Proof that $\frac{1}{2} + \frac{1}{2} = 1$ using just the algebraic properties of $\mathbb R$

Question

Like the title says, can you prove rigorously that $\frac{1}{2} + \frac{1}{2} = 1$ using only the nine field properties of $\mathbb R$? I don't know if addition and multiplication are supposed to be pre-defined, for instance, if $+$ says that $5 + 6 = 11$ automatically without any proof. This is true in common sense math, but the question I'm really wanting to know is are equations like those all just derived solely from those nine algebraic field properties of $\mathbb R$?

Like it seems obvious that $5\cdot0 = 0$, but that's because $0 =5\cdot 0 - 5\cdot0= 5\cdot(0+0)-5\cdot0 = (5\cdot0 + 5\cdot0)-5\cdot0$

$=5\cdot0 + (5\cdot0 - 5\cdot0)=5\cdot0+0=5\cdot0$, using additive identities, additive inverses, distributive laws, and additive associativity. Can a similar proof be formed for $\frac{1}{2} + \frac{1}{2} = 1$?

Thanks in advance.

Answer 1

How about this:

\begin{align} & \frac12 \cdot 2 = 1 \qquad \qquad\text{(definition of $\frac12$)} \\ \implies& \frac12 \cdot (1 + 1) = 1 \qquad\qquad \text{(definition of $2$)} \\ \implies& \frac12 \cdot 1 + \frac12 \cdot 1 = 1 \qquad \qquad \text{(distributive property)} \\ \implies& \frac12 + \frac12 = 1 \qquad \text{(because $1$ is a multiplicative identity)}. \end{align}

Answer 2

As already said by others, $1/2 +1/2=1$ is basically the definition of $1/2$ as the inverse of the number $2$ which, in $\mathbb R$, is a synonym of $1+1$.

But No, you cannot prove that by using only the "nine properties of fields". This is because you cannot prove that $1+1$ is invertible.

In fact there exist fields where $1+1=0$. For example $\mathbb Z/2\mathbb Z$ Is a field, it has only two elements: $0,1$ and there $1+1=0$.