Re: 1 = -1 math puzzle

On Jun 11, 5:45 pm, "michalc...@xxxxxxx" <michalc...@xxxxxxx> wrote:
Where is the mistake?

1 =
sqrt (1) =
sqrt (-1*-1) =
sqrt(-1) * sqrt(-1) =
i * i =
-1

Lines 3-4. You cannot go from sqrt(-1*-1) to sqrt(-1) * sqrt(-1).

Why?

EXPLANATION:

Every number has two (2) square roots, not one. Although this
is ordinarily not a problem on the positive real numbers, when
we start introducing imaginary and complex numbers it comes
back to bite us.

For one, we've defined "i" to equal "sqrt(-1)", or conversely
we define "sqrt(-1)" to equal "i". But this definition, or the
opposite that it equals "-i", both destroy the rule sqrt(ab) =
sqrt(a)sqrt(b).

How though does this relate to there being two square roots?
Because the rule "sqrt(a)sqrt(b) = sqrt(ab)" holds when you
think of it as saying "*A* square root of a times *A* square root
of b equals *A* square root of ab.". And that is completely
true. *A* square root of -1 times *A* square root of -1 does
equal *A* square root of -1 times -1. i*i equals -1, which is
*A* square root of -1*-1, i.e. 1. But when you talk about "THE"
square root of a times "THE" square root of b -- that is the
extension of the conventional positive square root function into
the complex domain -- these rules no longer hold. "THE" square
root of -1 times "THE" square root of -1, that is, i times i, does
not equal "THE" square root of 1, that is, 1.
.

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