Real analysis

• Real numbers


  An ordered field
• Real numbers


  • Represent points on a straight line extending indefinitely in both directions
  • Denoted by the set R
• Subsets of real numbers

  • Natural numbers N = {1, 2, 3, ...}
  • Integers Z = {..., -2, -1, 0, 1, 2, ...}
  • Rational numbers Q = {p/q | p ∈ Z, q ∈ Z, q ≠ 0}
• Irrational numbers


  Real numbers that are not rational
• Irrational numbers


  • √2
  • π
• Real numbers can be constructed rigorously from set theory axioms
• Real numbers satisfy the properties of a field
• Field properties of real numbers

  • Addition: 0 element, associative, inverses, commutative
  • Multiplication: 1 element, associative, inverses, commutative
  • Distributive law: (a + b)c = ac + bc
• The zero element, unit element, additive inverses, and multiplicative inverses are unique
• Subtraction and division can be defined in terms of addition, multiplication, and inverses
• Certain arithmetic rules like a·0 = 0 = 0·a can be deduced from the field axioms
• Real numbers are an ordered field


  • There is a natural ordering < on the real numbers
  • This ordering satisfies certain properties
• The ordering < on real numbers satisfies the following properties:

  1. For any real numbers a and b, either a < b, a = b, or a > b
  2. If a < b and b < c, then a < c (transitivity)
  3. For any real numbers a and b, either a < b, a = b, or a > b (trichotomy)
  4. If a < b, then a + c < b + c for any real number c
  5. If a < b and 0 < c, then ac < bc
• aa^-1


  Equal to 1
• 1 ≠ 0
• aa^-1 ≠ 0
• a^-1 ≠ 0
• a


  Multiplicative inverse of a^-1
• • a


  Unique number satisfying a + (-a) = 0
• (−1) · a = a · (−1)
• (−a)b = −(ab)
• a(−b) = −ab
• −(−1) = 1
• Real numbers


  • Totally ordered
  • Compatible algebraic and order properties
• x > 0 if and only if -x < 0
• 1 > 0
• Real numbers


  • If x > 0, then 1/x > 0
  • If x > y > 0, then 1/y > 1/x > 0
• If x > y and z < 0, then xz < yz
• x > y if and only if x - y > 0
• If x > 0 and y > 0, then x > y if and only if x^2 > y^2
• x ≥ y


  x > y or x = y
• x ≤ y


  x < y or x = y
• If x ≥ y and y ≥ z, then x ≥ z
• If x ≥ y and z ≥ 0, then xz ≥ yz
• For any real number y ≥ 0 and any positive integer n, there exists a unique real number x ≥ 0 such that y = x^n
• n√y


  1. th root of y
• √2 is irrational
• |x|


  Absolute value or modulus of x
• |x| ≥ 0
• • |x| ≤ x ≤ |x|