Question

a
$a - (b - 2a)$

Answer 1

I was browsing the Math Forum and saw the "proof" that 2=1. I read how, since two does not equal one, somewhere along the line an error was made. It was then concluded that the source of error was division by zero. Here is the proof I am discussing: a=b Given a^2=ab Multiplication Property of Equality a^2-b^2=ab-b^2 Subtraction Property of Equality (a+b)(a-b)=b(a-b) Sum and Difference Pattern/Distributive Property a+b=b Division Property of Equality b+b=b Substitution Property 2b=b Combination of like terms 2=1 Division Property of Equality I suppose division by zero makes sense as a valid reason for fallacy, but how do we know for sure that it is the cause? The Sum and Difference Pattern also makes sense, but when a=b I found that the solution actually changes: (a+b)(a-b)=a^2-b^2 Sum and Difference Pattern (a+a)(a-a)=a^2-a^2 Substitution Property a(1+1)a(1-1)=a^2(1-1) Distributive Property 2a^2*0=a^2*0 Combination of Like Terms, Inverse Axiom of Addition Notice that here the LHS is exactly twice the RHS if you disregard the Property of Zero in Multiplication, which states that 0a=0. If you momentarily assume that 0a may not equal zero for all a, then the fact that the LHS is twice the RHS accounts for the fact that the solution to the false proof has an LHS twice the RHS. Therefore, if you assume that 0x=0y is only true when x=y, the Sum and Difference Pattern is revised to (a+b)(a-b)=2(a^2-b^2) when a=b. If you substitute this into the equation in step four, the solution is a true identity. Is it possible that this is the reason that the proof is false? Otherwise it would seem hard to believe that the fact that the Sum and Difference Pattern's "identity" when a=b and the "identity" of the false proof are proportional is just a coincidence. Perhaps the difference between 0x and 0y where x does not equal y is so minute that it cannot be perceived from our perspective and is impossible to tell apart, but can be viewed from our perspective and made apparent by cancelling out, or dividing, by zero? Is it possible to assume that the error in this false proof may have resulted from something other than division by zero? Thank you.