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Congratulations; you have correctly identified the fallacious step!

This innocent-looking step is in fact quite wrong. The problem is that there is no rule that guarantees sqrt(a/b) = sqrt(a) / sqrt(b) , except in the case in which a and b are both positive.

If this surprises you, think about the question

Why should sqrt(a/b) equal sqrt(a)/sqrt(b)?
If you were to try to convince someone of this, you'd have to start with the definition of what a "square root" is: it's a number whose square is the number you started with. So, from first principles, all that has to be true is that sqrt(a) squared is a, sqrt(b) squared is b, and sqrt(a/b) squared is a/b.

So, when you square sqrt(a/b), you will get a/b, and when you square sqrt(a)/sqrt(b), you will also get a/b. That's all that the definition of square root tells you.

Now, the only way two numbers x and y can have the same square is if x = + or -y. So, what is true is that

sqrt(a/b) = + or -sqrt(a) / sqrt(b) ,
but in general there's no reason it has to be sqrt(a/b) = + sqrt(a) / sqrt(b) rather than sqrt(a/b) = - sqrt(a) / sqrt(b) , unless a and b are both positive: for then (because by convention we take the positive square root) everything in the above equation is positive, and that's why we obtain sqrt(a/b) = + sqrt(a) / sqrt(b) . But remember it's only because everything is positive that we obtain it!

In our case, it is true that sqrt(-1/1) = sqrt(-1)/sqrt(1), but sqrt(1/-1) is - sqrt(1)/sqrt(-1) not sqrt(1)/sqrt(-1). The fallacy comes from using the latter instead of the former.

In fact, the whole proof really boils down to the fact that (-1)(-1) = 1, so sqrt((-1)(-1)) = 1, but sqrt(-1)sqrt(-1) = i^2 = -1 (not 1). The proof tried to claim that these two were equal (but in a more disguised way where it was harder to spot the mistake).

This fallacy is a good illustration of the dangers of taking a rule from one context and just assuming it holds in another. When you first learned about square roots you had never encountered complex numbers, so the only objects that had sqare roots were positive numbers. In this case, sqrt(a/b) = sqrt(a) / sqrt(b) is always true, and you were probably taught it as a "rule". But it is only a mathematical truth in that original context, and fails to remain true after you extend the definition of "square root" to allow the square roots of negative and complex numbers.

To see the explanations for the other steps, finding out exactly why they're correct (or, in some cases, finding out which other steps have slight mistakes in them), go back to the list of steps in the proof. To try your hand at finding the fallacy in a different problem, go back to the Classic Fallacies index page.
This page last updated: May 26, 1998
Original Web Site Creator / Mathematical Content Developer: Philip Spencer
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