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What is a real number system in mathematics?

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  • Difficulty Level : Easy
  • Last Updated : 07 Jan, 2022
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The number system includes different types of numbers for example prime numbers, odd numbers, even numbers, rational numbers, whole numbers, etc. These numbers can be expressed in the form of figures as well as words accordingly. For example, the numbers like 40 and 65 expressed in the form of figures can also be written as forty and sixty-five.

A Number system or numeral system is defined as elementary system to express numbers and figures. It is the unique way of representation of numbers in arithmetic and algebraic structure.

Numbers are used in various arithmetic values applicable to carry out various arithmetic operations like addition, subtraction, multiplication, etc which are applicable in daily lives for the purpose of calculation. The value of a number is determined by the digit, its place value in the number, and the base of the number system.

Numbers generally are also known as numerals are the mathematical values used for counting, measurements, labeling, and measuring fundamental quantities.

Real Numbers are the mathematical values or figures used for the purpose measuring or calculating the quantities. It is represented by numerals as 2,4,7, etc. Some examples of numbers are integers, whole numbers, natural numbers, rational and irrational numbers, etc.

What are Real numbers?

Real numbers are the set numbers that do not include any imaginary value. It includes all the positive integers, negative integers, fractions, and decimal values. It is generally denoted by ‘R’.

All the negative and positive integers, decimal and fractional numbers without imaginary numbers are called real numbers. 

Real numbers are represented by the “R” symbol. Real numbers can be explained as the union of both rational and irrational numbers. They can be both negative or positive and are denoted by the symbol “R”. All the decimals, natural numbers, and fractions come under this category. The examples below show the classification of real numerals.


Rational numbers  ⇢   – {5/3 , 0 .63 , -6/5 O.7116 ….}    

Irrational numbers  ⇢  -{√3, √5, √11, √21, π(Pi)}                                            

Integers  ⇢   – {-3, -2,-1,0,1,2 , 3….}

Whole numbers  ⇢   -{ 0,1,2,3,4..}

Natural numbers  ⇢   –  {1,2,3,4….}

Types Of Numbers

There are different types of numbers categorized into sets by the real number system. The types are described below:

Rational numbers: Rational numbers are the numbers that can be expressed as the ratio of two integers. It includes all the integers and can be expressed in terms of fractions or decimals. It is denoted by ‘Q’.

It can be expressed as 4/3 , 0 .83 , -7/5,  O.711

Irrational numbers: Irrational numbers are numbers that cannot be expressed in fractions or ratios of integers. It can be written in decimals and have endless non-repeating digits after the decimal point. It is denoted by ‘P’. 

It can be expressed as √5, √11, √21,

Integers: Integers are the set of numbers including all the positive counting numbers, zero as well as all negative counting numbers which count from negative infinity to positive infinity. The set doesn’t include fractions and decimals. The set of integers is denoted by ‘Z’. 

The set of integers can be represented as Z=………..,-5.-4,-3,-2,-1,0,1,2,3,4,5,………….

Decimal numbers: Any numeral value that consists of a decimal point is a decimal number. 

It can be expressed as 2.5 ,0.567, 9.08 etc. 


Whole numbers: Whole numbers are positive numbers including zero, which counts from 0 to infinity. Whole numbers do not include fractions or decimals. The set of whole numbers is represented by ‘W’. 

The set can be represented as W=0,1,2,3,4,5,……………… 


Natural numbers: Natural numbers are the positive numbers that count from 1 to infinity. The set of natural numbers is represented by ‘N’. It is the numbers we generally use for counting. 

The set of natural numbers can be represented as N=1,2,3,4,5,6,7,……………

Sample Examples

Example 1:  What will be the number if we Add  two irrational numbers √7 and √5


(√7 + √5) answer will be an irrational number.

Example 2: if we Multiply two irrational numbers √4 and √9.


√4 × √9 

= 2 x 3 

 = 6 

Now answer is a rational and whole number.

Example 3: What will be the result if we add rational numbers with irrational numbers 4 and √5?


(4 + √5)

Now answer will be an irrational number.

Example 4: Simplify the following expression: (4 + √3)(6 + √3)


(4 + √3)(6 + √3)

= 24 + 4√3 + 6√3 + 3

= 27 + 10√3

here answer is an irrational number.

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