In this video we introduce Brahmagupta's celebrated formula for the area of a cyclic quadrilateral in terms of the four sides. This is an obvious extension of Heron's formula. We are interested in finding a rational variant of it, that will be independent of a prior theory of `real numbers', `square roots' and `lengths'. For motivation we look at the situation of four affine 1-points. Is there an analog of the Triple Quad Formula? Yes there is: it is the more mysterious and complicated Quadruple Quad Formula. To get it, we will examine some important manipulations for a pair of quadratic equations which are of independent interest.This lecture has some more serious algebra in it: a great place to practice your manipulation and organizational skills. If you are teaching college mathematics, please consider doing yourself and your students a favour: teach them some of the material of this lecture carefully and explicitly!

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

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.