# 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.

$\blacksquare$