The Collaborative International Dictionary of English v.0.48:

irrational \ir*ra"tion*al\ ([i^]r*r[a^]sh"[u^]n*al), a. [L.
   irrationalis: cf. F. irrationnel. See In- not, and
   Rational.]
   [1913 Webster]
   1. Not rational; void of reason or understanding; as, brutes
      are irrational animals.
      [1913 Webster]

   2. Not according to reason; having no rational basis; clearly
      contrary to reason; easily disproved by reasoning; absurd;
      -- of assertions and beliefs. Hence, of actions: foolish;
      unreasonable.
      [1913 Webster +PJC]

            It seemed utterly irrational any longer to maintain
            it.                                   --I. Taylor.
      [1913 Webster]

   3. (Math.) Not capable of being exactly expressed by an
      integral number, nor by a ratio of integral numbers; surd;
      -- said especially of roots. See Surd.

   Syn: Absurd; foolish; preposterous; unreasonable; senseless.
        See Absurd.
        [1913 Webster]

WordNet (r) 3.0 (2006):

irrational
    adj 1: not consistent with or using reason; "irrational fears";
           "irrational animals" [ant: rational]
    2: real but not expressible as the quotient of two integers;
       "irrational numbers" [ant: rational]
    n 1: a real number that cannot be expressed as a rational number
         [syn: irrational number, irrational]

Moby Thesaurus II by Grady Ward, 1.0:

185 Moby Thesaurus words for "irrational":
   abnormal, absonant, algorismic, algorithmic, aliquot,
   bereft of reason, bigoted, brainless, brainsick, cardinal,
   contradictory, contrary to reason, crackbrained, cracked, crazed,
   crazy, daft, decimal, deluded, demented, deprived of reason,
   deranged, differential, digital, disoriented, distraught, even,
   exponential, extravagant, extreme, extremist, fallacious, fanatic,
   faulty, figural, figurate, figurative, finite, flawed, flighty,
   foolish, fractional, haggard, hallucinated, headless, ignorant,
   ill-advised, ill-considered, ill-contrived, ill-devised,
   ill-gauged, ill-judged, illogical, imaginary, impair, impolitic,
   impossible, imprudent, inadvisable, inauthentic, inconclusive,
   incongruous, inconsequent, inconsequential, inconsiderate,
   inconsistent, indiscreet, inept, inexpedient, infinite,
   injudicious, inordinate, insane, insensate, integral, invalid,
   lackbrained, lean-minded, lean-witted, loco, logarithmic,
   logometric, loose, lunatic, mad, maddened, manic, mazed, mental,
   mentally deficient, meshuggah, mindless, misadvised, misguided,
   moon-struck, myopic, negative, non compos, non compos mentis,
   nonrational, nonscientific, not all there, not bright,
   not following, not right, numeral, numerary, numerative, numeric,
   odd, of little brain, of unsound mind, off, ordinal,
   overenthusiastic, overreligious, overzealous, pair, paralogical,
   pea-brained, perfervid, pin-brained, positive, possible, prime,
   psycho, queer, rabid, radical, rational, real, reasonless,
   reciprocal, reckless, self-annulling, self-contradictory,
   self-refuting, senseless, shortsighted, sick, slackminded,
   slackwitted, sophistic, stark-mad, stark-staring mad, strange,
   submultiple, surd, tetched, thoughtless, touched, transcendental,
   ultrazealous, unadvised, unauthentic, unbalanced, unconnected,
   unconsidered, undiscerning, unforeseeing, ungifted, unhinged,
   unintellectual, unintelligent, unphilosophical, unreasonable,
   unreasoning, unreflecting, unreflective, unsane, unscientific,
   unseeing, unsensible, unsettled, unsound, untalented, unthinking,
   unthoughtful, unwise, wandering, wild-eyed, wild-looking,
   without reason, witless, zealotic