Math 2.0

Learn Year 1

Number systems

103 ideas in teaching order. Each idea comes after the ideas that it builds on. 102 of them have a page so far; the others are on the list in grey.

Start with Natural numbers →

  1. Natural numbers object
  2. Peano axioms concept
  3. Successor function definition
  4. The successor function is injective statement
  5. Zero is not a successor statement
  6. Definition by recursion concept
  7. Addition of natural numbers definition
  8. Multiplication of natural numbers definition
  9. Commutativity of addition statement
  10. Associativity of addition statement
  11. Distributive law statement
  12. Order on the natural numbers definition
  13. Mathematical induction concept
  14. Induction from an arbitrary base case statement
  15. Strong induction statement
  16. well-ordering principle statement
  17. Sum of the first n natural numbers statement
  18. Sum of a finite geometric progression statement
  19. Bernoulli's inequality statement
  20. Integers object
  21. Construction of the integers from the natural numbers concept
  22. Ring of integers statement
  23. Order on the integers definition
  24. Induction on the integers statement
  25. Rational numbers object
  26. Construction of the rationals as fractions concept
  27. Fraction in lowest terms statement
  28. The rationals form a field statement
  29. Ordered field definition
  30. The rationals form an ordered field statement
  31. Irrational number definition
  32. Irrationality of the square root of 2 statement
  33. Irrationality of the square root of a prime statement
  34. A root of an integer is an integer or irrational statement
  35. Real numbers object
  36. The reals form a field statement
  37. Order on the reals definition
  38. The reals form an ordered field statement
  39. Set bounded above definition
  40. Supremum definition
  41. Infimum definition
  42. Uniqueness of the supremum statement
  43. Completeness axiom statement
  44. Existence of the infimum statement
  45. Approximation property of the supremum statement
  46. Complete ordered field concept
  47. Uniqueness of the complete ordered field statement
  48. Archimedean ordered field definition
  49. Archimedean property statement
  50. The reals are Archimedean statement
  51. There is a natural number n with 1/n below any positive epsilon statement
  52. Density of the rationals statement
  53. Density of the irrationals statement
  54. Floor function definition
  55. Ceiling function definition
  56. Characterization of the floor statement
  57. Square root definition
  58. Existence of square roots of nonnegative reals statement
  59. Dedekind cut definition
  60. Construction of the reals from Cauchy sequences concept
  61. Decimal expansion definition
  62. Every real number has a decimal expansion statement
  63. 0.999... equals 1 example
  64. Repeating decimals are rational statement
  65. Interval definition
  66. Absolute value definition
  67. Absolute value is multiplicative statement
  68. Triangle inequality statement
  69. Reverse triangle inequality statement
  70. Inequality of arithmetic and geometric means statement
  71. Extended real number line object
  72. Complex numbers object
  73. Imaginary unit object
  74. i squared is minus one statement
  75. Real and imaginary parts definition
  76. Equality of complex numbers statement
  77. Complex numbers as pairs of reals concept
  78. The complex numbers form a field statement
  79. Complex plane concept
  80. Complex conjugate definition
  81. Modulus of a complex number definition
  82. Formula for the modulus statement
  83. A complex number times its conjugate statement
  84. Inverse of a complex number statement
  85. Modulus is multiplicative statement
  86. Modulus of the conjugate statement
  87. The complex numbers cannot be ordered statement
  88. Argument of a complex number definition
  89. Principal value of the argument statement
  90. Polar form statement
  91. Euler's formula statement
  92. Euler's identity statement
  93. Product of exponentials statement
  94. Arguments add under multiplication statement
  95. de Moivre's formula statement
  96. Root of unity definition
  97. The nth roots of unity statement
  98. Number of nth roots of unity statement
  99. Primitive root of unity definition
  100. exp(2 pi i/n) is a primitive root of unity statement
  101. Sum of the roots of unity statement
  102. Existence of nth roots of complex numbers statement
  103. Quadratic formula statement