Matej Balog

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Theorem. $i^2 = 1$

Proof. We have $-1 = -1$, which can be written as \begin{equation} \frac{1}{-1} = \frac{-1}{1} \end{equation} Taking square roots and simplifying yields \begin{equation} \sqrt{\frac{1}{-1}} = \sqrt{\frac{-1}{1}} \end{equation} \begin{equation} \frac{\sqrt{1}}{\sqrt{-1}} = \frac{\sqrt{-1}}{\sqrt{1}} \end{equation} \begin{equation} \frac{1}{i} = \frac{i}{1} \end{equation} Multiplying both sides by the non-zero $i$ gives $1 = i^2$, as required.

Theorem. $0 = 1$

Proof. Integrating by parts \begin{equation} \int{\frac{1}{x \log x} dx} = \int{\frac{1}{x} \frac{1}{\log x} dx} = \log x \frac{1}{\log x} - \int{\log{x} \frac{-1}{\log^2{x}} \frac{1}{x} dx} = 1 + \int{\frac{1}{\log{x}} \frac{1}{x} dx} = 1 + \int{\frac{1}{x \log{x}}} dx \end{equation} Subtracting the integral from both sides gives $0 = 1$ as required.