Real Numbers

Any number that can be found in the real world is a real number. We find numbers everywhere around us. Natural numbers are used for counting objects, rational numbers are used for representing fractions, irrational numbers are used for calculating the square root of a number, integers for measuring temperature, and so on. These different types of numbers make a collection of real numbers. In this lesson, we will learn all about real numbers and their important properties.

Table of Contents

Definition of Real Numbers

Any number that we can think of, except complex numbers, is a real number. The set of real numbers, which is denoted by R, is the union of the set of rational numbers (Q) and the set of irrational numbers ( \(\overline{Q}\)). So, we can write the set of real numbers as, R = Q ∪ \(\overline{Q}\) . This indicates that real numbers include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. For example, 3, 0, 1.5, 3/2, ⎷5, and so on.

Now, which numbers are not real numbers? The numbers that are neither rational nor irrational are not real numbers, like, ⎷-1, 2+3i and -i. These numbers include the set of complex numbers, C.

Observe the following table to understand this better. The table shows the sets of numbers that come under real numbers.

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

Natural Numbers

Whole Numbers

Integers

Rational Numbers

Irrational Numbers

Complex Numbers

Symbol of Real Numbers

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

  • N - Natural numbers
  • W - Whole numbers
  • Z - Integers
  • Q - Rational numbers
  • \(\overline{Q}\) - Irrational numbers

Real Number System

All numbers except complex numbers are real numbers. The real number system has the following five subsets:

  • Counting objects gives the set of natural numbersN = 1, 2, 3, ...
  • The set of natural numbers along with 0 represents the set of whole numbersW = 0, 1, 2, 3, ...
  • Measurement of debts, temperatures, etc., fall under the set of integersZ = ..., -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 ⎷2, shows the set of irrational numbers, ( \(\overline{Q}\))

Among these sets, the sets N, W, and Z are the subsets of Q. The following figure shows the relationship between all the numbers mentioned above.

Real Number System

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.

Rational Numbers

Any number which is defined in the form of a fraction 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, a whole number, a decimal or an integer. For example, 1/2, -2/3, 0.5, 0.333 are rational numbers.

Irrational Numbers

Irrational numbers are the set of real numbers that cannot be expressed in the form of a fraction p/q where p and q are integers and the denominator q is not equal to zero (q≠0.). For example: π (pi) is an irrational number. π = 3.14159265...In this case, the decimal value never ends at any point. Therefore, ⎷2 is an irrational number.

Properties of Real Numbers

Just like the set of natural numbers and integers, the set of real numbers also satisfies the closure property, the associative property, the commutative property and the distributive property. The important properties of real numbers are mentioned below.
  • 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 ∈ R, a + b ∈ R and ab ∈ R

  • Associative Property: The sum or product of any three real numbers remains the same even when the grouping of numbers is changed. The associative property of R is stated as follows: For all a,b,c ∈ R, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c

  • 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 ∈ R,  a + b = b + a and a × b = b × a

  • Distributive Property: The distributive property of multiplication over addition is a × (b + c) = (a × b) + (a × c) and the distributive property of multiplication over subtraction is a × (b - c) = (a × b) - (a × c)

Real Numbers on Number Line

A number line helps you to display real numbers by representing them by a unique point on the line. When we represent a real number by a point, the point is called a coordinate. When we represent the point on a real number line representing a coordinate, the real line is called its graph. Every point on the number line shows a unique real number. Note the following steps to represent real numbers on a number line:

  • Draw a horizontal line with arrows on both ends and mark the number 0 somewhere in the middle. The number 0 is called the origin.
  • Mark an equal length on both sides of the origin and label it with a definite scale.
  • Remember that the positive numbers lie on the right side of the origin and the negative numbers lie on the left side of the origin.

Observe the numbers highlighted on the number line. It shows the real numbers -5/2, 0, 3/2, and 2.

Represent Real Numbers on Number Line


Important Topics

Given below is the list of topics that are closely connected to real numbers. These topics will also give you a glimpse of how such concepts are covered in Cuemath. 


FAQs on Real Numbers

What is the Set of all Real Numbers?

The set of real numbers is a set containing all the rational and irrational numbers. It includes natural numbers (N), whole numbers (W), integers (Z), rational numbers (Q) and irrational numbers ( \(\overline{Q}\)).

How to Represent Real Numbers on Number Line?

Real numbers can be represented on a number line by following the steps given below:

  • Draw a horizontal line with arrows on both ends and mark the number 0 somewhere in the middle. The number 0 is called the origin.
  • Mark an equal length on both sides of the origin and label it with a definite scale.
  • Remember that the positive numbers lie on the right side of the origin and the negative numbers lie on the left side of the origin.

Is the Square Root of a Negative Number a Real Number?

No, the square root of a negative number is not a real number. For example, ⎷-2 is not a real number. However, if the number inside the ⎷ symbol is positive, then it is a real number.

Is 0 a Real Number?

Yes, 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.

Is 9 a Real Number?

Yes, 9 is a real number because it belongs to the set of natural numbers that comes under real numbers.

Which Numbers are Not Real Numbers?

Complex numbers, like ⎷-1, are not real numbers. In other words, the numbers that are neither rational nor irrational, are not real numbers.


Solved Examples

Example 1: Anna was asked to identify the real numbers among the following numbers: ⎷6,  -3,   3.15,  -1/2,  ⎷-5. Can you help her?

Solution:

Among the given numbers, ⎷-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, the real numbers from the list are ⎷6,  -3,  3.15, and -1/2

Example 2: John says ⎷3 is a rational number. Emma says ⎷3 is an irrational number. Who is right?

Solution:

The decimal representation of ⎷3 = 1.732020...

This decimal value neither repeats nor terminates at any point. So, ⎷3 is an irrational number. Therefore,  Emma is right.

Practice Questions

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

 
 
 
 
 
 

 


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