Difference Between Rational and Irrational Numbers

By Aina Parasher|Updated : June 20th, 2022

Difference Between Rational and Irrational Numbers: Let us learn the difference between rational and irrational numbers in detail. The positive whole numbers 1, 2, 3, and so on, which are used for counting, are the simplest numbers in the complete number system. Natural numbers have been with us for millennia. The introduction of common fractions such as 1/2, 2/3, 6/7, and so on were prompted by the basic needs of everyday life. These are known as rational numbers. Integers, whole numbers, natural numbers, real numbers, and complex numbers are all examples of numbers. Real numbers are divided into two categories:

  • Rational Numbers
  • Irrational Numbers

Rational numbers are integers that can be stated in the form of p/q (both numerator and denominator are integers), whereas irrational numbers are those that cannot be expressed in a fraction. Further, in this article, we have provided the difference between rational and irrational numbers, along with a brief introduction to rational numbers, and irrational numbers.

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What is the Difference Between Rational and Irrational Numbers?

The main difference between rational and irrational numbers is that a number that may be written in a ratio of two integers is known as a rational number whereas, a number that cannot be represented in the form of a ratio of two integers is called an irrational number. Further, let us look at the various differences mentioned in the table below.

Key Difference Between Rational and Irrational Numbers

Rational NumbersIrrational Numbers
Rational numbers can be expressed in the form of p/q.Irrational numbers cannot be expressed in the form of p/q.
Both the numerator and denominator are full numbers in rational numbers, and the denominator is not zero.A fraction cannot be written for an irrational number.
Rational numbers include perfect squares like 9, 16, 25, etc.Irrational numbers include surds like 2, 3, 5, 7, etc.
Only finite and repeating decimals are included in the rational number. Irrational numbers, are those whose decimal expansion is limitless, non-repetitive, and has no pattern.

What are Rational and Irrational Numbers?

If a number can be expressed as a fraction with both the numerator and denominator being integers and the denominator being a natural number, it is said to be rational (a non-zero number). All rational numbers are integers, fractions, including mixed fractions, recurring decimals, finite decimals, and so on.

When a number cannot be reduced to any fraction of an integer or a natural number, it is said to be an irrational number. It can also be interpreted as an irrational number. The irrational number's decimal expansion is neither finite nor recurrent. Surds and unusual numbers such as pi and e are examples of irrational numbers.

You can also check the number system here.

How Can you Tell the Difference Between Rational and Irrational Numbers?

All integers, fractions, and repeating decimals are rational numbers, and they can be stated in fractions. The following conditions can be used to identify rational numbers:

  • It's written as a/b, where b≠0.
  • The a/b ratio can also be simplified and represented in decimal form.

Numbers that are not reasonable are referred to as irrational. Irrational numbers can be expressed in decimal form but not in fractions, implying that they cannot be expressed as a ratio of two integers.

After the decimal point, rational numbers have an endless amount of non-repeating digits.

Examples of Rational and Irrational Numbers

Following the discussion of the difference between rational and irrational numbers. Let's look at some examples of these two to understand the concept of rational and irrational numbers completely.

  • 6 - Rational number, terminating, and non-repeating in nature.
  • 4/5 - Rational number, in the form p/q, and q≠0.
  • √7 - Irrational number, is the square root of a number that is not a perfect square.
  • √16 - Rational number, is the square root of a perfect square and the value is 4.


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FAQs on Difference Between Rational and Irrational Numbers

  • The difference between a rational and irrational number is that a rational number is one that can be written as p/q, where p and q are integers and q is not equal to zero. However, an irrational number cannot be expressed using simple fractions.

  • A number is rational if it is a terminating number or a repeating decimal, such as 1/2 = 0.5.

    Irrational numbers, such as 0.31545673..., are non-terminating and non-repeating decimals.

  • No, rational numbers are not always whole numbers. All numbers that finish or repeat are rational numbers. A whole number is any number that is larger than or equal to zero and does not have a fractional portion. For example, 2.9 is a rational but not a whole number.

  • If a number cannot be written in the form of p/q, then such types of numbers are known as irrational numbers. While the real number system has an infinite number of irrational numbers, the square roots of non-perfect squares, such as the square root of 2, and the constants pi and e are the most often utilized in mathematics.

  • 16 is a perfect square, so √16 is the answer.

    = 4, which is a rational number.

    The square root of prime numbers is an irrational number, as we all know. Prime numbers include 7, 5, and 11. As a result, the only number that is not irrational is 16.

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