Transcendence of Sum or Product of Transcendentals
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Theorem
Let $a$ and $b$ be two transcendental numbers.
Then at least one of $a + b$ and $a \times b$ is transcendental.
Proof
Aiming for a contradiction, suppose $a + b$ and $a \times b$ are both not transcendental.
Hence by definition, they are both algebraic.
Hence, $\left({z - a}\right) \left({z - b}\right)$ is a polynomial with algebraic coefficients.
Therefore, $a$ and $b$ must both be algebraic.
However, this contradicts with the assumption that $a$ and $b$ are both transcendental.
Hence by Proof by Contradiction it must follow that at least one of $a + b$ and $a \times b$ is transcendental.
$\blacksquare$