Real Numbers

In this fun lesson, you will learn about real numbers symbol, real numbers system, real numbers definition, and real numbers examples and also check out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

We see numbers everywhere around us.

  • For counting objects, we use natural numbers.
  • For representing fractions, we use rational numbers.
  • For calculating the square root of any number, we use irrational numbers.
  • For measuring temperature, we use integers, and so on.

These different types of numbers make a collection of real numbers.

Any number that can be found in the real world is, literally, a real number. 

In this lesson, you will learn all about real numbers and their important properties. Let's begin!

Lesson Plan 

Definition of Real Numbers

Any number we can think of, except complex numbers, is a real number.

Real Numbers Definition

  • The set of Real Numbers is the union of the set of Rational Numbers \(Q\) and the set of Irrational Numbers \(I \).
  • The set of real numbers is denoted by \(R\).
\(R = Q \cup I\)

This indicates that real numbers include natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

This leads us to the next question; which numbers are NOT real numbers?

The numbers that are neither rational nor irrational, say \(\sqrt{-1}\), are NOT real numbers.

These numbers include the set of complex numbers, \(C\).

We can understand the same thing better from the following table:

Number set Is it a part of the set of real numbers?

Natural Numbers

Whole Numbers

Integers

Rational Numbers

Irrational Numbers

Complex Numbers

Real Numbers Examples

By the above definition of real numbers, some examples of real numbers are \(3,  0,  1.5,  \dfrac{3}{2},  \sqrt{5},  \sqrt[3]{-9}\), etc.


Real Numbers: Symbol

Before we look at the real numbers symbol, let's discuss the symbols used for other types of numbers.

N - Natural numbers

W - Whole numbers

Z - Integers

Q - Rational numbers

I - Irrational numbers

Real numbers consist of both rational and irrational numbers.

Therefore, real numbers are represented as \(R\).

Real Numbers Symbol is \(R\)

Real Numbers System

The real number system \(R\) has the following five subsets within the set of real numbers.

  • Counting objects gives the set of natural numbers:
    \[N=\{1, 2, 3, ...\}\]
  • The set of natural numbers along with \(0\) would give the set of whole numbers:
    \[W=\{0, 1, 2, 3, ...\}\]
  • Dealing with debts, temperatures, etc. give the set of integers:
    \[Z = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}\]
  • If we cut a cake into equal pieces, then we may have a piece that represents a fraction. This is an element of the set of rational numbers, \(Q\)
  • The numbers that are square roots of positive rational numbers, cube roots of rational numbers etc., such as \(\sqrt{2}\), give the set of irrational numbers, \(I\)

Among these sets, the sets \(N, W\), and \(Z\) are the subsets of \(Q\).


What Are the Types of Real Numbers?

There are different types of real numbers.

From the definition of real numbers, we know that the set of real numbers is formed by both rational numbers and irrational numbers. 

Thus, there does NOT exist any real number that is neither rational nor irrational.

It simply means that if we pick up any number from \(R\), it is either rational or irrational.

1. Rational Numbers

Any number which is defined in the form of a fraction \(\frac{p}{q}\) or ratio is called a rational number.

The numerator is represented as \(p\) and the denominator as \(q\), where \(q\) is not equal to zero.

A rational number can be a natural number, whole number, or an integer.

Example : Divide \[\begin{equation}\frac{2}{3}\end{equation}\]

\[\begin{equation}\frac{2}{3}=0.6666=0.67\end{equation}\]

The above decimal value is recurring (repeating).

Thus, we approximate it to 0.67.

2. Irrational Numbers

Irrational numbers are the set of real numbers that cannot be expressed in the form of a fraction \(\frac{p}{q}\) where p and q are integers and the denominator q is not equal to zero (\(q≠0.\))

Example: (pi) is an irrational number.

\[\begin{align}\pi=3 \cdot 14159265\dots\end{align}\]

The decimal value never stops at any point.

\(\sqrt{2}\) is an irrational number.

Real Number Chart

The above explanations are summarized with examples in the following real numbers chart.

real numbers chart

We can use the following simulation to see how the set of real numbers is formed.

Click on the start button to begin the simulation.

 
tips and tricks
Tips and Tricks
  1. All numbers except complex numbers are real numbers.
    A complex number has \(i\) in it.
    For example, \(2+3i, -i\) etc. are complex numbers.

What Are the Properties of Real Numbers?

1. Closure Property:

The sum and product of two real numbers is always a real number.

The closure property of \(R\) is stated as follows:

For all \(a,b \in R\)

\[a + b \in R\\ {\text{  and  }} \\a \times b \in R\]

2. Associative Property:

The sum or product of any three real numbers remains the same though the grouping of numbers is changed.

The associative property of \(R\) is stated as follows:

For all \(a,b,c \in R\)

\[\begin{gathered}
  a + \left( {b + c} \right) = \left( {a + b} \right) + c \\
  {\text{and}} \\
  a \times \left( {b \times c} \right) = \left( {a \times b} \right) \times c \\
\end{gathered} \]

3. Commutative Property:

The sum and the product of two real numbers remain the same even after interchanging the order of the numbers.

The commutative property of \(R\) is stated as follows:

For all \(a,b \in R\)

\[\begin{gathered}
  a + b = b + a \\
  {\text{and}} \\
  a \times b = b \times a \\
\end{gathered} \]

Let us summarise these three properties of real numbers in a table.

Sign Closure Property Associative Property Commutative Property
+ yes yes yes
- no no no
* yes yes yes
/ no no no



4. Distributive Property:

The distributive property of multiplication over addition is

\[\begin{align}a \times (b+c) &= a \times b + a \times c\end{align}  \]

The distributive property of multiplication over subtraction is

\[\begin{align}a \times (b-c) = a \times b - a \times c\end{align}  \]

 
Thinking out of the box
Think Tank
  1. Is \(- \dfrac{3}{2}\) a real number?
  2. Which is a real number, \(\sqrt{4}\) or \(\sqrt{-5}\)?
  3. Is \(R\) closed under subtraction and division?
  4. Is \(R\) associative under subtraction and division?
  5. Is \(R\) commutative under subtraction and division?

Solved Examples

Example 1

 

 

Anna was asked to identify the real numbers among the following numbers:

\[\sqrt{6},  -3,   3.15,  \frac{-1}{2},  \sqrt{-5}\]

Can you help her?

Anna is asked to identify real numbers

Solution

Among the given numbers, \(\sqrt{-5}\) is a complex number. Hence, it cannot be a real number.

The other numbers are either rational or irrational. Thus, they are real numbers.

\(\therefore\) \(\ \sqrt{6},  -3,  3.15 \text{ and }  \frac{-1}{2}\)
Example 2

 

 

John and Emma are solving their math homework on real numbers.

John says \(\sqrt{3}\) is a rational number.

Emma says \(\sqrt{3}\) is an irrational number.

Who is right?

Children doing home work

Solution

\(\sqrt{3} = 1.732020\cdots\)

The decimal value neither repeats nor terminates at any point.

\(\sqrt{3}\) is an irrational number.

\(\therefore\) Emma is right.
Example 3

 

 

The teacher asks Charles to find the rational number between two given rational numbers written on the board.

The rational numbers are \( \dfrac{1}{2} \text{ and } \dfrac{2}{3}\)

Help Charles find the rational number between them.

A teacher is asking a question on rational numbers.

Solution

We know that the average of any two numbers lies between the two numbers.

Let's find the average of the given two rational numbers.

\[ \begin{aligned} \dfrac{ \dfrac{1}{2}+ \dfrac{2}{3}}{2} &= \dfrac{\dfrac{3}{6}+ \dfrac{4}{6}}{2}\\[0.3cm] &= \dfrac{ \left(\dfrac{7}{6} \right)}{2}\\[0.3cm] &= \dfrac{ \left(\dfrac{7}{6} \right)}{ \left(\dfrac{2}{1} \right)}\\[0.3cm] &= \dfrac{7}{6} \times \dfrac{1}{2}\\[0.3cm] &= \dfrac{7}{12} \end{aligned} \]

\(\therefore\) The rational number is \(\dfrac{7}{12}\)

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Let's Summarize

We hope you enjoyed learning about Real numbers with the simulations and practice questions. Now, you will be able to easily solve problems on real numbers symbol, real numbers system, real numbers definition, and real numbers examples.

About Cuemath

At Cuemath, our team of math experts are dedicated to make learning fun for our favorite readers, the students! Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic. Be it problems, online classes, videos, or any other form of relation, it’s the logical thinking and smart learning approach that we at Cuemath believe in.


Frequently Asked Questions (FAQs)

1. 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.

2. Is 9 a real number?

\(9\) 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.

3. Which numbers are not real numbers?

Complex numbers, say \(\sqrt{-1} \), 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, \(C\).

Download Numbers and Number Systems Worksheets
Numbers and Number Systems
grade 9 | Questions Set 2
Numbers and Number Systems
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Numbers and Number Systems
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Numbers and Number Systems
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