Real numbers are the numbers people use every day. They include any number you can place on a number line, whether it’s positive or negative. Here is the definition of a real number, a look at the sets and properties of real numbers, and specific examples of numbers that are real and imaginary.

### Real Number Definition

A **real number** is any number that can be placed on a number line or expressed as in infinite decimal expansion. In other words, a real number is any rational or irrational number, including positive and negative whole numbers, integers, decimals, fractions, and numbers such as pi (*π*) and Euler’s number (*e*).

In contrast, an imaginary number or complex number is *not* a real number. These numbers contain the number *i*, where *i*^{2} = -1.

Real numbers are represented by the capital letter “R” or double struck typeface ℝ. The real numbers are an infinite set of numbers.

### Set of Real Numbers

The set of real numbers includes several smaller (yet still infinite) subsets:

Set | Definition | Examples |
---|---|---|

Natural Numbers (N) | Counting numbers, starting from 1. N = {1,2,3,4,…} | 1, 3, 157, 2021 |

Whole Numbers (W) | Zero and the natural numbers. W = {0,1,2,3,…} | 0, 1, 43, 811 |

Integers (Z) | The whole numbers and the negative of all the natural numbers. Z = {..,-1,0,1,…} | -44, -2, 0, 28 |

Rational Numbers (Q) | Numbers that can be written as the fraction of integers p/q, q≠0. where Q = {p/q}, q≠0 | ^{1}/_{3}, ^{5}/_{4}, 0.8 |

Irrational Numbers (P or I) | Real numbers which cannot be expressed as the fraction of integers p/q. They are non-terminating and non-repeating decimals. | π, e, φ, √2 |

### Examples of Real Numbers and Imaginary Numbers

While it’s pretty easy to recognize familiar numbers natural numbers and integers as real numbers, many people wonder about specific numbers. Zero is a real number. Pi, Euler’s number, and phi are real numbers. All fractions and decimal numbers are real numbers.

Numbers which are not real numbers are either imaginary (e.g., √-1, *i*, 3*i*) or complex (*a + bi*). So, some algebraic expressions are real [e.g., √2, -√3, (1+ √5)/2] and some are not [e.g., *i*^{2}, (x + 1)^{2} = -9].

Infinity (∞) and negative infinity (-∞) are *not* real numbers. They are not members of mathematically-defined sets. Mainly, this is because infinity and negative infinity can have different values. For example, the set of whole numbers is infinite. So is the set of integers. But, the two sets are not the same size.

### Properties of Real Numbers

The four main properties of real numbers are the commutative property, associative property, distributive property, and identity property. If m, n, and r are real numbers, then:

#### Commutative Property

**Addition:**m + n = n + m. For example, 5 + 23 = 23 + 5.**Multiplication:**m × n = n × m. For example, 5 × 2 = 2 × 5.

#### Associative Property

**Addition:**The general form will be m + (n + r) = (m + n) + r. An example of additive associative property is 5 + (3 + 2) = (5 + 3) + 2.**Multiplication:**(mn) r = m (nr). An example of a multiplicative associative property is (2 × 5) 6 = 2 (5 ×6).

#### Distributive Property

- m (n + r) = mn + mr and (m + n) r = mr + nr. An example of the distributive property is: 2(3 + 5) = 2 x 3 + 2 x 5. Both expressions equal 16.

#### Identity Property

**For addition:**m + 0 = m. (0 is the additive identity)**For multiplication:**m × 1 = 1 × m = m. (1 is the multiplicative identity)

### References

- Bengtsson, Ingemar (2017). “The Number Behind the Simplest SIC-POVM”.
*Foundations of Physics*. - Borwein, J.; Borwein, P. (1990).
*A Dictionary of Real Numbers*. Pacific Grove, CA: Brooks/Cole. - Feferman, Solomon (1989). T
*he Number Systems: Foundations of Algebra and Analysis*. AMS Chelsea. ISBN 0-8218-2915-7. - Howie, John M. (2005).
*Real Analysis*. Springer. ISBN 1-85233-314-6. - Landau, Edmund (2001).
*Foundations of Analysis*. American Mathematical Society. ISBN 0-8218-2693-X.