Legal and Illegal Algebraic Manipulations

A very useful reduction for ratios makes use of crossmultiplication  

Complicated algebraic expressions are best handled nowadays using symbolic math programs such as Mathematical, Maple , or Mathcad . Cancellation is a wonderful way to simplify formulas. Consider 

For your amusement, here is a proof that 2=1. The following sequence of algebraic operations is entirely legitimate, except for one little item of trickery snuck in. Suppose we are given that 

Since a = b, this means that 2=1. Where did we go wrong  

Once you recover from shock, note that a — b = 0. Division by 0 is not legitimate. It is, in fact, true that 2x0= 1x0, but we cannot cancel out the zeros. 

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