Definition

1. Noun:
   • A number that can be expressed as the quotient or fraction p/q of two integers, where p (the numerator) and q (the denominator) are integers, and q is not zero. This includes all integers, as any integer n can be written as n/1.

Usage

• The term "rational number" is used in mathematics to describe a specific set of numbers within the real number system. It is used to distinguish numbers that can be expressed as a simple fraction from those that cannot (irrational numbers).
• It is typically used as a countable noun (e.g., "a rational number," "the set of rational numbers").

Examples

• Noun:
  • 1/2, 3/4, and 5 are all examples of a rational number.
  • The decimal expansion of a rational number either terminates or repeats.
  • Zero is a rational number because it can be written as 0/1.

Advanced Usage

• "Dense in the reals": A property of the set of rational numbers, meaning between any two real numbers, there exists a rational number.
  • The fact that rational numbers are dense is a fundamental concept in real analysis.
• "Lowest terms": A rational number is in lowest terms when the greatest common divisor of and is 1.
  • The fraction 2/4 represents a rational number, but it is not in its lowest terms; 1/2 is.

Variants and Related Words

• Rational (adj): Relating to or based on reason or logic. In mathematics, it specifically relates to rational numbers.
  • The solution was reached through a rational process. (general use)
  • π is not a rational number. (mathematical use)
• Irrational number (n): A real number that cannot be expressed as a ratio of two integers (e.g., √2, π).
  • The square root of 2 is a famous irrational number.

Synonyms

• Fraction (when not an integer): A numerical quantity that is not a whole number, expressed as a ratio.
  • The value was expressed as a simple fraction. (Note: While all fractions of integers are rational numbers, the term "fraction" can sometimes be used more broadly.)

Related Phrases and Concepts

• Field of rational numbers: The mathematical set of all rational numbers, often denoted by ℚ.
  • ℚ, the field of rational numbers, is closed under addition, subtraction, multiplication, and division (except by zero).
• Rational approximation: Using a rational number to closely estimate the value of an irrational number.
  • 22/7 is a common rational approximation for π.

Noun

1. an integer or a fraction