Home | Math | Difference between Rational and Irrational Numbers

Difference between Rational and Irrational Numbers

May 7, 2023
written by Azhar Ejaz

The key difference between rational and irrational numbers is that rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot be expressed as simple fractions.

image showing difference between rational and irrational number
The key difference between rational and irrational numbers is that rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot be expressed as simple fractions.

Rational number vs irrational numberSummary

Rational NumbersIrrational Numbers
Can be expressed as a simple fraction.Cannot be expressed as a simple fraction.
Have a finite or repeating decimal representation.Have an infinite non-repeating decimal representation.
Are closed under addition, subtraction, multiplication, and division.Are not closed under addition, subtraction, multiplication, and division.
Their sum, difference, product, and quotient are always rational.Their sum, difference, product, and quotient can be either rational or irrational.

5 Differences between Rational and Irrational Numbers  

  1. Rational numbers can be expressed as simple fractions, while irrational numbers cannot be expressed in simple fractions.
  2. Rational numbers have a finite or repeating decimal representation, while irrational numbers have an infinite non-repeating decimal representation.
  3. Rational numbers are closed under addition, subtraction, multiplication, and division, while irrational numbers are not.
  4. The sum, difference, product, and quotient of rational numbers are always rational, while the sum, difference, product, and quotient of irrational numbers can be either rational or irrational.
  5. Examples of rational numbers include 2/3, 5/4, 0.25 (1/4 in decimal form), and 0.6 (3/5 in decimal form). Examples of irrational numbers include √2 (the square root of 2), π (pi), e (Euler’s number), and φ (the golden ratio).