How to prove (-1)*u=-u for all real numbers $u$?

Question

How can we show that additive inverse of a real number equals the number multiplied by -1, i.e. how can we show that $(-1)*u = -u$ for all real numbers $u$?

Answer 1

$-u$ is the unique additive inverse of $u$. By distributivity we have $$(-1)u+u=(-1)u+(1)u=(-1+1)u=0u=0$$ and hence $(-1)u=-u.$ To prove $0u=0$, observe $0u=(0+0)u=0u+0u$ and the additive identity element is unique.

Answer 2

$(-1)*u = (-1)*u + u - u = (-1)*u + (1)*u - u = (-1+1)*u - u = -u$

Answer 3

\begin{align} && (1 + (-1)) &= 0 \\ &\implies& (1+(-1))\cdot u &= 0\cdot u = 0\\ &\implies& u + (-1)\cdot u &= 0\\ &\implies& (-u) + (u + (-1)\cdot u) &= -u\\ &\implies& (-u + u) + (-1)\cdot u &= -u\\ &\implies& (-1)\cdot u &= -u \end{align}