# Math Foundations with Norman Wildberger

### Course Description

Does mathematics make logical sense? No, it does not. Foundational issues have been finessed by modern mathematicians, 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. This course is aimed for a general audience, interested in mathematics, or willing to learn.

Prof. Wildberger discussing "The Stern-Brocot tree, matrices and wedges."
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### Video Lectures & Study Materials

# Lecture Play Lecture
1 What is a number? (9:55) Play Video
2 Arithmetic with numbers (10:07) Play Video
3 Laws of Arithmetic (9:36) Play Video
4 Subtraction and Division (10:07) Play Video
5 Arithmetic and Math education (9:23) Play Video
6 The Hindu-Arabic number system (8:16) Play Video
7 Arithmetic with Hindu-Arabic numbers (10:03) Play Video
8 Division (9:56) Play Video
9 Fractions (6:29) Play Video
10 Arithmetic with fractions (9:42) Play Video
11 Laws of arithmetic for fractions (6:30) Play Video
12 Introducing the integers (9:22) Play Video
13 Rational numbers (9:15) Play Video
14 Rational numbers and Ford Circles (9:43) Play Video
15 Primary school maths education (10:01) Play Video
16 Why infinite sets don't exist (7:38) Play Video
17 Extremely big numbers (9:58) Play Video
18 Geometry (8:14) Play Video
19 Euclid's Elements (9:05) Play Video
20 Euclid and proportions (9:39) Play Video
21 Euclid's Books VI--XIII (7:33) Play Video
22 Difficulties with Euclid (8:01) Play Video
23 The Basic Framework for Geometry I (8:49) Play Video
24 The Basic Framework for Geometry II (9:39) Play Video
25 The Basic Framework for Geometry III (9:41) Play Video
26 The Basic Framework for Geometry IV (6:52) Play Video
27 Trigonometry with rational numbers (9:34) Play Video
28 What exactly is a circle? (9:14) Play Video
29 Parametrizing circles (8:43) Play Video
30 What exactly is a vector? (9:53) Play Video
31 Parallelograms and affine combinations (9:23) Play Video
32 Geometry in primary school (9:25) Play Video
33 What exactly is an area? (7:55) Play Video
34 Areas of polygons (9:45) Play Video
35 Translations, rotations and reflections I (10:09) Play Video
36 Translations, rotations and reflections II (9:51) Play Video
37 Translations, rotations and reflections III (9:58) Play Video
38 Why angles don't really work I (9:20) Play Video
39 Why angles don't really work II (9:52) Play Video
40 Correctness in geometrical problem solving (9:50) Play Video
41 Why angles don't really work III (8:11) Play Video
42 Deflating Modern Mathematics: the problem with 'functions' - Part 1 (9:43) Play Video
43 Deflating Modern Mathematics: the problem with 'functions' - Part 2 (6:27) Play Video
44 Reconsidering `functions' in modern mathematics (9:52) Play Video
45 Definitions, specification and interpretation (9:54) Play Video
47 Introduction to Algebra (9:46) Play Video
48 Baby Algebra (9:16) Play Video
49 Solving a quadratic equation (8:11) Play Video
50 Solving a quadratic equation (7:08) Play Video
51 How to find a square root (10:06) Play Video
52 Algebra and number patterns (9:32) Play Video
53 More patterns with algebra (9:54) Play Video
54 Leonhard Euler and Pentagonal numbers (10:05) Play Video
55 Algebraic identities (8:48) Play Video
56 The Binomial theorem (9:58) Play Video
57 Binomial coefficients and related functions (10:04) Play Video
58 The Trinomial theorem (10:08) Play Video
59 Polynomials and polynumbers (9:47) Play Video
60 Arithmetic with positive polynumbers (9:52) Play Video
61 More arithmetic with polynumbers (9:20) Play Video
62 What exactly is a polynomial? (9:39) Play Video
63 Factoring polynomials and polynumbers (9:53) Play Video
64 Arithmetic with integral polynumbers (7:58) Play Video
65 The Factor theorem and polynumber evaluation (10:10) Play Video
66 The Division algorithm for polynumbers (45:17) Play Video
67 Row and column polynumbers (49:53) Play Video
68 Decimal numbers (28:20) Play Video
69 Visualizing decimal numbers and their arithmetic (44:23) Play Video
70 Laurent polynumbers (the New Years Day lecture) (39:52) Play Video
71 Translating polynumbers and the Derivative (37:15) Play Video
72 Calculus with integral polynumbers (36:00) Play Video
73 Tangent lines and conics of polynumbers (36:29) Play Video
74 Graphing polynomials (37:50) Play Video
75 Lines and Parabolas I (39:52) Play Video
76 Lines and Parabolas II (38:19) Play Video
77 Cubics and the prettiest theorem in calculus (28:09) Play Video
78 An introduction to algebraic curves (34:33) Play Video
79 Object-oriented versus expression-oriented mathematics (45:50) Play Video
80 Calculus on the unit circles (35:20) Play Video
81 Calculus on a cubic: the Folium of Descartes (31:56) Play Video
82 Inconvenient truths about Square Root of 2 (42:03) Play Video
83 Measurement, approximation and interval arithmetic I (45:38) Play Video
84 Measurement, approximation and interval arithmetic II (41:49) Play Video
85 Newton's method for finding zeroes (25:50) Play Video
86 Newton's method for approximating cube roots (29:31) Play Video
87 Solving quadratics and cubics approximately (36:18) Play Video
88 Newton's method and algebraic curves (30:28) Play Video
89 Logical weakness in modern pure mathematics (27:10) Play Video
90 The decline of rigour in modern mathematics (27:20) Play Video
91 Fractions and repeating decimals (48:44) Play Video
92 Fractions and p-adic numbers (53:29) Play Video
93 Difficulties with real numbers as infinite decimals I (51:01) Play Video
94 Difficulties with real numbers as infinite decimals II (52:06) Play Video
95 The magic and mystery of π (41:33) Play Video
96 Problems with limits and Cauchy sequences (28:42) Play Video
97 The deep structure of the rational numbers (35:42) Play Video
98 Fractions and the Stern-Brocot tree (36:07) Play Video
99 The Stern-Brocot tree, matrices and wedges (34:14) Play Video
100 What exactly is a sequence? (26:32) Play Video
101 "Infinite sequences": what are they? (36:41) Play Video
102 Slouching towards infinity: building up on-sequences (26:02) Play Video
103 Challenges with higher on-sequences (35:32) Play Video
104 Limits and rational poly on-sequences (48:28) Play Video
105 MF103: Extending arithmetic to infinity! (32:11) Play Video
106 Rational number arithmetic with infinity and more (36:56) Play Video
107 The extended rational numbers in practice (39:20) Play Video
108 What exactly is a limit? (35:03) Play Video
109 Inequalities and more limits (34:33) Play Video
110 Limits to Infinity (38:29) Play Video
111 Logical difficulties with the modern theory of limits I (36:17) Play Video
112 Logical difficulties with the modern theory of limits II (36:50) Play Video
113 Real numbers and Cauchy sequences of rationals I (21:06) Play Video
114 Real numbers and Cauchy sequences of rationals II (35:54) Play Video
115 Real numbers and Cauchy sequences of rationals III (30:24) Play Video
116 Real numbers as Cauchy sequences don't work! (52:19) Play Video
117 The mostly absent theory of real numbers (52:07) Play Video
118 Difficulties with Dedekind cuts (40:20) Play Video
119 The continuum, Zeno's paradox and the price we pay for coordinates (34:38) Play Video
120 Real fish, real numbers, real jobs (21:23) Play Video
121 Mathematics without real numbers (33:07) Play Video
122 Axiomatics and the least upper bound property I (29:11) Play Video
123 Axiomatics and the least upper bound property II (28:27) Play Video
124 Mathematical space and a basic duality in geometry (33:19) Play Video
125 Affine one-dimensional geometry and the Triple Quad Formula (26:56) Play Video
126 Heron's formula, Archimedes' function, and the TQF (46:04) Play Video
127 Brahmagupta's formula and the Quadruple Quad Formula I (41:29) Play Video