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In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is e, that is, e raised to the power π. Like both e and π, this constant is a transcendental number. This was first established by Gelfond and may now be considered as an application of the Gelfond–Schneider theorem, noting that
where i is the imaginary unit. Since −i is algebraic but not rational, e is transcendental. The constant was mentioned in Hilbert's seventh problem. A related constant is 2, known as the Gelfond–Schneider constant. The related value π + e is also irrational.
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