Real numbers are definitely the mixture of rational and irrational numbers, within the number system. In general, all the mathematics operations may be finished on these numbers and they may be represented in the number line, also.

Real Numbers Definition 

Real numbers may be defined as the union of both rational and irrational numbers. They may be each positive or negative and are denoted by using the image “R”. all the natural numbers, decimals and fractions come underneath this class.

A real number is a number that can be found on the number line. These are the numbers that we normally use and utilize them in real-world applications.

Set of real numbers 

The set of real numbers includes  different categories which are as follows , 

The classification of real numbers can be done in the following way -: 

• Natural numbers

• Whole numbers

• Integers

• Fractions

• Rational numbers

• Irrational numbers

Here is the classification chart of real numbers -: 

• Natural Numbers -: 

A natural number is defined as a counting number. It starts from 1 onwards. They are presented at the right side of the number line (after 0).

Examples -:   1,3,66,204,789,34,2499,……

• Whole Numbers -: 

A whole number is a collection of  zero (0) and natural Numbers. They are presented at the right side of the number line.

Examples -:   0, 1,9,80,23,79,…….

• Integers -: 

They are defined as the collective result of whole numbers and negative of all natural numbers.

Examples -: -infinity (-∞),……..-4, -3, -2, -1, 0, 1, 2, 3, 4, ……+infinity (+∞)

• Rational Numbers -: 

          Rational numbers are those numbers  that can be written or represented in the form of p/q, where q≠0.

Examples -:  -1/8 , 3/9 , 8/2, 8/5 , -5/6 ……

• Irrational Numbers -:  Irrational numbers are those numbers which are not rational and cannot be represented in the form of p/q.

 Examples. -:  Irrational numbers are non-terminating and non-repeating in nature like √2.

Examples of Real Numbers –

Example 1 -: Find five rational numbers between ½ and 3/5.

Solution -:  We will make the denominator same for both the given rational number (1 × 5)/(2 × 5) = 5/10  and (3 × 2)/(5 × 2) = 6/10

Now, multiply both the numerator and denominator of both the rational number by 6, 

(5 × 6)/(10 × 6) = 30/60  and (6 × 6)/(10 × 6) = 36/60

Five rational numbers between ½ = 30/60 and 3/5 = 36/60 are -: 

31/60, 32/60, 33/60, 34/60, 35/60.  

Example 2  -: Write the decimal equivalent of the following:

1. ¼  (ii) 5/8 (iii) 3/2

Solution -:  (i) ¼ = (1 × 25)/(4 × 25) = 25/100 = 0.25

2. 5/8 = (5 × 125)/(8 × 125) = 625/1000 = 0.625

3. 3/2 = (3 × 5)/(2 × 5) = 15/10 = 1.5

Facts of Real Numbers 

•  The addition and the subtraction of a rational number and an irrational number is irrational.

• The multiplication or quotient of a non-zero rational number with an irrational number is irrational.

• If we perform addition, subtraction, multiplication or division of two irrational numbers then the result may be rational or irrational.

• Some numbers have more than one real nth root.

• For instance 36 has two real roots 6 and -6.

• The real numbers include or consist of integers, rational, and irrational numbers.

• The number line contains all the real numbers and nothing else.

• Every real number has a decimal representation.

• Real numbers can do arithmetic operations.

• There are numbers that are not real (imaginary, complex).

Properties of Real Number-

• Commutative Property

It states that if  x and y are the numbers, then the general form will be x + y= y + x for  and  x.y = y.x for multiplication.

• Addition: x+ y = y + x. 

• Multiplication: x × y = y × x . 

• Associative Property 

If we have the numbers x,  and z. The general form will be x + (y + z) = (x + y) + z for addition(x y) z = x (y z) for multiplication.

• Distributive Property 

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

              m (n + r) = m n + m r and (m + n) r = m r + n r.

• Identity Property 

Here  the  additive and multiplicative identities -: 

• For addition: x + 0 = x. (0 is the additive identity)

• For multiplication: n × 1 = 1 × n = n. (1 is the multiplicative identity)

  Conclusion 

  Real numbers are defined as the aggregate of rational and irrational numbers, within the number system. In general, all the mathematics operations may be achieved on these numbers and they can be represented within the number  line, also. We have discussed facts related to the real numbers which should be kept in mind. We have also discussed the examples related to real numbers to understand the concept in a better way. We have also gone through the properties of the real numbers. The properties of real numbers are the identity property, distributive property, associative property and commutative property.