Problem #PRU-100344-3

Problem

The Hatter has a peculiar ancient device, which can perform the following three operations: for each $x$ and $y$ it calculates $x+y$, $x-y$ and $\frac{1}{x}$ (for $x\ne 0$).

(a) The Hatter claims that he can square any positive real number using the device by performing not more than 6 operations. How can he do it?

(b) Moreover, the Hatter claims that he can multiply any two positive real numbers with the help of the device by performing not more than 20 operations. Can you show how?

(All intermediate results are allowed to be written down, and can be used in further calculations.)