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A transcendental number is a complex number that is not an algebraic number. No nonzero polynomial with coefficients in the integers vanishes at it. Familiar examples are and ; proving their transcendence requires substantially more than proving their irrationality.
There are only countably many integer-coefficient polynomials, and each nonzero one has finitely many roots. Consequently the algebraic numbers are countable, while the real numbers are uncountable; transcendental real numbers therefore exist in abundance. This counting proof establishes existence without identifying a particular example. Every real transcendental number is an irrational number, but shows that the converse fails.
Sometimes mathematicians go a little overboard with their naming.