rationals

Definition

  1. Noun (plural form of rational, used in mathematics):

    • Numbers expressible as a ratio: "rationals" refers to rational numbers — any number that can be expressed as the quotient or fraction ( \frac{p}{q} ) of two integers, where ( q \neq 0 ). This includes integers, fractions, and terminating or repeating decimals.
    • Example in context: The set of all rationals is denoted by ( \mathbb{Q} ).

  2. Noun (archaic or rare, from the reference dictionary):

    • A type of women's knee-length trousers: In historical or dialectal use, "rationals" may refer to a garment worn by women, specifically trousers that are fastened or gathered at the knee. This meaning is obsolete in modern English and is not part of standard WordNet definitions.
Usage Examples

  • Mathematical meaning:

    • The class discussed whether the square root of 2 is among the rationals. (They questioned if (\sqrt{2}) can be expressed as a fraction of integers.)
    • All integers are rationals because they can be written as fractions with denominator 1. (For example, 5 = 5/1.)

  • Archaic meaning (rarely used):

    • She wore rationals for her daily work in the fields. (She wore knee-length trousers, a now-uncommon garment.)
Advanced Usage
  • "the rationals" as a set: In mathematics, "the rationals" is a countable, dense subset of real numbers.
    • The rationals are dense in the real line, meaning between any two real numbers there is a rational. (This property is fundamental in analysis.)
Variants and Related Words

  • Rational (adj): based on reason; also, capable of being expressed as a ratio of integers.

    • Her decision was entirely rational. (Based on logic, not emotion.)
    • A rational number is one that can be written as a fraction. (Mathematical sense.)

  • Rationality (n): the quality of being reasonable or logical.

    • His rationality helped solve the complex problem. (His logical thinking.)
Synonyms
  • Mathematical sense: Fractions (though not all fractions are rationals in simplest form), quotients (of integers).
  • Archaic sense: Breeches, knee-breeches (historical trousers).
Phrasal Verbs
  • None directly associated with "rationals." The word "rational" does not form common phrasal verbs.
Related Idioms

  • "To be rational about something": to think logically and sensibly.

    • Try to be rational about the loss; emotions won't help. (Think clearly and reasonably.)

  • "The rational choice": the most logical or sensible option.

    • Given the data, the rational choice was to invest cautiously. (The decision based on reason.)