Prove: (-1) x (-1) = 1 Most of us use this fact without thinking about it. To prove the formula, we use the "Whole equals the sum of its parts" idea developed in the previous example. By geometry, we see that the result is true. Some facts of thermodynamics are less clear.

♦  To start, we identify A and B as:

A = 2 - 1      and       B = 2 - 1

Then since A  and    B both equal 1 we have:

A x B = A x B      and      (2 - 1) x ( 2 - 1 ) = 1

The expression above right can be written as:

[ (2) + (- 1) ]² = 1

This result has the form  (A + B) ²    with     A = 2    and     B = -1.

By a previous proof we have:

(A + B)² = A² + 2 AB + B²

Therefore, using the right side we have,

(2 x 2) + 2 x [(2) x (-1)] + (-1) x (-1) = 1

Which equals:

4 - 4 + [(-1) x (-1)] = 1

Collected, we obtain:

( -1 ) x ( -1 ) = 1     Q.E.D.

This proof depended upon a previous proof. In thermodynamics (as in algebra) the small undertandings are combined to become larger understandings.

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