Real Number Definition:
A continuous quantity:
Represents a distance along a line.
Real Number Notation
We represent the set of real numbers as ℝReal numbers extend from but do not include (-∞, ∞)
What do Real Numbers Include?
Natural (Counting) Numbers (ℕ) (1, 2, 3, 4, ...)Whole Numbers (0, 1, 2, 3, 4, ...)
Integers (ℤ) (..., -3, -2, -1, 0, 1, 2, 3, ...)
Rational Numbers (ℚ) (1/2, 0.8, 3.5, ... )
Irrational Numbers (like π, Euler's Number (e) = 2.718281828459, √2, etc)
What do Real Numbers NOT Include?
Positive Infinity ∞Negative Infinity -∞
Imaginary Numbers like i = √-1
Properties of Real Numbers
UncountableExtend infinitely (but do not include infinity)
Any non-zero real number is either negative or positive.
Real Numbers can be ordered
The sum of two non-negative real numbers is a non-negative real number:
x ∈ ℝ, y ∈ ℝ → x + y ∈ ℝ
The product of two non-negative real numbers is a non-negative real number:
x ∈ ℝ, y ∈ ℝ → xy ∈ ℝ