Illogical vs. Irrational

Contrary to logic; lacking sense or sound reasoning.

Example: ‘I received an illogical reply and that left me standing there feeling confused.’;

Ignorant or negligent of the rules of logic or correct reasoning; as, an illogical disputant; contrary of the rules of logic or sound reasoning; as, an illogical inference.

lacking in correct logical relation

lacking orderly continuity;

Example: ‘a confused set of instructions’; ‘a confused dream about the end of the world’; ‘disconnected fragments of a story’; ‘scattered thoughts’;

Not rational; unfounded or nonsensical.

Example: ‘an irrational decision’;

Of a real number, that cannot be written as the ratio of two integers.

Example: ‘The number π is irrational.’;

A real number that can not be expressed as the quotient of two integers, an irrational number.

Not rational; void of reason or understanding; as, brutes are irrational animals.

Not according to reason; having no rational basis; clearly contrary to reason; easily disproved by reasoning; absurd; - of assertions and beliefs.

Example: ‘It seemed utterly irrational any longer to maintain it.’;

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

not consistent with or using reason;

Example: ‘irrational fears’; ‘irrational animals’;

real but not expressible as the quotient of two integers;

Example: ‘irrational numbers’;