The Square Root Fallacy

My National Review Online "Diary" column for October 2003 included the following brainteaser.

The Square Root Fallacy

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Readers of Prime Obsession will know that, contrary to what your schoolteachers told you, the number “minus one” does so have a square root, the friendly little number i.   In fact, not only is the square of i equal to -1, so, by the rule of signs, is the square of -i.

If you are still with me, see if you can find the flaw in the following argument.  Out of consideration for Aaron the Webbie, I am going to use “Sqr(x)” to indicate the square root of x and an asterisk to indicate multiplication.

Start from this obvious truth:  Sqr(x — y) = i * Sqr(y — x).

Since this is plainly the case for any numbers x and y, put x = a and y = b.

Then:  Sqr(a — b) = i * Sqr(b — a).

On the other hand, since that obvious truth I started with is the case for any numbers x and y, I could equally well put x = b and y = a, to get another, equally true statement:
Sqr(b — a) = i * Sqr(a — b).

Now, if P = Q and R = S, then obviously P * R = Q * S.

So:  Sqr(a — b) * Sqr(b — a) = i ² * Sqr(b — a) * Sqr(a — b).

Since i ² = -1 and the other components of each side are identical, it follows that 1 = -1.  It easily follows that every minus sign can be replaced by a plus, all debits are credits, all liabilities are assets, and the Pope is Jewish.  (Proof of the latter: Since 1 = -1, adding 1 to each side gives 2 = 0. Halving both sides, 1 = 0. Adding 1 to both sides, 2 = 1. Now, the Pope and Jackie Mason — who is Jewish — are two people, therefore they are one person, therefore the Pope is Jewish.)

Where is the logical flaw?

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The flaw is right up there at the beginning.  Since, for any number X whatsoever — positive, negative, real or complex — there are two numbers whose square is equal to X, we have to settle on a precise definition of the term "square root."  The usual definition is as follows:  The square root of X is the number whose square is equal to X, and whose amplitude (Prime Obsession, Figure 11-2) is between zero (inclusive) and pi (exclusive).

This means that the square root of 4 is 2, not -2;  and the square root of -4 is 2i, not -2i.

If you put x = 13 and y = 9, you will now see that the "obvious truth" I started with is, in fact, false! 

(Because the left-hand side is the square root of 4, which is 2, while the right-hand side is i times the square root of -4, which is i times 2i, which is -2.)