Multiplying using reciprocal, addition and subtraction

Question

Let $a,b\in\mathbb{R}$. How can we compute $a\times b$ using only the following operations only (with any reals) :

• $\frac{1}{*}$ (inverse)
• $*+*$ (addition)
• $*-*$ (subtraction)

?

Answer 1

Build first the square function, as follows.

If $x\neq -1,0$, then $\displaystyle x^2= \frac{1}{\frac{1}{x}-\frac{1}{x+1}}-x$

Squaring $0$ and $-1$ is trivial.

Now, $\displaystyle xy=\frac{(x+y)^2-(x-y)^2}{4}$

Answer 2

I claim it is impossible. Let $a = b = \sqrt{2}$. Then any number you can make with your operations is a rational multiple of $\sqrt{2}$, and in particular you cannot make $ab = 2$.

edit: this answer was given when it was unclear whether the use of arbitrary real constants was allowed. I assumed they were not, but the question has since been edited to indicate they are. I'm keeping this answer anyway because people seem to like it.