Formal proof for (−1)×(−1)=1(-1) \times (-1) = 1

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

=(11)×(11)

=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

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