## 5. Math Puzzle 10/65

The German mathematician Gottfried Wilhelm Leibniz once managed to show that 4 equals 5. Of course, there is a catch, as the proof is not entirely accurate.

Who can find the error?

$$
\begin{aligned}
& 16-36=25-45 \quad \left\lvert\,+\frac{81}{4}\right. \\
& 16-36+\frac{81}{4}=25-45+\frac{81}{4}
\end{aligned}
$$

Both sides are transformed:

$$
\left(4-\frac{9}{2}\right)^{2}=\left(5-\frac{9}{2}\right)^{2}
$$

The square root is taken, and we get

$$
\begin{gathered}
\left.4-\frac{9}{2}=5-\frac{9}{2} \quad \right\rvert\,+\frac{9}{2} \\
4=5
\end{gathered}
$$
Every square has two roots, namely the positive $x_{1}$ and the negative $x_{2}$. In this problem, Leibniz used the solution $x_{1}$ on the right side of the equation and the solution $x_{2}$ on the left side. Correctly, with the proper application of the absolute value,

$$
\begin{gathered}
\left(4-\frac{9}{2}\right)^{2}=\left(5-\frac{9}{2}\right)^{2} \\
\left.-\left(4-\frac{9}{2}\right)=5-\frac{9}{2} \quad \right\rvert\,+\frac{9}{2} \\
-4+9=5
\end{gathered}
$$