Real Numbers Definition
Real numbers can be defined as the union of both the 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.
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Set of Real Numbers
The set of real numbers consist 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 number.
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  All the numbers which are not rational and cannot be written in the form of p/q.  Irrational numbers are nonterminating and nonrepeating 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 four main properties of real numbers are commutative property, associative property, distributive property and 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)
Learn More About Real Number Properties  

Commutative Property  Associative Property 
Distributive Property  Additive Identity and Multiplicative Identity 
Practice Questions
 Which is the smallest composite number?
 Prove that any positive odd integer is of the form 6x + 1, 6x + 3, or 6x + 5.
 Evaluate 2 + 3 × 6 – 5
 What is the product of a nonzero rational number and irrational number?
 Can every positive integer be represented as 4x + 2 (where x is an integer)?
Read More:
Rational numbers on a number line  Operations On Real Numbers 
Laws Of Exponents  Euclid’s Division Lemma 
Fundamental Theorem Of Arithmetic  Properties Of Integers 
Frequently Asked Questions
What are Natural and Real Numbers?
Natural numbers are all the positive integers starting from 1 to infinity. All the natural numbers are integers but not all the integers are natural numbers. These are the set of all counting numbers such as 1, 2, 3, 4, 5, 6, 7, 8, 9, …….∞.
Real numbers are numbers that include both rational and irrational numbers. Rational numbers such as integers (2, 0, 1), fractions(1/2, 2.5) and irrational numbers such as √3, π(22/7), etc., are all real numbers.
Is Zero a Real or an Imaginary Number?
Zero is considered as both a real and an imaginary number. As we know, imaginary numbers are the square root of nonpositive real numbers. And since 0 is also a nonpositive number, therefore it fulfils the criteria of the imaginary number. Whereas 0 is also a rational number, which is defined in a number line and hence a real number.
Are there Real Numbers that are not Rational or Irrational?
No, there are no real numbers that are neither rational nor irrational. The definition of real numbers itself states that it is a combination of both rational and irrational numbers.
Is the real number a subset of a complex number?
Yes, because a complex number is the combination of a real and imaginary number. So, if the complex number is a set then the real and imaginary number are the subsets of it.
What are the properties of real numbers?
Commutative Property
Associative Property
Distributive Property
Identity Property