The difference between Irrational and Transcendental
When used as nouns, irrational means a real number that can not be expressed as the quotient of two integers, an irrational number, whereas transcendental means a transcendentalist.
When used as adjectives, irrational means not rational, whereas transcendental means concerned with the a priori or intuitive basis of knowledge, independent of experience.
check bellow for the other definitions of Irrational and Transcendental
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Irrational as an adjective:
Not rational; unfounded or nonsensical.
Examples:
"an irrational decision"
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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."
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Irrational as a noun:
A real number that can not be expressed as the quotient of two integers, an irrational number.
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Transcendental as a noun (obsolete):
A transcendentalist.
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Transcendental as a noun (philosophy, metaphysics, Platonism, Christian theology, usually plural):
Any one of the three transcendental properties of being: truth, beauty or goodness, which respectively are the ideals of science, art and religion and the principal subjects of the study of logic, aesthetics and ethics.
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Transcendental as an adjective (philosophy):
Concerned with the a priori or intuitive basis of knowledge, independent of experience.
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Transcendental as an adjective:
Superior; surpassing all others; extraordinary; transcendent.
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Transcendental as an adjective:
Mystical or supernatural.
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Transcendental as an adjective (algebra, number theory, field theory, of a [[number]] or an [[element]] of an [[extension field]]):
Not algebraic (i.e., not the root of any polynomial that has positive degree and rational coefficients).
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Transcendental as an adjective (algebra, field theory, of an [[extension field]]):
That contains elements that are not algebraic.