Transcendental numbers

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Transcendental numbers are irrational numbers which are not algebraic numbers, i.e. they are not a solution of some polynomial equation of any finite degree (they transcend the algebraic numbers, so to speak). “Most” irrational numbers are transcendental (an uncountable infinity) while “few” irrational numbers are algebraic numbers (a countable infinity).

The transcendence of a given number is much harder to prove than the irrationality of said number. In 1882, Ferdinand von Lindemann published a proof that the number

π

is transcendental. He first showed that

e

to any nonzero algebraic power is transcendental, and since

e i π =  − 1

is algebraic (see Euler’s identity),

i π

is therefore transcendental and

π

must be transcendental. Although the irrationality of Apéry’s constant was proved in the late 20^th century, its transcendence is still an open problem. Whether or not the Euler–Mascheroni constant is transcendental or at least irrational is another open problem.

References

• Ivan Niven, Numbers: Rational and Irrational. New York: Random House for Yale University (YEAR). ^{(add YEAR)}