What is a number? 
What is a number?
by UNSW / N.J. Wildberger
Video Lecture 1 of 127
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Date Added: January 19, 2015

Lecture Description

The first of a series that will discuss foundations of mathematics. Contains a general introduction to the series, and then the beginnings of arithmetic with natural numbers. This series will methodically develop a lot of basic mathematics, starting with arithmetic, then geometry, then algebra, then analysis (calculus) and will also treat so called set theory. It will have a lot of critical things to say once we get around to facing squarely up to the many logical weaknesses of modern pure mathematics.

Course Index

  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

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.


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