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In mathematics, a transcendental number is a number (possibly a complex number) which is not algebraic—that is, it is not a root of a nonconstant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e. Though only a few classes of transcendental numbers are known (in part, because it can be extremely difficult to show that a given number is transcendental) transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental, eg the square root of 2 is irrational but is an algebraic number (therefore, not transcendental).
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History
Euler was probably the first person to define transcendental numbers in the modern sense.^{[1]} The name "transcendentals" comes from Leibniz in his 1682 paper where he proved sin x is not an algebraic function of x.^{[2]}^{[3]}
Joseph Liouville first proved the existence of transcendental numbers in 1844,^{[4]} and in 1851 gave the first decimal examples such as the Liouville constant
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