2018-02-13
Math Foundations A (1-79) by Norman J Wildberger
source: 2017年2月1日
Does modern pure mathematics make logical sense? No, unfortunately there are serious problems! Foundational issues have been finessed by modern mathematics, and this series aims to turn things around. And it will have interesting things to say also about mathematics education---especially at the primary and high school level. The plan is to start right from the beginning, and to define all the really important concepts of basic mathematics without any waffling or appeals to authority. Roughly we discuss first arithmetic, then geometry, then algebra, then analysis, then set theory. Aimed for a general audience, interested in mathematics, or willing to learn.
9:55 1: What is a number?
10:07 2: Arithmetic with numbers
9:36 3: Laws of Arithmetic
10:07 4: Subtraction and Division
9:23 5: Arithmetic and maths education
8:16 6: The Hindu-Arabic number system
10:03 7: Arithmetic with Hindu-Arabic numbers
9:56 8: Division
6:29 9: Fractions
9:42 10: Arithmetic with fractions
6:30 11: Laws of arithmetic for fractions
9:22 12: Introducing the integers
9:15 13: Rational numbers
9:43 14: Rational numbers and Ford Circles
10:01 15: Primary school maths education
7:38 16: Why infinite sets don't exist
9:58 17: Extremely big numbers
8:14 18: Geometry
9:05 19: Euclid's Elements
9:39 20: Euclid and proportions
7:33 21: Euclid's Books VI--XIII
8:01 22: Difficulties with Euclid
8:49 23: The basic framework for geometry (I)
9:39 24: The basic framework for geometry (II)
9:41 25: The basic framework for geometry (III)
6:52 26: The basic framework for geometry (IV)
9:34 27: Trigonometry with rational numbers
9:14 28: What exactly is a circle?
8:39 29: Parametrizing circles
9:53 30: What exactly is a vector?
9:23 31: Parallelograms and affine combinations
9:25 32: Geometry in primary school
7:55 33: What exactly is an area?
9:45 34: Areas of polygons
10:09 35: Translations, rotations and reflections (I)
9:51 36: Translations, rotations and reflections (II)
9:58 37: Translations, rotations and reflections (III)
9:20 38: Why angles don't really work (I)
9:52 39: Why angles don't really work (II)
9:50 40: Correctness in geometrical problem solving
8:11 41: Why angles don't really work (III)
9:43 42a: Deflating modern mathematics: the problem with `functions'
6:27 42b: Deflating modern mathematics: the problem with `functions'
9:52 43: Reconsidering `functions' in modern mathematics
9:54 44: Definitions, specification and interpretation
9:59 45: Quadrilaterals, quadrangles and n-gons
9:46 46: Introduction to Algebra
9:16 47: Baby Algebra
8:11 48a: Solving a quadratic equation
7:08 48b: Solving a quadratic equation
10:06 49: How to find a square root
9:32 50: Algebra and number patterns
9:54 51: More patterns with algebra
10:05 52: Leonhard Euler and Pentagonal numbers
8:48 53: Algebraic identities
9:58 54: The Binomial theorem
10:04 55: Binomial coefficients and related functions
10:08 56: The Trinomial theorem
9:47 57: Polynomials and polynumbers
9:52 58: Arithmetic with positive polynumbers
9:20 59: More arithmetic with polynumbers
9:39 60: What exactly is a polynomial?
9:53 61: Factoring polynomials and polynumbers
7:58 62: Arithmetic with integral polynumbers
10:10 63: The Factor theorem and polynumber evaluation
45:17 64: The Division algorithm for polynumbers
49:53 65: Row and column polynumbers
28:20 66: Decimal numbers
44:23 67: Visualizing decimal numbers and their arithmetic
39:52 68: Laurent polynumbers (the New Years Day lecture)
37:15 69: Translating polynumbers and the Derivative
36:00 70: Calculus with integral polynumbers
36:29 71: Tangent lines and conics of polynumbers
37:50 72: Graphing polynomials
39:52 73: Lines and parabolas I
38:19 74: Lines and parabolas II
28:09 75: Cubics and the prettiest theorem in calculus
34:33 76: An introduction to algebraic curves
45:50 77: Object-oriented versus expression-oriented mathematics
35:20 78: Calculus on the unit circles
31:56 79: Calculus on a cubic: the Folium of Descartes
Math Foundations B (80-149) by Norman J Wildberger
source: njwildberger 2017年12月20日
Math Foundations B deals quite a lot with the problems involved with real numbers, measurement and alternative ways of thinking about geometry. We really need to address the logical weakness of our thinking about the continuum. A better more solid mathematics is around the corner!
42:03 80: Inconvenient truths about sqrt(2)
45:38 81: Measurement, approximation and interval arithmetic I
41:49 82: Measurement, approximation and interval arithmetic II
25:50 83: Newton's method for finding zeroes
29:31 84: Newton's method for approximating cube roots
36:18 85: Solving quadratics and cubics approximately
30:28 86: Newton's method and algebraic curves
27:10 87: Logical weakness in modern pure mathematics
27:20 88: The decline of rigour in modern mathematics
48:44 89: Fractions and repeating decimals
53:29 90: Fractions and p-adic numbers
51:01 91: Difficulties with real numbers as infinite decimals I
52:06 92: Difficulties with real numbers as infinite decimals II
41:33 93: The magic and mystery of "pi"
28:42 94: Problems with limits and Cauchy sequences
35:42 95: The deep structure of the rational numbers
36:07 96: Fractions and the Stern-Brocot tree
34:14 97: The Stern-Brocot tree, matrices and wedges
26:32 98: What exactly is a sequence?
36:41 99: "Infinite sequences": what are they?
26:02 100: Slouching towards infinity: building up on-sequences
35:32 101: Challenges with higher on-sequences
48:28 102: Limits and rational poly on-sequences
32:11 103: Extending arithmetic to infinity!
36:56 104: Rational number arithmetic with infinity and more
39:20 105: The extended rational numbers in practice
35:03 106: What exactly is a limit??
34:33 107: Inequalities and more limits
38:29 108: Limits to Infinity
36:17 109: Logical difficulties with the modern theory of limits I
36:50 110: Logical difficulties with the modern theory of limits II
21:06 111: Real numbers and Cauchy sequences of rationals I
35:54 112: Real numbers and Cauchy sequences of rationals II
30:24 113: Real numbers and Cauchy sequences of rationals III
52:19 114: Real numbers as Cauchy sequences don't work!
52:07 115: The mostly absent theory of real numbers
40:20 116: Difficulties with Dedekind cuts
34:38 117: The continuum, Zeno's paradox and the price we pay for coordinates
21:23 118: Real fish, real numbers, real jobs
33:07 119: Mathematics without real numbers
29:11 120: Axiomatics and the least upper bound property I
28:27 121: Axiomatics and the least upper bound property II
33:19 122: Mathematical space and a basic duality in geometry
26:56 123: Affine one-dimensional geometry and the Triple Quad Formula
46:04 124: Heron's formula, Archimedes' function, and the TQF
41:29 125: Brahmagupta's formula and the Quadruple Quad Formula I
41:01 126: Brahmagupta's formula and the Quadruple Quad Formula II
29:19 127a: The Cyclic quadrilateral quadrea theorem
22:09 127b: The Cyclic quadrilateral quadrea theorem (cont.)
34:25 128: Robbins' formulas, the Bellows conjecture, and volumes of polyhedra
45:28 129: The projective line, circles, and a proof of the CQQ theorem
37:01 130: The projective line, circles and a proof of the CQQ theorem II
45:48 131: Ptolemy's theorem and generalizations
30:50 132: The Bretschneider von Staudt formula for the quadrea of a quadrilateral
29:05 133: Higher dimensions and the roles of length, area and volume
30:53 134: Absolute versus relative measurements in geometry
38:46 135: NJ's pizza model for organizing geometry
25:46 136: The projective Triple Quad Formula
40:41 137: Algebraic structure on the Euclidean projective line
34:26 138: Isometry groups of the projective line I
35:04 139: Isometry groups of the projective line II
30:45 140: Isometry groups of the projective line III
23:38 141: The three-fold symmetry of chromogeometry
33:01 142: Relativistic velocity addition, core circles and Paul Miller's protractor I
39:15 143: Relativistic velocity addition, core circles, and Paul Miller's protractor II
39:03 144: Relativistic velocity addition, core circles and Paul Miller's protractor III
48:41 145: Relativistic velocity, core circles and Paul Miller's protractor IV
23:58 146: The Triple spread formula, circumcircles and curvature
23:50 147: The curvature of a parabola, without calculus
31:27 148: The projective Quadruple quad formula
36:03 149: The circumquadrance of a cyclic quadrilateral
Math Foundations C (150-- ) by Norman J Wildberger
source: njwildberger 2017年3月20日
34:40 150: What exactly is a set?
35:18 151: Sets and other data structures in mathematics
30:13 152: Fun with lists, ordered sets, multisets and sets I
17:07 153: Fun with lists, ordered sets, multisets and sets II
19:06 154: Fun with lists, ordered sets, multisets and sets III
24:49 155: The realm of natural numbers
29:47 156: The realm of natural number multisets
23:05 157: The algebra of natural number multisets
32:42 158: An introduction to the Tropical calculus
47:34 159: Inclusion/Exclusion via multisets
31:08 160: Unique factorization, primes and msets
19:13 161: Fun with lists, ordered sets, multisets and sets IV
28:11 162: Four basic combinatorial counting problems
27:09 163: Higher data structures
27:49 164: Arrays and matrices I
30:31 165: Arrays and matrices II
33:03 166: Maxel theory: new thinking about matrices I
26:54 167: Maxel theory: new thinking about matrices II
40:49 168: Maxel theory: new thinking about matrices III
35:21 169: Maxel algebra! I
20:30 170: Maxel algebra! II
28:16 171: Singletons, vexels, and the rank of a maxel I
36:27 172: Singletons, vexels, and the rank of a maxel II
32:06 173: A disruptive view of big number arithmetic
32:00 174: Complexity and hyperoperations
37:35 175: The chaotic complexity of natural numbers
31:51 176: The sporadic nature of big numbers
31:46 177: Numbers, the universe and complexity beyond us
25:49 178: The law of logical honesty and the end of infinity
22:27 179: Hyperoperations and even bigger numbers
43:15 180: The successor - limit hierarchy
21:52 181: The successor-limit hierarchy and ordinals I
25:18 182: The successor-limit hierarchy and ordinals II
42:17 183: Limit levels and self-similarity in the successor-limit hierarchy
31:42 184: Reconsidering natural numbers and arithmetical expressions
20:56 185: The essential dichotomy underlying mathematics
27:29 186: The curious role of "nothing" in mathematics
30:15 187: Multisets and a new framework for arithmetic
24:38 188: Naming and ordering numbers for students
30:05 189: The Hindu Arabic number system revisited
24:10 190: Numbers, polynumbers and arithmetic with vexels I
20:24 191: Numbers, polynumbers, and arithmetic with vexels II
29:38 192: Arithmetic with base 2 vexels
33:38 193: A new look at Hindu Arabic numbers and their arithmetic
27:22 194: Arithmetical expressions as natural numbers
26:37 195: Divisibility of big numbers
26:38 196: Back to Gauss and modular arithmetic
37:21 197: Modular arithmetic with Fermat and Euler
34:26 198: Unique factorization and its difficulties I
27:53 199: Unique factorization and its difficulties II
30:02 200: Mission impossible: factorize the number z
23:32 201: A celebration of 200 videos of Math Foundations
24:27 202: Reciprocals, powers of 10, and Euler's totient function I
25:33 203: Reciprocals, powers of 10, and Euler's totient function II
27:48 204: Euclid and the failure of prime factorization for z
22:01 205: Negative numbers, msets, and modern physics
21:45 206: A new trichotomy to set up integers
23:41 207: Integral vectors and matrices via vexels and maxels I
23:40 208: Integral vectors and matrices via vexels and maxels II
21:20 209: A broad canvas: algebra with maxels from integers
21:06 210: Numbers as multipliers and particle/antiparticle duality I
28:57 211: Numbers as multipliers and particle/antiparticle duality II
23:18 212: The anti operation in mathematics
25:38 213: An introduction to abstract algebra
41:58 214: Logical challenges with abstract algebra I
19:49 215: Logical challenges with abstract algebra II
27:19 216: The fundamental dream of algebra
28:27 217: What is the Fundamental theorem of Algebra, really?
29:26 218: Why roots of unity need to be rethought
24:14 219: Linear spaces and spans I
34:19 220: Linear spaces and spans II
29:12 221: Bases and dimension for integral linear spaces I
45:27 222: Bases and dimension for integral linear spaces II
44:18 223: Integral row reduction and Hermite normal form
34:21 224: Lattice relations and Hermite normal form
27:53 225: Relations between msets
Universal Hyperbolic Geometry A (1-32) by Norman J Wildberger
source: njwildberger 2017年2月1日
Hyperbolic geometry, in this new series, is made simpler, more logical, more general and... more beautiful! The new approach will be called `Universal Hyperbolic Geometry', since it extends the subject in a number of directions. It works over general fields, it extends beyond the usual disk in the Beltrami Klein model, and it unifies hyperbolic and elliptic (and other) geometries.
23:13 0: Introduction
40:38 1: Apollonius and polarity
38:22 2: Apollonius and harmonic conjugates
21:38 3: Pappus' theorem and the cross ratio
37:14 4: First steps in hyperbolic geometry
35:54 5: The circle and Cartesian coordinates
50:38 6: Duality, quadrance and spread in Cartesian coordinates
37:40 7a: The circle and projective homogeneous coordinates
24:17 7b: The circle and projective homogeneous coordinates (cont.)
44:32 8: Computations with homogeneous coordinates
33:11 9: Duality and perpendicularity
44:06 10: Orthocenters exist!
37:28 11: Theorems using perpendicularity
36:20 12: Null points and null lines
26:31 13: Apollonius and polarity revisited
31:23 14: Reflections in hyperbolic geometry
50:29 15: Reflections and projective linear algebra
36:41 16: Midpoints and bisectors
34:09 17: Medians, midlines, centroids and circumcenters
29:35 18: Parallels and the double triangle
42:27 19: The J function, sl(2) and the Jacobi identity
38:44 20: Pure and applied geometry--understanding the continuum
35:54 21: Quadrance and spread
36:14 22: Pythagoras' theorem in Universal Hyperbolic Geometry
39:11 23: The Triple quad formula in Universal Hyperbolic Geometry
34:34 24: Visualizing quadrance with circles
25:22 25: Geometer's Sketchpad and circles in Universal Hyperbolic Geometry
20:20 26: Trigonometric laws in hyperbolic geometry using Geometer's Sketchpad
24:21 27: The Spread law in Universal Hyperbolic Geometry
35:26 28: The Cross law in Universal Hyperbolic Geometry
42:35 29: Thales' theorem, right triangles and Napier's rules
32:46 30: Isosceles triangles in hyperbolic geometry
42:05 31: Menelaus, Ceva and the Laws of proportion
35:47 32: Trigonometric dual laws and the Parallax formula
Universal Hyperbolic Geometry B (33- ) by Norman J Wildberger
source: njwildberger 2017年1月23日
32:14 33: Spherical and elliptic geometries: an introduction
44:19 34: Spherical and elliptic geometries (cont.)
32:14 35: Areas and volumes for a sphere
34:48 36: Classical spherical trigonometry
31:54 37: Perpendicularity, polarity and duality on a sphere
38:43 38: Parametrizing and projecting a sphere
33:08 39: Rational trigonometry: an overview
31:20 40: Rational trigonometry in three dimensions
42:24 41: Trigonometry in elliptic geometry
40:58 42: Trigonometry in elliptic geometry II
33:38 43: Applications of rational spherical trigonometry I
42:33 44: Applications of rational spherical trigonometry II
19:41 45: The geometry of the regular tetrahedron
24:49 46: Eight ninths and the geometry of A4 paper
26:07 47: The remarkable Platonic solids I
31:18 48: The remarkable Platonic solids II: symmetry
27:13 49: Canonical structures inside the Platonic solids I
23:30 50: Canonical structures inside Platonic solids II
22:25 51: Canonical structures inside the Platonic solids III
26:25 52: Petrie polygons of a polyhedron
27:59 53: The classification of Platonic solids I
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