The Whole is the Sum of the Parts This principle, known for very many years, is important to thermodynamics and all disciplines. Below we use the idea with geometric areas to prove:
(A - B) 2 = A 2 - 2AB + B 2
♦ A common approach to any problem is "postulate somthing similar but simpler." Solve the simpler then adapt the method to a "next level." We start with "A + B" squared.
This proof was made by Euclid (circa 300 B.C.). By the figure to the right, it is apparent that:
(A + B)2 = A2 + 2 AB + B2
But we need (A - B)2. Suppose we use the identical square areas but assign new lengths for A and B such that (A - B) is a factor. Again the area of the outer square equals the sum of the areas interior:
A2 = (A - B)2 + 2(A - B)B + B2
A2 = (A - B)2 + 2AB - 2B2 + B2
Now, rearrange to obtain:
(A - B)2 = A2 - 2AB + B2 Q.E.D.
In thermodynamics, like geometry, it is important to remember to break things apart then put them back together when necessary.