[Math] Multiplying using reciprocal, addition and subtraction

arithmetic

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)

?

Best Answer

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.

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