Why do things break in a 2 element set?

The math jargon for a set where you can implement a 4 function calculator \((+,-,*,/)\) is a field. Fields can have a finite number of elements, and such fields always have a prime power number of elements, aka \(p^k\) where \(p\) is prime and \(k\) is a natural number.

Many results are not true if the underlying field has exactly 2 elements. The reason why:

\(x = -x\) \(x + x = -x + x\) \(2x = 0\)

The final formula is taken to mean that \(x\) is zero. But there’s another interpretation.

What if 2 = 0?

In a field with 2 elements, this is true. Moreover, \(x = -x\) is a tautology AKA always true AKA worthless. \(-1 \mod 2 = 1\) and \(0 \mod 2 = 0\).

This is only when the underlying field has 2 elements. A 2 element set is exceptional because negation doesn’t actually do anything.

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