Real Numbers System

Table of Contents

Numbers can be classified on the basis of

What are Real Numbers?

Rational and irrational numbers combine to form real numbers. These numbers can be positive as well as negative and are detonated by the symbol- “R”. In this blog we will understand the real numbers definition and examples. 

Real Number System

To put it another way, a real number is any number you can think of, excluding complex numbers. Examples of real numbers include 3, 0, 1.5, 3/2, 5, and so forth. The numbers that may be stated in the form of p/q, where q0, are considered rational numbers.

An integer is a whole number that is not a fraction and can be positive, negative, or zero. Whole numbers are the sum of all natural numbers plus zero.

Natural numbers start from 1 and go all the way to infinity. Irrational numbers cannot be expressed as p/q, are non-terminating and non-repeating.

Real Numbers Examples

A list of real numbers is a mathematical object that represents a sequence of numbers. The set of all real numbers is usually denoted by the symbol R, and so the list (5, 2, 4) would be written as 5, 2, 4 ∈ R.

What is the Difference Between an Integer and a Real Number?

Real Numbers


Real numbers consist of rational numbers, integers, whole numbers, natural numbers and irrational numbers. 

Integers only consist of positive and negative whole numbers and natural numbers. 

Fractions like rational and irrational numbers can be included in real numbers. 

Fractions, however, cannot be integers.

Real numbers can be used to prove the Archimedean Property, which holds that there exists a natural number that is equal to or greater than every real number. 

In contrast, integers are not subject to the Archimedean Property.

Real numbers cannot be counted as a set. 

Integers can be counted as a set. 

Real numbers also include decimal numbers. 

There are no decimals in integers. 

Real numbers are denoted by “R.”

A group of integers is denoted by “Z”. 

What are the Properties of Real Numbers?

Following are the four main features of real numbers.

  1. Commutative Property
  2. Associative Property
  3. Distributive Property
  4. Identity Property

Assume that there are three real numbers, “a, b, and c.” Then, as illustrated below, a, b, and c can be used to describe the following properties:

1. Commutative Property

If a and b are the numbers then; 

Addition: a + b = b + a; for example, 2 + 3 = 3 + 2. 

Multiplication: a × b = b × a; for example, 2 × 3 = 3 × 2.

2. Associative Property

If the numbers are a, b, and c, then the standard form of addiction is a + (b + c) = (a + b) + c; for example, the additive associative property is 1 + (2 + 3) = (1 + 2) + 3.

Multiplication: (ab) c = a (bc); for example, multiplicative associative property is (2 × 3) 4 = 2 (3 × 4).

3. Distributive Property

If a, b and c are the numbers then; 

a (b + c) = ab + ac and (a + b) c = ac + bc.

Example of distributive property is: 5(2 + 3) = 5 × 2 + 5 × 3 = 25

4. Identity Property

If a is the number then; 

Addition: a + 0 = a. (0 is the additive identity)

Multiplication: a × 1 = 1 × a = a. (1 is the multiplicative identity)

How to Represent Real Numbers in Various Formats?

Real numbers can be represented on a number line. All real numbers can be shown on the number line, including whole numbers, integers, rationals, and irrationals.

How to Add and Subtract Real Numbers?


(-3/7) + (-6/7) 

= 3/7 + 6/7

= 9/7

= (-9/7)

27.832 + (-3.06)

= 27.832 – 3.06 (different signs we will subtract)

= 24.772 (larger number is positive)

(-2) – (-3) 

= -2 + 3 (different signs we will subtract)

= 1


38 – (-23)

= 38 + 23

= 61


How to Multiply and Divide Real Numbers?


How to Convert from Decimal to Fractional Form?

How to change a decimal into a fraction. 

The specified decimal should first be written as a ratio (p/q), with one as the denominator.

Divide the denominator by multiples of 10 for each decimal point in order to convert the decimal to a whole number. (Divide by 100 if there are two numbers following the decimal point.)

Simply the outcome portion.



= 75/100

= 3/4

How to Convert from Fraction to Decimal Form?

We must divide the denominator by the numerator to convert a decimal to a fraction.

We must increase the numerator if the denominator is greater than the numerator.

By adding a 0 next to the numerator, we must make the numerator greater than the denominator.

It is also crucial to add a decimal (.) after the 0 in the quotient part before beginning the division.

Continue until either the residual is 0, or the quotient has at least three decimal places.



= 30 / 4 

= 0.75

How do Real Numbers Affect our Daily Lives?

1. Food Purchases

The supermarket is a great place to practice your math skills. You use mathematical ideas like multiplication, estimating, and percentages every time you go to the grocery shop. Let’s say you want to calculate the cost of your groceries—all you have to do is multiply the price per unit by the number of units in each product! Plus, percentage discounts might incentivize you to buy things that are on sale.

2. Cooking and Baking

Baking and cooking are a lot of fun and can be quite satisfying as both of them are a perfect combination of science and mathematics.

Recipes are merely elaborate task lists or mathematical algorithms that must be followed step by step to produce an excellent final product. 

3. Traveling

When you’re traveling, you might not be aware of how often you use math. But if you pay attention, you’ll see that it’s always with you.

When driving anywhere, for example, you must calculate how much fuel will be needed and plan your route accordingly.

4. Build Things

Calculating lengths, widths, and angles to create a house or flat-pack furniture requires math. Did you know that math is also needed for things like measuring the cost of a project and putting it all together when it’s finished?

5. Sports

Teenagers who want to be successful athletes could benefit from learning some geometry and trigonometry.

They can use it to figure out the most effective strategy for hitting a ball, making a basket, or running a lap of the track.

6. Astronomy

Mathematics has always been at the forefront of space exploration. Many of the most well-known mathematicians and vice versa have also been involved in astronomy. From when the space shuttle enters Earth’s atmosphere to how an astronaut drives a spacecraft, there are precise mathematical equations that power it.

Real Numbers System Questions


Yes, real numbers are integers thus they can be negative such as -4, 15, -⅘. 

An endless decimal representation is a possibility that can be used to determine any real integer, and an integer is also a real number. Thus the set of real numbers is uncountable.


Richard Dedekind, a mathematician, was the first to define real numbers.

The number 0 may or may not be considered a natural number, but it is an integer, and hence a rational number and a real number. The number 0 is neither positive nor negative, and is usually displayed as the central number in a number line, so it can be said that 0 is the first real number.


Numbers are the foundation of our day and mark its conclusion because they have the power to alter our lives, numbers dominate every minute of our existence.

Math stimulates the use of logic, critical analysis, creativity, abstract or spatial thinking, problem-solving abilities, and even excellent communication skills, all of which help us to stay organized and avoid chaos in our lives.

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