Real Numbers

 Real numbers are simply the combination of rational and irrational numbers, in the number system. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also. At the same time, the imaginary numbers are the un-real numbers, which cannot be expressed in the number line and are commonly used to represent a complex number. Some of the examples of real numbers are 23, -12, 6.99, 5/2, π, and so on. In this article, we are going to discuss the definition of real numbers, the properties of real numbers and the examples of real numbers with complete explanations

Real Numbers Definition

Real numbers can be defined as the union of both rational and irrational numbers. They can be both positive or negative and are denoted by the symbol “R”. All the natural numbers, decimals and fractions come under this category. See the figure, given below, which shows the classification of real numerals.

Set of Real Numbers

The set of real numbers consists of different categories, such as natural and whole numbers, integers, rational and irrational numbers. In the table given below, all the real numbers formulas (i.e.) the representation of the classification of real numbers are defined with examples.

Category	Definition	Example
Natural Numbers	Contain all counting numbers which start from 1. N = {1, 2, 3, 4,……}	All numbers such as 1, 2, 3, 4, 5, 6,…..…
Whole Numbers	Collection of zero and natural numbers. W = {0, 1, 2, 3,…..}	All numbers including 0 such as 0, 1, 2, 3, 4, 5, 6,…..…
Integers	The collective result of whole numbers and negative of all natural numbers.	Includes: -infinity (-∞),……..-4, -3, -2, -1, 0, 1, 2, 3, 4, ……+infinity (+∞)
Rational Numbers	Numbers that can be written in the form of p/q, where q≠0.	Examples of rational numbers are ½, 5/4 and 12/6 etc.
Irrational Numbers	The numbers which are not rational and cannot be written in the form of p/q.	Irrational numbers are non-terminating and non-repeating in nature like √2.

Real Numbers Chart

The chart for the set of real numerals including all the types are given below:

Properties of Real Numbers

The following are the four main properties of real numbers:

• Commutative property
• Associative property
• Distributive property
• Identity property

Consider “m, n and r” are three real numbers. Then the above properties can be described using m, n, and r as shown below:

Commutative Property

If m and n are the numbers, then the general form will be m + n = n + m for addition and m.n = n.m for multiplication.

• Addition: m + n = n + m. For example, 5 + 3 = 3 + 5, 2 + 4 = 4 + 2.
• Multiplication: m × n = n × m. For example, 5 × 3 = 3 × 5, 2 × 4 = 4 × 2.

Associative Property

If m, n and r are the numbers. The general form will be m + (n + r) = (m + n) + r for addition(mn) r = m (nr) for multiplication.

• Addition: The general form will be m + (n + r) = (m + n) + r. An example of additive associative property is 10 + (3 + 2) = (10 + 3) + 2.
• Multiplication: (mn) r = m (nr). An example of a multiplicative associative property is (2 × 3) 4 = 2 (3 × 4).

Distributive Property

For three numbers m, n, and r, which are real in nature, the distributive property is represented as:

m (n + r) = mn + mr and (m + n) r = mr + nr.

• Example of distributive property is: 5(2 + 3) = 5 × 2 + 5 × 3. Here, both sides will yield 25.

Identity Property

There are additive and multiplicative identities.

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