What are real numbers? Explain

Real numbers are only a mixture of rational and irrational numbers in the number system. All the calculation operations can be performed with these numbers’ help, and they are represented on the number line. The concepts linked to real numbers are explained in this article in detail.

What are Real Numbers?

Real numbers can be described as the combination of both rational and irrational numbers. They can be both positive as well as negative and are represented by the symbol “R”. All the natural numbers, decimals, and fractions in the number system come under this category.

Set of Real Numbers

The set of real numbers consist of diverse categories, such as natural and whole numbers, integers, rational, irrational numbers, and all others included in the number system.

Properties of Real Numbers

There are four principal properties which comprise commutative property, associative property, distributive property, and identity property. Let’s consider “m, n, and r” as three real numbers.

Commutative Property:

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

  • In Addition we use m + n = n + m. The given example: 5 + 3 = 3 + 5, 2 + 4 = 4 + 2
  • In multiplication we use m × n = n × m. The given example: 5 × 3 = 3 × 5, 2 × 4 = 4 × 2

Associative Property

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


  • In addition the standard form will be as m + (n + r) = (m + n) + r.
  • In multiplication we use (mn) r = m (nr).

Distributive Property

For three real numbers, m, n, and r, which are given. Here the distributive property is interpreted as the following:

m (n + r) = mn + mr as well as (m + n) r = mr + nr.

Example of distributive property is: 5(6 + 3) = 5 × 6 + 5 × 3. Here, both sides will give us the answer 25.

Identity Property

In Real numbers, there are two additive and multiplicative identities.

For addition we use m + 0 = m. (0 is the additive identity here)

For multiplication we use m × 1 = 1 × m = m. (1 is the multiplicative identity here)

Is Zero a Real or an Imaginary Number?

Zero is recognized as both a real and an imaginary number. As we all know, imaginary numbers are the square root of non-positive real numbers in the number system. Here since 0 is also a non-positive number, hence it fulfills the principles of the imaginary number. Whereas 0 is also a rational number, which is determined in a number line and hence is a real number.

Why are they called “Real” Numbers?

Real numbers are named real because they are not imaginary in nature.

The Real Numbers in the past had no name and were thus considered as Imaginary Numbers only. They were termed as “Real” because they are not imaginary in nature. That is the exact and only answer!

Symbol of Real Numbers

Since the collection of real numbers is the set of all rational and irrational numbers, real numbers and are symbolized by the symbol, R. Here is a list of the symbols of the other kinds of numbers.

N – Natural numbers

W – Whole numbers

Z – Integers

Q – Rational numbers

Irrational numbers



Real numbers are a very fundamental concept in the number system. Real numbers help in the main four operations in mathematics. They will help students to master the tricks of mental math as well. To ace the concept of Real numbers, students can take the help of online courses and grade 5 math worksheets that offer helpful guides and worksheets for practice. Cuemath is an online learning platform that can help you understand real numbers in the most interesting way. Cuemath online classes are the best way through which students can explore the core concepts of real numbers.

Written by Crystal Rae

Leave a Reply

Your email address will not be published. Required fields are marked *

Eyelash Serum

Try Careprost Eyelash Serum for Long and Thick Eyelashes

Identity Verification

NFC Identity Verification: A Product of Technological Evolution