Rational Number vs Irrational Number: Difference and Comparison

Perfect square numbers are classified as rational numbers. In the case of rational numbers, which can be represented as fractions, there is a concept of numerators and denominators.

The numbers 25, 36, 49, 64, and so on are examples of perfect squares that come under the category of Rational numbers. Irrational Numbers include surds. Surds like 7, 5, 3, 2, and so on are examples of irrational numbers.

Key Takeaways

  1. Rational numbers can be expressed as a fraction with integers as numerators and denominators, whereas irrational numbers cannot be represented as exact fractions.
  2. Rational numbers include integers, fractions, and repeating or terminating decimals, while irrational numbers have non-repeating, non-terminating decimal expansions.
  3. Examples of irrational numbers are the square root of 2 and the mathematical constant pi, while examples of rational numbers are 1/2, -3, and 0.25.

Rational Number vs Irrational Number

Rational numbers are any numbers that can be expressed as a fraction, such as 3/2 or 4.5. Irrational numbers cannot be expressed in fractions, including the decimal expansions of irrational roots. Rational numbers have finite representations, while irrationals go on forever without repeating.

Rational Number vs Irrational Number
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Only those decimals that are characterized by recurring and finite numbers belong to the set of rational numbers. The numbers that are perfect squares come into the category of rational numbers.

Perfect squares that fall within the category of Rational numbers are 25, 36, 49, 64, and so on. Rational numbers can be expressed as fractions.

Rational numbers include 1/9, 7/3, 17/13, and so on. Rational numbers have numerators and denominators because they may be expressed as fractions.

Only nonrecurring and non-terminating numbers are included in the set of irrational numbers. Surds are classified as irrational numbers.

Surds that come into the category of Irrational Numbers are 7, 5, 3, 2, and so on. Irrational numbers can’t be represented as fractions.

Irrational numbers include √7, √23, √17, √5, pi (π), and many others. Irrational numbers have no denominators or numerators since they cannot be represented or expressed as fractions.

Comparison Table

Parameters of ComparisonRational NumberIrrational Number
Numerator-denominator conceptExistsDoes not exist
Depicted asFractionsAnything other than fractions
Consists ofRecurring and finite.Nonrecurring and non-terminating.
InvolvesPerfect SquaresSurds
Examples2/5, 5/9√7, π

What is Rational Number?

The capacity to represent rational numbers as fractions is a property of rational numbers. 5/9, 7/13, 7/3, and so on are all examples of rational numbers.

In the case of rational numbers, which can be expressed as fractions, there is a concept of numerators and denominators.

Only those decimals that are characterized by recurring and finite numbers are included in the set of rational numbers. The numbers that are perfect squares are classified as rational numbers.

25, 36, 49, 64, and so on are some examples of perfect squares that fall within the category of Rational numbers. Any two numbers can be represented in the form of x/y to get the concept of rational numbers for two numbers.

There is a condition where the numerator and denominator are both integers in this case. The denominator, on the other hand, should not be zero.

What is Irrational Number?

Irrational numbers are incapable of being represented as fractions. The digits √23, √17, √5, pi (π), and many others are examples of irrational numbers.

In the case of irrational numbers, there is no idea of denominators or numerators because they cannot be represented or displayed as fractions.

Only those numbers that are nonrecurring and non-terminating are included in the set of irrational numbers. Surds come within the category of Irrational Numbers.

7, 5, 3, 2, and so on are some examples of surds that fall under the category of Irrational Numbers.

The inability of two numbers to be represented in the form of x/y gives rise to the concept of irrational numbers. In this case, both x and y are integers, and y is not equal to zero.

Main Differences Between Rational Number and Irrational Number

  1. The concept of Rational numbers for two numbers can be achieved by representing any two numbers in the Form of x/y. Here there exists a condition where both the numerator and denominators are integers. However, the denominator should not be equal to zero. On the other hand, the concept of irrational numbers can be achieved by the incapability of two numbers to be represented in the form of x/y. Where both x and y are considered as integers and y is not equivalent to zero.
  2. The set of rational numbers club only that set of decimals that are characterized by those numbers which are recurring and finite. On the other hand, the set of irrational numbers clubs only those set of numbers that are characterized as nonrecurring and non-terminating.
  3. Usually, the numbers which are the perfect squares fall under the category of Rational numbers. Some of the examples of perfect squares which fall under the category of Rational numbers are 25, 36, 49, 64, and so on. On the other hand, usually, the numbers which are the surds fall under the category of Irrational Numbers. Some of the examples of surds that fall into the category of Irrational Numbers are 7, 5, 3, 2, and so on.
  4. Rational numbers possess the capability to be represented in the form of fractions. On the other hand, Irrational numbers do not possess the capability to be represented in the form of fractions.
  5. Some of the general examples of rational numbers are 1/9, 7/3, 17/13, etc. On the other hand, some of the general examples of irrational numbers are √7, √23, √17, √5, pi (π), and many more.
  6. There exists a concept of numerators and denominators in the case of Rational numbers, as they can be portrayed in the form of fractions. On the other hand, there does not exist any concept of denominators or numerators in the case of irrational numbers, as they can not be portrayed or depicted in the form of fractions.
References
  1. https://link.springer.com/article/10.1007/BF01273899
  2. https://www.jstor.org/stable/pdf/10.4169/j.ctt19b9mgs.12.pdf

Last Updated : 20 July, 2023

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