Proof that 1 is less than 0
In mathematics, there are numerous "proofs" that show impossible results. These proofs, while seemingly valid to the casual observer, always contains an invalid step where a principle of mathematics is violated. These are normally regarded as mere curiosities, but can be used to show the importance of rigour in mathematics.
Most of these proofs depend on some variation of the same error.
The error is to take a function f that is not one-to-one, to
observe that f(x) = f(y) for some x and y, and to (erroneously) conclude that therefore x = y. Division by zero is a special case of this; the function f is x → x × 0, and the erroneous step is to start with x×0 = y×0 and to conclude that therefore x=y.
Examples
Proof that 1 equals −1
We start with
Then we convert these into fractions
Applying square roots on both sides gives
Which is equal to
But since  (see imaginary number), we can substitute, obtaining
By rearranging the equation to remove the fractions, we get
And since  , we therefore have
Q.E.D.
This proof is invalid since it applies the following principle for square roots wrongly:
This principle is only correct when both x and y are positive numbers.
In the "proof" above, one is a negative number, thus making the whole proof invalid.
Proof that 1 is less than 0
Let us suppose that
Now we will take the logarithm on both sides. As long as x > 0, we can do this because logarithms are monotonically increasing. Observing that the logarithm of 1 is 0, we get
Dividing by ln x gives
Q.E.D.
The violation is found in the last step, the division. This step is wrong because the number we are dividing by is negative, which in turn is because the argument to the logarithm is less than 1, our original assumption. A multiplication with or division by a negative number flips the inequality sign; in other words, we should obtain 1 > 0, which is indeed correct.
Proof that 2 equals 1
Suppose that
Multiplying by  gives
Subtracting  gives
Factoring out  ,
then cancelling the common factor gives
But since , we can substitute for 
 .
As is arbitrary it can be cancelled from both sides, therefore,
Q.E.D.
The violation is found in the step where the common factor is cancelled. This step is wrong because that factor is equal to zero, and in cancelling it, an implicit division by zero is made. This invalidates the succeeding steps and the proof is no longer valid. Before that last step, it can be said that we proved x×0=0.
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