Real Numbers

Real Number Definition:

A real number is a value of a continuous quantity that can represent 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 = √-1

Properties 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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Real Numbers Calculator:

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Real Numbers Calculator Video

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• number
• property
• real numbers
• real

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