late 15c., "not endowed with reason" (of beasts, etc.), from Latin irrationalis/inrationalis "without reason, not rational," from assimilated form of in- "not, opposite of" (see in- (1)) + rationalis "of or belonging to reason, reasonable" (see rational (adj.)).
Meaning "illogical, absurd" is attested from 1640s. Related: Irrationally. The mathematical sense "inexpressible in ordinary numbers" is from late 14c. in English, from use of the Latin word as a translation of Greek alogon in Euclid.