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The Paradox of the digits
It's official: 1=2
And here is the proof...
(1) X = Y Given
(2) X2 = XY Multiply both sides by X
(3) X2 - Y2 = XY - Y2 Subtract Y2 from both sides
(4) (X+Y)(X-Y) = Y(X-Y) Factor both sides
(5) (X+Y) = Y Cancel out common factors
(6) Y+Y = Y Substitute in from line (1)
(7) 2Y = Y Collect the Y's
(8) 2 = 1 Divide both sides by Y
Can you spot the error that has led to this curious
conclusion?
The error arises in lines (4) and (5).
Because X = Y, the common factor (X-Y) is equal to
zero. Therefore by cancelling out the common factor, one is dividing
both sides by zero, the results of which are undefined. The result
of this calculation can therefore be seen to be a simple fallacy.
(1) X = Y Given
(2) X2 = XY Multiply both sides by X
(3) X2 - Y2 = XY - Y2 Subtract Y2 from both sides
(4) (X+Y)(X-Y) = Y(X-Y) Factoring both sides by (x-y), ie zero, is
nonsensical
(5) (X+Y) = Y Dividing both sides by zero leads to undefined results
(6) Y+Y = Y Substitute in from line (1)
(7) 2Y = Y Collect the Y's
(8) 2 = 1 Divide both sides by Y
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