# Difference between Rational and Irrational Numbers

May 7, 2023

**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 number**– **Summary**

Rational Numbers | Irrational 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 **

- Rational numbers can be expressed as simple fractions, while irrational numbers cannot be expressed in simple fractions.
- Rational numbers have a finite or repeating decimal representation, while irrational numbers have an infinite non-repeating decimal representation.
- Rational numbers are closed under addition, subtraction, multiplication, and division, while irrational numbers are not.
- 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.
- 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).

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