How do we know that there are infinitely many transcendental numbers?
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Well, a simple proof would be that any number has an infinite amount of multiples, and the multiple of a transcendental number is still transcendental.
Also, we know that the real numbers are uncountably infinite, but the algebraic numbers are only countably infinite. And since the real numbers are the algebraic numbers *plus* the transcendental numbers, the transcendental numbers would also have to be uncountably infinite.
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