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Webster 1913 Edition

Adj.

Incapable of being reduced, or brought into a different state; incapable of restoration to its proper or normal condition;

as, an

irreducible

hernia

(Math.)

Incapable of being reduced to a simpler form of expression;

as, an

irreducible

formula

–

Irˊre-du′ci-ble-ness

Noun.

– –

Irˊre-du′ci-bly

adv.

irreducible ‎(comparative more irreducible, superlative most irreducible)

Not able to be reduced or lessened.
Not able to be brought to a simpler or reduced form.
(mathematics, of a polynomial) Unable to be factorized into polynomials of lower degree, as (x^2 + 1).
(mathematics, of an integer) Unable to be factored into smaller integers; prime.
(topology, of a manifold) Not containing a sphere of codimension 1 that is not the boundary of a ball.
(group theory, of a representation) Impossible to divide further into representations of lower dimension by means of any similarity transformation.

irreducible ‎(plural irreducibles)