Difference Between Rational and Irrational Numbers

Mathematics is nothing but a game of numbers. A number is an arithmetical value that can either be an object, word or symbol representing a quantity that has multiple implications in counting, measurements, labelling etc. Numbers can either be integers, whole numbers, natural numbers, real numbers. or complex numbers. Real numbers are further categorized into rational and irrational numbers. Rational numbers are those numbers that are integers and can be expressed in the form of x/y where both numerator and denominator are integers whereas irrational numbers are those numbers which cannot be expressed in a fraction. In this article, we will discuss rational numbers, irrational numbers, Rational and irrational numbers examples, the difference between irrational and rational numbers etc.

Rational Numbers

The term ratio came from the word ratio which means the comparison of any two quantities and represented in the simpler form of a fraction. A number is considered as a rational number if it can be expressed in the form of a/b where both a (numerator) and b(denominator) are integers. The denominator of a rational number is a natural number(a non-zero number). Integers, fractions including mixed fraction, recurring decimals, finite decimals etc all come under the category of rational numbers.

Irrational Numbers

A number is considered as an irrational number if it cannot be able to simply further to any fraction of a natural number and an integer. The decimal expansion of irrational numbers is neither finite nor recurring. Irrational numbers include surds and special numbers such as π. The most common form of an irrational number is pi (π). A surd is a non-perfect square or cube which cannot be simplified further to remove square root or cube root.

Rational and Irrational Numbers Examples

Some of the examples of rational numbers

  • Number 4 can be written in the form of 4/1 where 4 and 1 both are integers.

  • 0.25 can also be written as 1/4, or 25/100 and all terminating decimals are rational numbers.

  • √64 is a rational number, as it can be simplified further to 8, which is also the quotient of 8/1.

  • 0.888888 is a rational number because it is recurring in nature.

Some of the Examples of Irrational Numbers

  • 3/0 is an irrational number, with the denominator equals to  zero.

  • π is an irrational number that  has value 3.142 and it is non-recurring and non terminating in nature.

  • √3 is an irrational number, as it cannot be able to simplify further.

  • 0.21211211 is an irrational number as it is non-recurring and non terminating in nature.

The important difference between rational numbers and irrational numbers are given below in the tabulated form.

What are the Important Differences Between Rational and Irrational Numbers?

Rational Numbers

Irrational Numbers

The numbers that can be represented  as a ratio of two numbers i.e. in the form of a/b is known as rational numbers.

The numbers that cannot be represented  as a ratio of two numbers  i.e. in the form of a/b is known as an irrational number.

The rational number includes only those decimals that  are finite and  are recurring in nature.

The irrational numbers include all those numbers that are non-terminating or  non-recurring in nature.

Rational Numbers consist of numbers that are perfect squares such as 4, 9, 16, 25, etc.

Irrational Numbers consist of surds such as 2, 3, 5, 7 and so on.

Both the numerator and denominator of rational numbers are whole numbers, in which the denominator of rational numbers is not equivalent  to zero.

Irrational numbers cannot be represented in fractional form.

Example: 5/3 = 1.66, 1/ 7 =0.1428..

Example: √7, √17


How to Classify Rational and Irrational Numbers?

Let us now study how to identify rational and irrational numbers on the basis of the below examples.

As we know, the rational numbers can be expressed in fraction and it includes all integers, fractions, and repeating decimals.

Rational numbers can be identified with the below conditions:

  • It is expressed in the form of a/b, where b≠0.

  • The ratio of a/b can further be simplified and illustrated in the decimal form.

Irrational numbers are those numbers that are not rational numbers. Irrational numbers can be represented in the decimal form but not in fractions which implies that the irrational numbers cannot be expressed as the ratio of two integers.

rational numbers have infinite non-repeating digits after the decimal point. 

Below are Some Examples of Rational and Irrational Numbers.

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Solved Examples

1. Find any 4 Rational Numbers Between 2/5 and 1/2.

Solution: To find the 5 rational numbers between -⅖ and ½, we will first make the denominator similar.

Hence, -⅖ = (2 * 10) /(5 * 10)  = 20/50

And, ½ = (1* 25) /(2* 25) = 25/50

 4 rational number between ⅖ and ½. =  5 rational numbers between 20/50 and 25/50.

Hence, the 4 rational numbers between -⅖ and ½ are 21/50,22/50, 23/50, and 24/50.

2. Which of the Numbers Given Below is not an Irrational Number?

\[\sqrt{7}\] , \[\sqrt{5}\] , \[\sqrt{16}\] , \[\sqrt{11}\] 

Solution: 16 is a perfect square i.e. \[\sqrt{16}\] =  4 which is a rational number

As we know the square root of prime numbers are irrational numbers. 7, 5 , and 11 are prime numbers.  Hence, the only number which is not an irrational number is \[\sqrt{16}\].

Quiz Time

1. Which of the Following Numbers is Irrational?

  1. 21/99

  2. \[\sqrt{100}\]

  3. \[\sqrt{36/3}\]

  4. 2/94

2. Is the Square Root of 225

  1. Rational number

  2. Irrational number

3. Which of the Following Numbers is Rational?

  1. ¼

  2. 3.7

  3. .25

  4. 1.2314

4. 9.0 is a

  1.  Rational number

  2.  Irrational number

Facts

  • Hippassus introduced irrational numbers when attempting to write the square root of 2 in the fraction form ( using geometry, it is thought). He proved that the square root of 2 cannot be written in the fraction form, so it is irrational.

FAQ (Frequently Asked Questions)

1. What are the Rules of Rational and Irrational Numbers?

Here are some rules that rely on mathematical operations such as addition and multiplication carried out on rational numbers and irrational numbers.

Rule 1: The addition of two rational numbers is always a rational number.

Example: 1/2 + 1/3 = (3+2)/6 = 5/6

Rule 2: The multiplication of two rational numbers is  always a rational number

Example: 1/4 x 1/3 = 1/12

Rule 3: The addition of two irrational numbers is not always irrational.

Example: √3+√3 = 3√3 is irrational

2+2√5+(-2√5) = 2  is rational

Rule 4: The multiplication of two irrational numbers is not always irrational.

Example: √2 x √3 = √6 (Irrational number)

√2 x √2 = √4 = 2 (Rational number)

2. What are Some Real-life Examples of Rational and Irrational Numbers?

Some of the real-life examples where rational numbers can be sued are

  1. Taxes can be represented in the form of fraction.

  2. When we divide the whole pizza into two or more than 2 parts.

  3. When you have completed half of your risk such as 50% of work is completed.

  4. Hockey players use rational numbers to represent their goals.

Basically, irrational numbers are used only in Mathematics, but it can also be used in real-world scenarios. Irrational numbers enable us to develop theories with important concepts such as derivatives, integrals, the multiple results of analytical geometry, the trigonometry rules, etc. These theories are used to solve real-world problems.