The difference between Irrational and Non-rational

When used as adjectives, irrational means not rational, whereas non-rational means contrary to reason.

Irrational is also noun with the meaning: a real number that can not be expressed as the quotient of two integers, an irrational number.

check bellow for the other definitions of Irrational and Non-rational

Irrational as an adjective:

Not rational; unfounded or nonsensical.

Examples:

"an irrational decision"
Irrational as an adjective (mathematics, arithmetic, number theory, not comparable):

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

Examples:

"The number π is irrational."

Irrational as a noun:

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

Non-rational as an adjective:

Contrary to reason; lacking an appropriate or sufficient reason; irrational.
Non-rational as an adjective:

Lacking the ability to reason.
Non-rational as an adjective (often, philosophy):

Not within the domain of what can be understood or analyzed by reason; outside the competence of the rules of reason.
Non-rational as an adjective (economics, social sciences, public policy):

Not based on one's own interests; inconsistent with utility maximization.