sin(x)	87
3  Linear Fractional Transformations
3.1  Introduction	93
3.2  Closed Form Expression for nth Iteration	95
3.3  General Periodicity Condition	99
3.4  The σ Form	101
3.5  Self-Similar Iterations	102
3.6  The λ Form	102
3.7  Sum and Difference Form	105
3.8  The Complement Form	106
3.9  Diagonal Form	106
3.10  Density of Real Iterations	108
4  Geometry
4.1  Constructing the Heptadecagon	115
4.2  Highly Heronian Ellipses	116
4.3  Iterative Isoscelizing	123
4.4  Forests or Trees On A Complex Plain	132
4.5  The Mystery of the Grazing Goat	134
4.6  Quarky Volume Formula for Parallelepiped	141
4.7  Factoring Convex Figures	142
4.8  Routh's Formula by Cross Products	146
4.9  Simplex Volumes and the Cayley-Menger Determinant	155
4.10  Reflecting on the Geometric Mean	163
4.11  Rotation Matrices	165
4.12  Rotations and Anti-Symmetric Tensors	172
5  Interpolation
5.1  Multiple Linear Regression and Fourier Series	179
5.2  Perpendicular Regression Of A Line	182
5.3  Cross-Linear Interpolation	185
5.4  Boolean Expansion as Linear Interpolation	187
5.5  Inverse Square Weighted Interpolation	189
5.6  Curvature of Linear Interpolation	191
6  Arrangements
6.1  The Cayley-Hamilton Theorem	194
6.2  Hamilton Cycles on McCauley Graphs	196
6.3  The Amanda Arrangement	204
6.4  Arranging the Solutions of f(x+y+z) = xyz	217
6.5  Convoluting Arrays	224
6.6  Asymptotic Approach to 2D Arrays	225
7  Curiosities
7.1  Napoleonic Vectors	227
7.2  Platonions	232
7.3  The Super-Symmetric Mean	237
7.4  Iterated Means	241
7.5  Quasi-Groups	243
7.6  Expulsion Sets	247
7.7  An Algebraic Duality	255
7.8  Generalized Mediant	258
8  Miscellaneous
8.1  Proving Algebraic Inequalities	263
8.2  Inverse Functions	268
8.3  Cyclic Redundancy Checks	272
8.4  Annuities With Inflation	279
8.5  On the Equation f(x^2 + y^2) = f(x)^2 + f(y)^2	281
8.6  Square Roots by Pencil and Paper	284
8.7  Loxodromic Aliasing	286
8.8  Iterated Logarithmic Functions	291
8.9  Recurrences For Harmonic Sums	292
8.10  Simple Complex Quadratic Fields	295
8.11  Eigenvalue Problems and Matrix Invariants	296
8.12  Do We Really Need Eigen Values?	300
8.13  Interleaving Fibonacci Numbers	302
9  History
9.1  Negative Numbers	306
9.2  Omar Khayyam On Cubics	307
9.3  Ancient Square Roots	308
9.4  Pythagoras On Dot and Cross Products	314
9.5  Why Calculus?	319
9.6  How Leibniz Might Have Anticipated Euler	322
10  Powers and Sums
10.1  Sums of Powers	327
10.2  Sums and Differences of Discrete Functions	332
10.3  Generating Functions and Recurrence Relations	338
10.4  Discrete Fourier Transforms	345
10.5  Phased Summations	350
11  Series
11.1  From the Geometric Series to Stirling Numbers	355
11.2  Mean Partial Sums of Non-Convergent Series	362
11.3  Meandering Convergence of a Dirichlet Series	364
11.4  Evaluate the Infinite Sum of n2/(1+n3)	365
11.5  Harmonic Sums of Integers With k Binary 1's	367
11.6  Decimal Representations as Power Series	370
11.7  Series Solutions of the Wave Equation	379
11.8  Series Solution of Non-Linear Equation	386
12  Rational and Irrational
12.1  Irrationality of Quadratic Sums	389
12.2  Proof That e is Irrational	391
12.3  Proof That π is Irrational	392
12.4  Rational Sines of Rational Multiples of π	495
12.5  Convergence of Series (How NOT to Prove π Irrational)	406
12.6  Mock-Rational Numbers	415
12.7  A Quasi-Periodic Sequence	421
13  Infinity
13.1  Cantor's Diagonal Proof	432
13.2  Interleaving Ad Infinitum	436
13.3  The Limit Paradox	440
13.4  Ptolemy's Orbit	441
13.5  Formal-Numeric Series	443
13.6  Interfering With π	445
13.7  Constructible Points and Coverable Points	446
13.8  Representing Sets of Pure Order	447
14  Calculus
14.1  Leibniz's Rule	451
14.2  Integrating Factors	452
14.3  Change of Variables in Multiple Integrals	456
14.4  Volume of n-Spheres and the Gamma Function	463
14.5  The Laplace Equation and Harmonic Functions	469
14.6  Differential Operators and the Divergence Theorem	476
14.7  High Order Integration Schemes	482
14.8  Invariance, Contravariance, Covariance	483
14.9  The Euler-Maclaurin Formula	486
14.10  Laplace Transforms	493
14.11  Analytic Continuation	497
14.12  The Zeta Function	511
14.13  The Dirac Delta Function	522
14.14  Fractional Calculus	532