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(-1)(-1) = 1

Proposed in Mathematics: A Very Short Introduction. Given the following rules:

A1 Commutative law for addition: \forall a,b\quad a + b = b + a

A2 Associative law for addition: \forall a,b,c\quad a + (b + c) = (a + b) + c

A3 Additive identity: \forall a\quad 0 + a = a

A4 Additive inverse: \forall a,\exists b\quad a + b = 0

A5 Cancellation law for addition: \forall a,b,c\quad if\ a + b = a + c,\ then\ b = c

M1 Commutative law for multiplication: \forall a,b\quad ab = ba

M2 Associative law for multiplication: \forall a,b,c\quad a(bc) = (ab)c

M3 Multiplicative identity: \forall a\quad 1a = a

M4 Multiplicative inverse: \forall a\ne0,\ \exists b\quad ab = 1

M5 Cancellation law for multiplication:

\forall a,b,c\quad if\ a\ne0\ and\ ab = ac,\ then\ b = c

D Distributive law: \forall a,b,c\quad (a + b)c = ac + bc

Let’s prove that (-1)(-1) = 1:

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