NUFFNANG

Saturday, April 05, 2008

Proof that 2 = 1

Let a and b be equal non-zero quantities

a = b \,

Multiply through by a

a^2 = ab \,

Subtract b^2 \,

a^2 - b^2 = ab - b^2 \,

Factor both sides

(a - b)(a + b) = b(a - b) \,

Divide out (a - b) \,

a + b = b \,

Observing that a = b \,

b + b = b \,

Combine like terms on the left

2b = b \,

Divide by the non-zero b

2 = 1 \,

The fallacy is in line 5: the progression from line 4 to line 5 involves division by (ab), which is zero since a equals b. Since division by zero is undefined, the argument is invalid. Deriving that the only possible solution for lines 5, 6, and 7, namely that a = b = 0, this flaw is evident again in line 7, where one must divide by b (0) in order to produce the fallacy (not to mention that the only possible solution denies the original premise that a and b are nonzero). A similar invalid proof would be to say that 2(0) = 1(0) (which is true) therefore, by dividing by zero, 2 = 1.


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