A Perplexing Paradox

WARNING: The author takes no responsibility for any cranium explosion that might occur due to the baffling nature of the following content...

Just joking, although this seemingly impossible occurrence may be confusing at first, I assure you there is a completely logical explanation for this phenomenon.

We begin by defining the relationship of two variables a and b as:
                                 a = b
          Now,
                            a - b = 0
             a - b + a² - ab = a² - ab
          a² + (1 - b)a - b = a(a - b)
               (a + 1)(a - b) = a(a - b)
          * Dividing by (a - b)
                            a + 1 = a
                                  1 = 0
          But it cannot be!  Therefore should we abandon the principles of algebra?

          What went wrong??

          * The Answer: We divided by a - b, which is not a defined operation as
                      we also said that a - b = 0


          To construct a paradox like this one, begin with a false expression such as:
                      2 = 4
          Now add any expression possible (including as many variables as you want)
              3a + 2 = 3a + 4     (I use expressions of a here for simplicity)
          Multiply by (a - b)   ---> remember the opposite operation, division,
                                                       is the bit that isn't allowed
          3a² - 3ab + 2a - 2b = 3a² - 3ab + 4a - 4b
          Now cancel equal expressions
          2a - 2b = 4a - 4b
          2b - 2a = 0
                   b = a
          Write these steps backwards and Voila!

          To be tricky you can also use a different ratio of a : b, in which case, multiply instead by (na - mb), where m : n = a : b (the final expression in the working should be na = mb.  The original paradox is a special case where m : n = 1 : 1.

          Proof:
          Where c ≠ d, we begin with the obviously false expression:
                                 c = d
           following the previous method, where e is any expression possible
                           c + e = d + e
         (c + e)(na - mb) = (d + e)(na - mb)
cna -cmb + ena - emb = dna - dmb + ena - emb
   (c - d)na - (c - d)mb = 0
                      na - mb = 0    (this division actually is permissible as c - d ≠ 0)
                              na = mb