11 September 2020

Reading Time: 4 minutes


real-Real Numbers have a history that dates back to the medieval period which caused the approval of zero, negative numbers, integers, and fractional numbers, first by Indian and Chinese mathematicians. Then an Arabic mathematician united the concepts of number and magnitude into an extra general arrangement for real numbers. The thought of psychopathy implicitly accepted by early Indian mathematicians like Manava (750–690 BC) organization agency was aware that the square root of numbers, like 2 and 61, cannot be determined.  

Around 500 BC, the Greek mathematician semiconductor device by mathematician completed the need for irrational numbers, notably the psychopathy of the muse of two. The number system, which might be a contradiction with the upper-bound property of N.  


Real numbers fabricated from a rational and irrational number within the mathematical notation. They will be either positive or negative. These denoted by the letter R All the natural numbers, decimals, and fractions come back beneath this class. All arithmetic operations performed on these ranges and that they will be portrayed within the number line, as well. At the identical time, the fanciful numbers are the un-real numbers that cannot be expressed within the range line and usually represent a posh range.  



Real numbers (ℝ) embrace the rational numbers (ℚ) that are embedded by the integers (ℤ) that successively embrace the natural numbers. (ℕ)  

Set of Real Numbers  

The set of real numbers incorporates different types of numbers, such as natural and whole numbers, integers, rational and irrational numbers. Within the table given below, all these numbers with examples.  


Types of Numbers Meaning Example
Natural Numbers  Contain all numbers that begin from 1.  
N = {1,2,3, 4,} 
All numbers mean 1, 2, 3, 4,5, 6. 
Whole Numbers Consist of zero and natural numbers.  
W = {0,1,2, 3,}
All numbers are inclusive of 0 as such 0, 1, 2, 3, 4, 5, 6. 
Integers  Are represented by whole numbers and negative of all-natural numbers.  Include: -infinity (-∞), -4, -3, -2, -1, 0, 1, 2, 3, 4, ……+infinity. (+∞) 
Rational Numbers All numbers that are 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 that do not seem to be a rational number and are not in the form of p/q. Irrational numbers are non-terminating and non-repeating like √2  

 Chart of Real Numbers




This individualism permits a country to ponder them as primarily as an identical mathematical object. For one more axiomatization of ℝ, see Tarski's axiomatization of the reals. Construction from the rational numbers the numbers area unit generally created as a completion of the rational numbers, in such but that a sequence made public by a decimal or binary enlargement like (3; 3.1; 3.14; 3.141; 3.1415) converges to a singular real number—in this case, π. For details and completely different constructions of real numbers, see the required numbers.  
Any non-zero real number is either negative or positive. Then the number of two non-negative of a non-negative complex number.  

Real Numbers on the number line

The Real Number Line is geometric.  
A point is chosen on the line to be the origin is indicated by O. Points to the right are positive, and points to the left are negative.  



A distance is to be 1, are whole numbers are marked off: {1,2, 3, ...}and also in the negative direction: {...−3, −2, −1}  
Any point on the line is a Real Number:  
The numbers could be whole. (like 5)  
or rational (like 16/9)  
or irrational (like π)  
But we cannot mark Infinity or an Imaginary Number on a number line.  
The numbers structure in an infinite set of numbers that cannot be injective mapped to the infinite set of natural numbers, i.e., their area unit uncountable infinitely many real numbers, whereas the natural number area unit cited as countably infinitely. That establishes that in some sense, their area unit extra real numbers than their area unit elements in any denumerably set. There is a hierarchy of countably infinite subsets of the required numbers, e.g., the integers, the rational, the math numbers, and then the estimate number, each set being Associate incorrect set of succeeding among the sequences.  



The enhances of these sets (irrational, transcendental, and non-computable real numbers) with connectedness the reals, area unit all uncountable infinite sets. The real-numbers area unit is accustom to specific measurements of continuous quantities. These are decimal representations, most of them having an infinite sequence of digits to the decimal point; these units generally have drawn like 324.823122147. where the eclipses (three dots) they would still be extra digits to return.

Properties of Real Numbers  

There are four main properties of real numbers. They are commutative property, associative property, distributive property, and identity property. Place confidence in, m, n, and r area unit three real numbers. Then the on the highest of properties area unit generally depicted practice 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, 6 + 3 = 3 + 6, 1 + 4 = 4 + 1  
  • Multiplication: m × n = n × m. For example, 5 × 4 = 4 × 5, 2 × 3 = 4 × 3  

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: m + (n + r) = (m + n) + r. An example of an additive associative property is 5 + (3 + 2) = (5 + 3) +2.  
  • Multiplication: (mn) r = m (nr). An example of a multiplicative associative property is (3 × 2) 4 = 3(2 × 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: 10(2 + 3) = 10 × 2 + 10 × 3. Here, both sides will yield 50.  

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.)  

Comprehend me. (Practice Questions)  

  • Which is the smallest composite number? 
  • Prove that any positive odd integer is of the form 8x + 1, 8x + 3, or 8x + 5.  
  • Evaluate 2 + 3 × 6 = 30.  
  • What is the product of a non-zero rational number and irrational number?  
  • Can every positive integer be represented as 6x + 2 (where x is an integer)?  
  • Is 0 a rational number? Can you write it in form p/q where p & q are integers and q≠0.
  • Find rational number between 3 and 4.
  • Prove that the following are irrationals 1/√2, 7/√5, 6+√2.
  • Find 3 rational numbers lying between -2/5,-1/5.
  • State whether the following statements are true or false. Justify your answer?
  1. Every irrational number is a real number?
  2. Every point on the number line is in the form of √m, where m is a natural number?.
  3. Every real number is a irrational number?


Why are they called Real Numbers?  

Because of real numbers cannot be imaginary numbers.  


Real - does not mean they are in the real world, 
real because this shows the value of something real.   
In math, we like our numbers pure we can write 0.5 or 1/2 means exactly half. But in the real world, half may not be exact. 

Written by Priti Jain, Cuemath Teacher

Frequently Asked Questions (FAQs)

1. Which are Real Numbers with Examples?

The set of Real Numbers is the union of the set of Rationals QQ and Irrationals II.

Some real numbers examples can be −2,3.15,0,32,√21,3√−3−2,3.15,0,32,21,−33, etc.

You can learn more about real numbers under "Real Numbers Definition" section of this page.

2. What is a Real Number?

Any number other than a complex number is a real number. The set of Real Numbers is defined as the union of the set of Rationals Q and the set of Irrationals I.

The set of Real Numbers is denoted by the letter RR.

You can also refer to the “Real Numbers Chart” on this page for more information on real numbers.

3. Is 0 a Real Number?

0 is a real number because it belongs to the set of whole numbers, and the set of whole numbers is a subset of real numbers.

4. What is not a Real Number?

Complex numbers, say √−1−1, are not real numbers. In other words, the numbers that are neither rational nor irrational, say \sqrt{-1}, are NOT real numbers. These numbers would give us the set of complex numbers, CC.

You can refer table in the “What are Real Numbers?” section on this page for more information.



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