Rational And Irrational Numbers: Differences, Examples

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Rational and irrational numbers are both considered to be real numbers, but they have their own distinctive properties. On the one hand, rational numbers are numbers that can be expressed in the form of fractions of two integers, which means that a number that can be written as a fraction is considered to be a rational number. For instance, ½ and 3/4 are some examples of rational numbers. On the other hand, irrational numbers are numbers that cannot be expressed in the form of fractions of two integers, for instance, the square root of any number.

Let us understand the concept of rational and irrational numbers with the help of this article.

What are Rational Numbers?

“Rational” is derived from the word “ratio,” which means comparing two or more values, which are also known as a fraction; in simpler words, it is the ratio of two integers.

Rational numbers are numbers that can easily be expressed as fractions of positive numbers, negative numbers, or zeros. They can also be written as a/b, where b does not equal zero.

These numbers can be written as a/b, where a and b are integers and b does not equal zero.

For instance, the number 4 can be written as 4/1, 8/2, or even -12/-3.

These fractions can be written as decimals as well; for example, 15/100 means 0.15 as 

5 is the hundredth decimal place, and 1 is the tenth decimal place. This is an example of a terminating decimal.

What are Irrational Numbers?

Real numbers that cannot be expressed in the form of a fraction are called irrational numbers. For instance, if a/b where a and b are integers, b does not equal zero, and a distinctive feature of an irrational number is that the decimal expansion of an irrational number is neither terminating nor repeating.

Irrational numbers are an essential component of the real number system and are used in many different mathematical formulas and computations. In many other areas of mathematics, including geometry and trigonometry, they are utilised. Irrational numbers are distinguished from rational numbers by the decimal form, which does not terminate or repeat. Furthermore, there are several ways to express irrational numbers, including continuing fractions.

Some notable examples of irrational numbers are:

  • Pi is an irrational number, and the value of pi is 3.14159265.
  • Square roots of any number
  • Euler’s number, e, is an irrational number.

Difference between Rational Numbers and Irrational Numbers

There are several differences between rational and irrational numbers, which are classified in the table below.

Rational NumbersIrrational Numbers
Ratio where both numerator and denominator are whole numbersIt is impossible to express irrational numbers  in a ratio of two integers
It includes perfect squaresIt does not include perfect squares.
For rational numbers, the decimal expansion carries out repeating or finite decimals.Decimals that are non-terminating and non-recurring are carried out.

Properties of a Rational Number 

Many properties make rational numbers a key component of mathematics, as they are used in various branches of the subject and even have practical applications in fields such as science, engineering, economics, and finance. These properties are as follows:

  • Closure property: According to this property, when one adds or subtracts two rational numbers, the result will always be a rational number For instance, if we add two rational numbers, say ½ and 4/2 then ½ + 4/2 will also be rational. The same goes if we subtract them both.
  • Closure property under Multiplication and division: When one multiplies or divides two rational numbers, except when they are divided by zero, the result will always be a rational number, For instance, take a/b and c/d as two rational numbers, where b and d are not 0, then a/b x c/d and a/b ÷ c/d are also rational numbers.
  • Commutative Property:  For any two rational numbers, say, a and b, 

a+b = b+a

a x b = b x a

  • Associative Property of Addition and Multiplication: For any two rational numbers say a and b, then,

a + (b + c) = (a + b) + c

a x (b x c) = (a x b) x c

  • Identity Property: 0, which is also a rational number, serves as an additive identity element that serves the same for any rational number, say for example the rational number p, p+0 = 0 +p =p  on the other hand, the number serves as a multiplicative identity element for any rational number say p, p x 1= 1 x p =p.
  • Additive and Multiplicative Inverses: For any rational number, say p there exists an additive and multiplicative inverse.
  • Distributive property: The distributive property states that for any three rational integers, r, s, and t, 𝑟×(𝑠 + 𝑡) = (𝑟× 𝑠) + (𝑟× 𝑡) r×(s+t)=(r×s)+(r×t).

Properties of Irrational Numbers 

Just like rational numbers, irrational numbers also have distinctive properties, some of which are as follows:

  • The addition and subtraction of two or more irrational numbers may or may not be irrational.
  • The multiplication of two or more irrational numbers may or may not be irrational.
  • Since the set of irrational numbers is uncountably limitless, it is not possible to map the set of irrational numbers onto the set of natural numbers
  • The decimals of irrational numbers are non-terminating and non-repeating.

 Examples of rational and irrational numbers

On the one hand, while rational numbers can be used in our day-to-day lives, irrational numbers are mostly confined to mathematics, Some of the examples are as follows: 

Rational Number Examples:

  • Taxes can be shown as fractions.
  • When any food item is divided into two equal parts, say a pizza, it is divided into two or more sections.
  • When a person has done 50% of his or her work,

Irrational Numbers 

Although they can also be utilised in practical situations, these numbers are primarily used in mathematics. These numbers enable us to construct models that incorporate concepts such as trigonometric rules, derivatives, integrals, and many results from analytical geometry. 

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