Real Number Definition:
A continuous quantity:
Represents a distance along a line.
Real Number Notation
We represent the set of real numbers as ℝ
Real 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 = √
-1Properties of Real Numbers
Uncountable
Extend 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
Properties of Real Numbers:
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