Is there a formal proof for (−1)×(−1)=1? It’s a fundamental formula not only in arithmetic but also in the whole of math. Is there a proof for it or is it just assumed?

**Answer**

We use only the usual field axioms for the real numbers. First we prove an intermediate result.

0×0

=0×(0+0)

=0×0+0×0

Subtract 0×0 from each side to get 0=0×0. Now we are ready for the final kill.

0

=0×0

=(1−1)×(1−1)

=1×1+1×(−1)+(−1)×1+(−1)×(−1)

=1+(−1)+(−1)+(−1)×(−1)

=(−1)+(−1)×(−1)

Add 1 to each side to get 1=(−1)×(−1).

**Attribution***Source : Link , Question Author : tvamsisai , Answer Author : Ilmari Karonen*