Definition

1. Noun:
   • A real number that cannot be expressed as a ratio of two integers: An irrational number is a real number that cannot be written as a simple fraction (a/b, where a and b are integers and b is not zero). Its decimal representation is non-terminating and non-repeating.

Usage

• The term "irrational number" is used in mathematics to distinguish numbers like π (pi) and √2 (the square root of 2) from rational numbers, which can be expressed as fractions.
• It is typically used as a countable noun (e.g., "an irrational number," "several irrational numbers").

Examples

• Noun:
  • Pi (π) is a famous example of an irrational number.
  • The square root of 2 was one of the first irrational numbers discovered.
  • The set of irrational numbers is uncountable.

Advanced Usage

• "To prove a number is irrational": A common phrase in mathematical discourse, referring to demonstrating that a number cannot be expressed as a fraction of integers.
  • Mathematicians worked for centuries to prove that π is an irrational number.

Variants and Related Words

• Irrational (adj): Describes a number that is not rational. While "irrational" can have a general meaning of "not logical," in mathematics, it specifically refers to this property of numbers.
  • The proof showed the value was irrational.
• Rational number (n): The direct counterpart; a number that be expressed as a ratio of two integers.
  • 1/2 and 0.75 are rational numbers.

Synonyms

• Non-rational number: A less common but technically accurate synonym.
• Incommensurable number: A historical and more specific term, often used to describe lengths that cannot be expressed as a ratio of integers.

Related Phrases

• Algebraic irrational number: An irrational number that is a root of a non-zero polynomial equation with integer coefficients (e.g., √2).
• Transcendental irrational number: An irrational number that is not a root of any such polynomial equation (e.g., π, e).

Noun

1. a real number that cannot be expressed as a rational number