What is the difference between rational and irrational numbers?

Rational and irrational numbers refer to mathematical concepts that confuse many students. These numbers help to prove and applied in many structural changes.

The lesson provides insights into the difference between rational and irrational numbers in a tabular form. The distinction helps to make their application easier.

**What Are Rational Numbers?**

Rational numbers are those numbers that can be an integer or expressed as a fraction such as p/q form. The nature of the numbers is finite or recurring. Examples of rational numbers are 1/9, 7, √16, 0.5 and 0.33333

**What Are Irrational Numbers?**

Irrational numbers are those numbers that cannot be simplified into a fraction of integers and a natural number. The decimal expansion of such numbers is neither finite nor recurring.

The common examples of irrational numbers are surds and special numbers like π. Examples of irrational numbers are √2, √7/5, 3/0, π, and 0.3131131113.

**Comparison Chart: Rational Vs Irrational Numbers**

Basic Terms |
Rational Numbers |
Irrational Numbers |

Meaning | Refers to numbers which can be expressed as a ratio of two integers | This refers to numbers that cannot be expressed as a ratio of two integers. |

Decimal expansion | Finite and recurring | Non-termination and non-repeating |

Nature | Perfect squares | Surds |

Fraction | Can be expressed as a fraction | Cannot be expressed as a fraction |

Examples | 3/2 = 1.5, 1/ 6 =0.1666… | √5, √11 |

**Core Differences between Rational and Irrational Numbers**

- Rational numbers are those that can be expressed as a ratio of two integers whereas irrational numbers are those which cannot be expressed as a ratio of two integers.
- Rational numbers can be written in the form of a fraction whereas irrational numbers cannot be written in the form of fractions.
- Rational numbers comprise of those perfect squares whereas irrational numbers comprise of surds.
- The decimal expansion of rational numbers is either finite or recurring while that of an irrational number is either non-terminating or non-repeating
- Examples of rational numbers are 3/2 = 1.5, 1/ 6 =0.1666… while those of irrational numbers are √5, √11

**Similarities between Rational and Irrational Numbers**

- Both belong to real numbers
- There exist rational numbers between any two rational numbers similarly there exist irrational numbers between any two irrational numbers.
- The sum of two rational numbers is a rational number and the sum of two irrational numbers is an irrational number.
- The difference between two rational numbers is a rational number and the difference of two irrational numbers is an irrational number.

**Comparison Video**

**Summary**

The core difference between rational and irrational numbers is that rational numbers are perfect square whereas irrational numbers are surds.