Difference Between Rational and Irrational Numbers

Difference Between Rational and Irrational Numbers

The rational numbers are expressed in the form of x/y, where both x and y are integers and y ≠ 0. But irrational numbers can not be represented in such a form. Let us know more about these numbers and their differences.

Rational Numbers

A number is considered a rational number if we can express it in the form of x/y where both x (numerator) and y(denominator) are integers and y ≠ 0. 

It consists of integers, simple fractions, mixed fractions, recurring decimals, finite decimals etc.

Some examples of rational numbers are – 

• We can write the number 8  in the form of 8/1, where 8 and 1 both are integers.

• 0.5 can also be written as 1/2, or 50/100 and all the terminating decimals are rational numbers.

• √49 is a rational number, as we can simplify it further to 7, which is also the quotient of 7/1.

• 0.777777 is a rational number because it is recurring in nature.

Irrational Numbers

An irrational number cannot be expressed in the form of the ratio of two integers. Irrational numbers decimal expansion is non-terminating and non-recurring. 

Surds and special numbers such as π are examples of irrational numbers. Pi (π) is the most common form of an irrational number. A surd is a non-perfect square or cube, and we cannot simplify it further to eliminate square or cube roots.

Some examples of irrational numbers are – 

• 5/0 is an irrational number since the denominator is zero.

• Pi (π) has a value of 3.14159…….., which is non-recurring and non terminating in nature. Therefore, (π) is an irrational number.

• √5 is an irrational number, as we cannot simplify it further.

• 0.03003000300003… is an irrational number since it is non-recurring and non-terminating in nature.

Rational and Irrational Numbers Difference

• Rational numbers can be expressed as a ratio of two numbers (in the form of x/y and y ≠ 0), but irrational numbers can not be defined as a ratio of two integers.
• Rational numbers are finite or are recurring in nature, whereas irrational numbers are non-terminating and non-recurring.
• Perfect squares such as 4, 9, 16, 25, 36 etc., are rational numbers, but non-perfect squares or cubes such as √2, √3, √5 , √7 etc., are irrational numbers.

Examples

1. Which of these numbers is not an irrational number?

√3, √13, √9, √11

9 is a perfect square, i.e. √9 = 3, a rational number.

Square roots of prime numbers are always irrational numbers. 3, 13 and 11 are prime numbers.  Therefore, √9 is the only number that is not irrational.

2. Is √2 a rational or irrational number?

The decimal expansion of √2 = 1.4142135… is infinite because it is non-terminating and non-recurring. And we know that Irrational numbers decimal expansion is non-terminating and non-recurring. That is the reason why √2 is an irrational number.