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.
word-forming element making abstract nouns from adjectives and meaning "condition or quality of being ______," from Middle English -ite, from Old French -ete (Modern French -ité) and directly from Latin -itatem (nominative -itas), suffix denoting state or condition, composed of -i- (from the stem or else a connective) + the common abstract suffix -tas (see -ty (2)).
Roughly, the word in -ity usually means the quality of being what the adjective describes, or concretely an instance of the quality, or collectively all the instances; & the word in -ism means the disposition, or collectively all those who feel it. [Fowler]
<a href="https://www.etymonline.com/word/irrationality">Etymology of irrationality by etymonline</a>
Harper, D. (n.d.). Etymology of irrationality. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/irrationality