Is $\mathbb{R^+}$ a field?

Question

I can't seem to violate any of the axioms, if I take the number $0$ to be in $\mathbb{R^+}$. Then the additive and multiplicative identities are in the set, the numbers all have a multiplicative inverse, etc.

However, I was told that I was wrong. What have I overlooked?

Thanks

Answer 1

Hint: What is 10's inverse under addition?

Answer 2

What about the equation $x+7=2$?

Answer 3

@ Everyone: What if we use $\log$ to identify points with whole $\mathbb{R}$ and then do both addition and multiplication there.