 # 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

## Real Numbers Calculator:

To see if a number is real, you can ask the search engine questions like is 15 a real number

### How does the Real Numbers Calculator work?

This lesson walks you through what real numbers are, how to write real numbers, real numbers notation, and what‘s included in real numbers

### What 2 formulas are used for the Real Numbers Calculator?

1. ℝ → (-∞, ∞)
2. Represents a distance along a line

For more math formulas, check out our Formula Dossier

### What 4 concepts are covered in the Real Numbers Calculator?

number
an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification. A quantity or amount.
property
an attribute, quality, or characteristic of something
real
real numbers