rational

Pluralrationals

1. 1
   adjective

   Mentally sound and balanced, showing good sense and reasonableness.

   Despite the frightening news, she remained rational and made careful decisions.

   A rational person weighs the facts before reacting to them.

   Similarsane, sensible, reasonable, level-headed, lucid

   Oppositeirrational, unreasonable

   Note

   Here the word describes a mind in working order rather than an argument that holds together, so it attaches to people and their behavior: rational conduct, a rational person. Clinical writing used it for lucidity. C. Handfield Jones, reporting a fever case in 1867, describes a patient as "quite conscious and rational at times." Compare the "logically sound" meaning, which judges statements, not the people who make them.

2. 2
   adjective

   Having the capacity to reason and draw conclusions.

   Philosophers long described human beings as rational animals.

   A rational creature can weigh options before choosing among them.

   Similarintelligent

   Oppositeirrational

   Note

   The root idea sits closest here: Latin rationalis, "of or belonging to reason," from ratio, which meant both "reason" and "calculation." That double meaning is why one adjective serves both philosophers and mathematicians. Here it names a capacity, so it pairs with kinds of being, as in a rational creature or rational beings. Mark Sainsbury's Logical Forms (2001) uses it to admit that we are "not completely rational beings."

3. 3
   adjective

   Sound in its logic, containing nothing self-contradictory or absurd.

   The committee accepted his proposal because it was rational and well structured.

   A rational explanation should hold together without contradicting itself.

   Similarlogical, reasonable, sound, coherent, sensible

   Oppositeirrational, illogical, absurd

4. 4
   adjective

   In mathematics, describing a number that equals some integer divided by a nonzero integer.

   Every integer is rational, since it can be written over a denominator of one.

   The value 0.75 is rational because it equals three-quarters.

   Oppositeirrational

   Note

   Mathematics takes the word back to the "calculation" side of Latin ratio: such a number can be written as a fraction p/q with q nonzero. Its decimal expansion either terminates or eventually repeats, and its opposite is irrational, as with √2. Unlike the senses about reasoning, this one has no degrees: no number is more rational than another.

5. 5
   nouncountable

   A number that can be expressed as the quotient of two integers.

   Between any two rationals there is always another.

   Every integer is also a rational.

   Oppositeirrational

   Note

   The noun is the adjective with its noun dropped: rational number is shortened to rational, plural rationals. It behaves like integer or real and pluralizes freely, as in "the quotient of two rationals is again a rational," provided the divisor is not zero. The rationals are countable, yet they are dense on the real line, so between any two of them there always lies another.

6. 6
   adjective

   Of an algebraic expression or function, able to be expressed as the ratio of two polynomials.

   Engineers often model systems with rational functions like (x²+1)/(x−3).

   The expression is rational because it is a ratio of two polynomials.

   Note

   The algebraic step up from the number sense: polynomials take the place of integers. A rational function such as (x²+1)/(x−3) is defined wherever its denominator does not vanish and undefined at that denominator's roots, which is why such functions can break at isolated points. The usual collocations are rational function and rational expression.

7. 7
   adjective

   Of a variety, birationally equivalent to projective space; roughly, geometrically simple almost everywhere.

   Every smooth conic over an algebraically closed field is rational.

   Proving that a given variety is rational can be extremely difficult.

   Note

   What earns a variety this label is a parametrization: away from some smaller exceptional set, its points can be described by rational functions of ordinary coordinates, just as the points of projective space can. That is the content of "birationally equivalent." The informal phrase "simple almost everywhere" says the same thing, and deciding whether a given variety qualifies is often very difficult.

8. 8
   adjective

   In algebraic geometry, describing a map from one variety to another that need not be defined everywhere but behaves as a morphism on an open part of its domain.

   A rational map need not be defined at every point of its domain.

   Projection from a point gives a rational map to the plane.

   Note

   Most often met in the phrase rational map, which may be undefined at some points. Such a map is a morphism on a dense open subset, and the places where it fails form a lower-dimensional locus. These are the geometric counterpart of the zeros of a denominator in a rational function. Projection from a point is the classic case, undefined only at the point itself.

9. 9
   adjective

   Of a point on a variety over a field, whose coordinates all belong to that field.

   Finding rational points on elliptic curves is a central problem in number theory.

   A point is rational when its coordinates lie in the base field.

   Note

   The phrase rational point carries this sense, and a point is rational only relative to a chosen field, never on its own. Over the rationals, it is a point whose coordinates are rational numbers. Over any other field, the coordinates must lie in that field. Points that need the algebraic closure are excluded, so a curve can have infinitely many points yet few or no rational ones.

10. 10
    adjective

    Of a formula, expressing the type, structure, relations, and reactions of a compound; graphic.

    The chemist wrote out the rational formula to display the compound's bonds.

    Older textbooks used rational formulas to show how a compound reacts.

    Similargraphic

    Note

    A chemist's distinction between formulae that list and formulae that show. A rational formula records not just which atoms a compound contains but how they are arranged and bonded, and how the compound tends to react. It is a graphic or structural formula, set against an empirical one that gives only proportions. The usual phrase is rational formula.

11. 11
    adjective

    Expressing or representing a physical object.

    In this scheme the expression is rational, standing for a physical object.

    Physicists distinguished rational symbols, which name objects, from purely abstract ones.

    Note

    A narrow physicist's use that sorts expressions by what they point to: a rational expression stands for an actual object rather than an abstract quantity or relation. The contrast is drawn sharply against writing itself: "a rational table is physical, a written table is neither." Keep it apart from the chemist's formula sense, which concerns structure rather than reference.

12. 12
    nounhistoricalcountable

    The jeweled breastplate worn by the high priests of ancient Israel.

    The high priest fastened the rational across his chest before entering the sanctuary.

    The Rationale of iudgement alſo thou shalt mke with embrodered worke of diuers colours, according to the workmanship of the Ephod of gold, hyacinth, and purple, and ſcarlet twiſe died, and twiſted ſilke.

    1609, The Holie Bible, Douay–Rheims Bible

    Similarbreastplate, pectoral

    Note

    A separate word that only looks like the others. It comes from Old French rational and Medieval Latin rationale, a priestly ornament, and ultimately renders the Greek logeion, or perhaps logion, "oracle," of the Septuagint's Exodus 28. The breastplate, set with twelve engraved stones, was used in seeking divine judgment, hence "the Rationale of iudgement" in The Holie Bible of 1609. It names the historical object only.

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