Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 adj. Not endowed with reason.
 adj. Affected by loss of usual or normal mental clarity; incoherent, as from shock.
 adj. Marked by a lack of accord with reason or sound judgment: an irrational dislike.
 adj. Being a syllable in Greek and Latin prosody whose length does not fit the metric pattern.
 adj. Being a metric foot containing such a syllable.
 adj. Mathematics Of or relating to an irrational number.
 n. Mathematics An irrational number.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adj. Not rational; unfounded or nonsensical.
 adj. Of a real number, that cannot be written as the ratio of two integers.
 n. A real number that can not be expressed as the quotient of two integers, an irrational number.
from the GNU version of the Collaborative International Dictionary of English
 adj. Not rational; void of reason or understanding.
 adj. Not according to reason; having no rational basis; clearly contrary to reason; easily disproved by reasoning; absurd;  of assertions and beliefs.
 adj. Not capable of being exactly expressed by an integral number, nor by a ratio of integral numbers; surd;  said especially of roots. See Surd.
from The Century Dictionary and Cyclopedia
 Not rational; without the faculty of reason; void of understanding; unreasoning.
 Without the quality of reason; contrary to reason; illogical; unreasonable: as, irrational motives; an irrational project.
 In mathematics: In arithmetic, not capable of being exactly expressed by a vulgar fraction, proper or improper; surd.
 In translations of Euclid, and cognate writings, at once incommensurable with the assumed unit and not having its square commensurable with that of the unit. This is the peculiar meaning given by Euclid to
α%27λογος , though Plato uses it in sense , above.  In algebra, noting a quantity involving a variable raised to a fractional power; or. in a wider sense, noting a quantity not rational, not a sum of products of constants and of variables into one another or into themselves.
 In Greek prosody, incapable of measurement in terms of the fundamental or primary time or metrical unit.
 n. That which is devoid of reason, as one of the lower animals.
 n. A prime number.
 n. In mathematics, an irrational number, that is, the mark of a cut which separates all rational numbers into two classes, the first having no greatest number, the second no least.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. a real number that cannot be expressed as a rational number
 adj. real but not expressible as the quotient of two integers
 adj. not consistent with or using reason
Etymologies
Examples

They are not irrational numbers according to Wittgenstein's criteria, which define, Wittgenstein interestingly asserts, “precisely what has been meant or looked for under the name ˜irrational number™” (PR Â§191).

Although Greenspan's use of the term "irrational exuberance" has been referred to quite often since the global financial meltdown of 2008, if you really want to see what "irrational exuberance" looks like you should start with this clip from Busby Berkeley's 1933 movie musical, 42nd Street in which Ginger Rogers sings one verse of "We're In The Money" in pig latin:
George Heymont: Follow The Money: Playwrights Tackle The Global Financial Crisis

I put it down to what I call the irrational power of television.

One of my classmates at seminary snorted contemptuously at this notion, which he termed irrational superstition.

He said Irish treasuries should be trading within a percentage point of German rates if what he called irrational market fears could be eased.

So, I must say that even with amendment that Arnold wrote as if he provided talking points for Republicans, that reflected what I call irrational affinity of many "left libertarians" toward Obama.
Why I am Not a Republican, Arnold Kling  EconLog  Library of Economics and Liberty

Chirac's response to concerns about the safety and environmental implications of the tests, which he described as irrational, demonstrated a "lamentable lack of sensitivity", Beahan said.

So, I think it's a very wide range of competitors some who think it looks so attractive, please get all the money long time and some who are just making what I define as irrational but judgment as to interest rate returns.

Former Fed Chairman Alan Greenspan once chided Wall Street for what he termed "irrational exuberance," in a time when stock prices were clearly racing ahead of reality, but I believe the exuberance we have seen this week is both rational and long overdue.

The deviations, which he classifies as irrational optimism or pessimism, accounted for more than 50 percent of actual business cycle fluctuations that occurred during this same period.
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